Отсчетная форма тел с конечными несовместными деформациями

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Аннотация

В статье развиваются дифференциально-геометрические методы моделирования конечных несовместных деформаций гиперупругих твердых тел. Несовместность деформаций может быть вызвана, к примеру, неоднородными температурными полями и распределенными дефектами. Как следствие, возникают внутренние напряжения и искажение геометрической формы тела. Эти факторы определяют критические параметры современных высокоточных технологий, в частности, в технологиях аддитивного изготовления. В этой связи развитие методов их количественного описания является актуальной проблемой современной механики деформируемого твердого тела.

Применение методов дифференциальной геометрии основано на представлении тела в виде гладкого многообразия, снабженного метрикой и неевклидовой связностью. Такой подход позволяет интерпретировать тело как глобальную, свободную от напряжений форму и сформулировать физический отклик и материальные уравнения баланса относительно этой формы. В рамках геометрического метода деформации характеризуются вложениями неевклидовой формы в физическое пространство, которое по-прежнему считается евклидовым. Меры несовместности отождествляются с инвариантами аффинной связности — кручением, кривизной и неметричностью, а сама связность определяется типом физического процесса. 

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Введение

В лекции "О гипотезах, лежащих в основании геометрии" Риман [1] впервые выдвинул идею n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ -мерного многообразия, снабженного метрическим тензором. Последний определяет на многообразии дифференциал расстояния между бесконечно близкими элементами, а также вводит на нем специфическое правило параллельного переноса. Спустя время, в конце XIX века, идея Римана нашла свое применение в физике и, в частности, в механике континуума. По-видимому, первым, кто применил неевклидову геометрию в теории упругости, был Бельтрами [2], который сформулировал линейные уравнения баланса в пространстве с неевклидовым метрическим тензором. Появление теории относительности инициировало дальнейшие исследования в механике континуума, в которых рассматривалось движение тела в плоском или искривленном пространстве-времени [3]. Следует отметить, что перенос классических положений механики континуума на релятивистский случай таит в себе ряд принципиальных трудностей, поскольку привычные понятия длины угла и времени зависят от наблюдателя. В частности, определение жесткого движения требует особого подхода, развитого в работах Борна [4; 5] и Герглотца [6] для случая специальной теории относительности и в работе Нордстрёма [7] для случая общей теории относительности. Таким образом, первоначально неевклидовой структурой снабжалось пространство, в котором наблюдалось движение деформируемого континуума.

Позже, в работах Билби [8] и Кондо [9 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFtaca aa@3C72@ 12] была предложена и реализована идея об использовании неевклидовой геометрии для моделирования нелинейно-упругого тела с остаточными напряжениями. Билби и Кондо предполагали физическое пространство евклидовым, а отсчетное состояние тела MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ неевклидовым. В последующих работах Нолла [13] и Вана [14] была построена математически строгая геометрическая теория тел с остаточными напряжениями, которая получила дальнейшее развитие в работах Можена [15], Марсдена [16] и Эпштейна [17; 18]. В настоящее время исследования, использующие неевклидово описание отсчетного состояния, образуют область, называемую геометрической механикой континуума. Современное состояние исследований в этом направлении отражено в работах зарубежных [19 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFtaca aa@3C72@ 22] и отечественных [23 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFtaca aa@3C72@ 26] школ.

Изучение астрофизических феноменов MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ аккреции массивных тел [27] и звездотрясений (starquakes) внешней оболочки нейтронных звезд [28] MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ привело к развитию релятивистской теории упругости, в которой как физическое пространство, так и отсчетное состояние моделируются неевклидовыми многообразиями [29; 30]. Причиной неевклидовости физического пространства являются наличие больших гравитирующих масс и необходимость в этой связи использовать положения общей теории относительности. Причиной же использования неевклидова отсчетного состояния является тот факт, что отсчетное состояние зависит от наблюдателя и то, что релятивистское упругое тело оказывается самонапряженным. В этом проявляется методологическое сходство релятивистской теории упругости и геометрической механики континуума.

Поскольку использование неевклидовых пространств для моделирования отсчетного состояния тела не является общепринятым в механике континуума, дадим необходимые комментарии. В классической теории упругости деформирование тела рассматривается относительно некоторого его привилегированного положения в физическом пространстве MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ отсчетной формы. Как правило, предполагается, что отсчетная форма состоит из представительных объемов, каждый из которых свободен от напряжений. В линейном приближении существование такой формы характеризуется условиями совместности Сен-Венана, а в нелинейном MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ равенством нулю тензора кривизны Римана, построенного относительно метрики, индуцированной на отсчетную форму из объемлющего евклидова пространства. Вместе с тем в начале XX века выяснилось, что отсчетная форма, свободная от напряжений, существует далеко не всегда. Теоретическое исследование условий совместности для многосвязных областей, проведенное Вайнгартеном [31] и Вольтерра [32], привело к примерам тел с остаточными напряжениями MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ дисторсиям Вольтерра.

Далее оказалось, что возникновение остаточных напряжений сопутствует реальному физическому процессу. Открытие периодической атомарной структуры кристаллических тел в 1912 году (эксперименты Лауэ) инициировало многочисленные исследования в физике кристаллов. Одной из проблем, над которой работали исследователи, являлось объяснение экспериментального значения предела текучести кристалла: теоретические вычисления, проведенные Френкелем [33] в 1926 году, дали значения предела текучести, существенно превышающие экспериментальные.

Для объяснения несовпадения теоретических вычислений с экспериментальными данными Орован [34], Тейлор [35; 36] и Поляни [37] в 1934 году независимо друг от друга предложили модель линейного кристаллического дефекта, называемого краевой дислокацией. Предположив, что каждый кристалл содержит большое количество дислокаций, Тейлор смог вычислить предел текучести как напряжение, необходимое для движения дислокации через упругое поле всех других дислокаций. Результат был в согласии с экспериментальными данными.

К середине XX века окончательно сформировалось представление о том, что тела с остаточными напряжениями существуют и предположение о существовании отсчетной формы, свободной от напряжений, является идеализацией. Одним из первых, кто указал на этот факт, был Эккарт [38]. Несколькими годами позже Билби и Кондо предложили определять отсчетную форму в подходящем неевклидовом пространстве [8; 9]. В таком случае деформация интерпретируется как вложение неевклидова многообразия в евклидово физическое пространство MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ собственная деформация. Кондо показал, что такие собственные деформации полностью характеризуют поля дефектов как внутренних источников напряжений, которые, в свою очередь, могут быть определены правилом параллельного переноса в пространстве, вмещающем отсчетную форму. Они количественно характеризуются соответствующей кривизной и кручением аффинной связности.

В настоящей статье предлагается новый вариант построения неевклидовой отсчетной формы, который позволяет геометрически охарактеризовать несовместность деформаций несколькими альтернативными способами. Первый способ предполагает описание материального многообразия как риманова, кривизна связности на котором определяет меру несовместности деформаций. Второй способ задает на материальном многообразии плоское пространство с ненулевым кручением, инварианты которого также характеризуют меры несовместности деформаций. В рамках третьего способа несовместность определяется неметричностью пространства Вейля. Для задания специфической геометрии на материальном многообразии используются вычисления относительно некоторой напряженной промежуточной формы. Доказывается, что результат MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ связность и инварианты связности на материальном многообразии не зависят от выбора этой промежуточной формы.

В работе используются следующие обозначения. Символ Hom(U;V) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaiaab+gacaqGTbGaaGikamrr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae8hfXxLaaG4oaiaaykW7cq WFveVvcaaIPaaaaa@4BAF@ обозначает векторное пространство всех линейных отображений UV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFueFvcqGHsgIRcqWFveVvaaa@473A@ , а символ End(U):=Hom(U;U) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyraiaab6gacaqGKbGaaGikamrr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae8hfXxLaaGykaiaaiQdaca aI9aGaaeisaiaab+gacaqGTbGaaGikaiab=rr8vjaaiUdacaaMc8Ua e8hfXxLaaGykaaaa@531C@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ векторное пространство линейных операторов VV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFveVvcqGHsgIRcqWFveVvaaa@473C@ . Пусть M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytaaaa@38D5@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ гладкое многообразие. Алгебра гладких функций M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytaiabgkziUorr1ngBPrwtHrhAYaqeguuDJXwA KbstHrhAGq1DVbacfaGae8xhHifaaa@457A@ на нем обозначается символом C (M) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4qamaaCaaaleqabaGaeyOhIukaaOGaaGikaiaa d2eacaaIPaaaaa@3CAA@ . Касательное пространство к M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytaaaa@38D5@ в точке p MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiCaaaa@38F8@ обозначается через T p M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaacaWGWbaabeaakiaad2eaaaa@3AD9@ и в зависимости от ситуации рассматривается либо как класс эквивалентных кривых, либо как пространство дифференцирований скалярных функций. Кокасательное пространство, являющееся векторным пространством, дуальным к T p M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaacaWGWbaabeaakiaad2eaaaa@3AD9@ , обозначается символом T p M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaDaaaleaacaWGWbaabaGaey4fIOcaaOGa amytaaaa@3BC9@ . Символ Vec(M) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOvaiaabwgacaqGJbGaaGikaiaad2eacaaIPaaa aa@3CE1@ означает C (M) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4qamaaCaaaleqabaGaeyOhIukaaOGaaGikaiaa d2eacaaIPaaaaa@3CAA@ -алгебру векторных полей на M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytaaaa@38D5@ . Если EM MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyraiabgkziUkaad2eaaaa@3B8C@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ векторное расслоение, то Sec E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4uaiaabwgacaqGJbWaaeWaaeaacaWGfbaacaGL OaGaayzkaaaaaa@3CFA@ обозначает C (M) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4qamaaCaaaleqabaGaeyOhIukaaOGaaGikaiaa d2eacaaIPaaaaa@3CAA@ -модуль всех его сечений (тензорных полей). Более подробно эти обозначения и связанная с ними техника изложены в руководствах [39 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFtaca aa@3C72@ 41].

Используется общее понятие структуры как упорядоченного набора, состоящего из множества MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ носителя структуры, и дополнительных элементов, характеризующих эту структуру. Таким образом, если X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ множество, а Struct MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4uaiaabshacaqGYbGaaeyDaiaabogacaqG0baa aa@3D9A@ обозначает объекты, характеризующие структуру на X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ , то структура в целом записывается как (X,Struct) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadIfacaaISaGaaGPaVlaabofacaqG0bGa aeOCaiaabwhacaqGJbGaaeiDaiaaiMcaaaa@421D@ .

1. Тело и его формы

1.1. Физическое пространство

Геометрическая механика континуума основана на идее, что тело и физическое пространство могут быть формализованы в терминах гладких многообразий, снабженных специфическими метриками и аффинными связностями, а деформация MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ в виде композиции вложений тела в физическое пространство. Настоящая статья следует этой методологии. Для формализации процедуры построения неевклидовой отсчетной формы, являющейся основной целью настоящей работы, вначале уточним, что понимается под физическим пространством, вмещающим образы тела, и что понимается под самим телом.

Будем полагать, что E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ является евклидовым точечным пространством размерности 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4maaaa@38C0@ , то есть структурой [42]

E=(E,V,vec,). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesrcaaI9aGaaGikaiaadweacaaISaGaaGPaVlab=vr8wj aaiYcacaaMc8UaaeODaiaabwgacaqGJbGaaGilaiaaykW7cqGHflY1 caaIPaGaaGOlaaaa@540B@ (1)

Здесь E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyraaaa@38CD@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ континуальное множество мест, V=(V,, + V , V ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFveVvcaaI9aGaaGikaiaadAfacaaISaGaaGPaVprr1ngBPr wtHrhAYaqehuuDJXwAKbstHrhAGq1DVbacgaGae4xhHiLaaGilaiaa ykW7cqGHRaWkdaWgaaWcbaGaamOvaaqabaGccaaISaGaaGPaVlabgw SixpaaBaaaleaacaWGwbaabeaakiaaiMcaaaa@5D3F@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ трансляционное векторное пространство над[1] MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb a8aacqWFDeIuaaa@42BB@ , имеющее размерность 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4maaaa@38C0@ , vec:E×EV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeODaiaabwgacaqGJbGaaGOoaiaaykW7caWGfbGa ey41aqRaamyraiabgkziUkaadAfaaaa@438C@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ отображение, удовлетворяющее аксиомам:

(a) для любых точек x,y z,E выполняется равенство

vec(x, y)+vec(y, z)vec(x, z)

(b) для любой точки x E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFaCpEcqGHiiIZcaWGfbaaaa@45CF@  и любого вектора vV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCODaiabgIGioprr1ngBPrwtHrhAXaqeguuDJXwA KbstHrhAG8KBLbacfaGae8xfXBfaaa@45F1@ существует единственная точка yE такая, что vec(x, y)=v,

а () MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabgwSixlaaiMcaaaa@3BB2@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ скалярное произведение на V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFveVvaaa@436E@ . Значение v=vec(x, y) интерпретируется как вектор с началом в точке x и концом в точке y.

Предположим, что фиксирован некоторый декартов репер (o,(ik)k=13, где oE  MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ начало отсчета, а ( i k ) k=1 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahMgadaWgaaWcbaGaam4AaaqabaGccaaI PaWaa0baaSqaaiaadUgacaaI9aGaaGymaaqaaiaaiodaaaaaaa@3EDC@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ ортонормированный базис пространства V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFveVvaaa@436E@ . Тогда точке oсоответствует поле радиус-векторов p  : xvec(o, x), а базис ( i k ) k=1 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahMgadaWgaaWcbaGaam4AaaqabaGccaaI PaWaa0baaSqaaiaadUgacaaI9aGaaGymaaqaaiaaiodaaaaaaa@3EDC@ позволяет определить координаты произвольной точки из E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyraaaa@38CD@ , которые полагаются равными координатам радиус-вектора этой точки.

Пара (o, (ik)k=13) в дальнейшем ассоциируется с инерциальной системой отсчета (наличие абсолютного времени неявно предполагается) [43]. За счет операции сопоставления точкам их декартовых координат, осуществляемой парой (o, (ik)k=13), на множестве E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyraaaa@38CD@ вводятся топология T E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFtepvdaWgaaWcbaGaamyraaqabaaaaa@4460@ и гладкая структура D E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFdeprdaWgaaWcbaGaamyraaqabaaaaa@4440@ , индуцированные соответствующими топологией и атласом из 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb a8aacqWFDeIudaahaaWcbeqaaiaaiodaaaaaaa@43A5@ . Тем самым приходим к структуре

E geom =(E, T E , D E ,g,,e) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesrdaWgaaWcbaGaae4zaiaabwgacaqGVbGaaeyBaaqaba GccaaI9aGaaGikaiaadweacaaISaGaaGPaVlab=nr8unaaBaaaleaa caWGfbaabeaakiaaiYcacaaMc8Uae83aXt0aaSbaaSqaaiaadweaae qaaOGaaGilaiaaykW7caWHNbGaaGilaiaaykW7cqGHhis0caaISaGa aGPaVlaakwgacaaIPaaaaa@5DCE@ (2)

трехмерного геометрического пространства над E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyraaaa@38CD@ . Здесь g MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4zaaaa@38F3@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ евклидова метрика, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ евклидова связность, а e MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaOyzaaaa@38F4@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ форма объема, согласованная с метрикой (тензор Леви-Чивита). Элементы g MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4zaaaa@38F3@ , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ и e MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaOyzaaaa@38F4@ структуры (2) выбраны раз и навсегда.

Структура (2) полностью определяется по структуре (1) и в этом смысле может быть названа производной по отношению к последней. Необходимость определения производной геометрической структуры связана с тем, что в дальнейшем предполагается рассматривать на E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyraaaa@38CD@ регионы и стирать с них геометрию, которая изначально индуцирована геометрией объемлющего пространства (2), для последующего определения на этих множествах геометрии более общего вида.

1.2. Тело и евклидовы формы

В рамках классической механики континуума под телом B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ понимается гладкое многообразие меток, которые идентифицируют представительные объемы, наделенные дополнительными атрибутами MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ массой и зарядом [44; 45]. Поэтому B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ является не просто чистым многообразием, а многообразием, снабженным некоторой мерой [46]. Вместе с тем, поскольку в настоящей работе рассматривается только кинематика самонапряженного тела, будем интерпретировать B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ лишь как гладкое n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ -мерное многообразие, т. е. как структуру

B=(B, T B , D B ), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cjaai2dacaaIOaGaamOqaiaaiYcacaaMc8+efv3ySL gznfgDOfdarCqr1ngBPrginfgDObYtUvgaiyaacqGFtepvdaWgaaWc baGaamOqaaqabaGccaaISaGaaGPaVlab+nq8enaaBaaaleaacaWGcb aabeaakiaaiMcacaaISaaaaa@5BC5@ (3)

в которой B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOqaaaa@38CA@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ континуальное множество меток, T B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFtepvdaWgaaWcbaGaamOqaaqabaaaaa@445D@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ топология на этом множестве [47, с.~20], а D B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFdeprdaWgaaWcbaGaamOqaaqabaaaaa@443D@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ гладкая структура на B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOqaaaa@38CA@ [39, с.~12--13]. Хотя в общем случае топология и гладкая структура на теле (3) могут быть произвольными, далее будем полагать, что они индуцированы топологией и гладкой структурой образа тела, реализованного в виде подмногообразия пространства (2). Заметим, что размерность тела n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ может принимать значения 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaaaa@38BE@ , 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGOmaaaa@38BF@ и 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4maaaa@38C0@ . В соответствующем случае будем говорить о материальных кривых, материальных поверхностях и материальных телах. Для обозначения элементов B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ используется фрактурный шрифт: p MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Lc8Wbaa@44D2@ , q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pc8Xbaa@44D4@ , r MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Tc8Ybaa@44D6@ .

Тело B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ наблюдается лишь посредством евклидовых форм, то есть образов гладких вложений [39, с.~85] ϰ:BE MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+caaI6aGaaGPaVprr1ngBPrMrYf2A0vNCaeXbfv3ySL gzGyKCHTgD1jhaiyaacqGFbaVqcqGHsgIRcqWFWesraaa@55CF@ тела B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ в евклидово физическое пространство E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ . Прилагательное <<евклидова>> использовано здесь для того, чтобы подчеркнуть отличие этих образов тела от более общих, рассматриваемых в рамках неевклидовой геометрии. Следуя терминологии, принятой в механике континуума [44], любое вложение ϰ:BE MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+caaI6aGaaGPaVprr1ngBPrMrYf2A0vNCaeXbfv3ySL gzGyKCHTgD1jhaiyaacqGFbaVqcqGHsgIRcqWFWesraaa@55CF@ будем называть конфигурацией.

Каждая евклидова форма S=ϰ(B) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=ucaaI9aGae8h8dKVaaGikamrr1ngBPrMrYf2A0vNCae Xbfv3ySLgzGyKCHTgD1jhaiyaacqGFbaVqcaaIPaaaaa@547F@ рассматривается как некоторое подмножество физического многообразия E geom MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesrdaWgaaWcbaGaae4zaiaabwgacaqGVbGaaeyBaaqaba aaaa@4688@ , из которого на S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ перенесены метрические свойства. В терминах структур последнее означает, что S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ является гладким вложенным n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ -мерным подмногообразием [39, с.~98] физического пространства E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ с индуцированной геометрией последнего:

S=(S, T E | S , D E | S ,g | S , | S ,e | S ), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=ucaaI9aGaaGikaiaadofacaaISaGaaGPaVlab=nr8un aaBaaaleaacaWGfbaabeaakiaaiYhadaWgaaWcbaGaam4uaaqabaGc caaISaGaaGPaVlab=nq8enaaBaaaleaacaWGfbaabeaakiaaiYhada WgaaWcbaGaam4uaaqabaGccaaISaGaaGPaVlaahEgacaaI8bWaaSba aSqaaiaadofaaeqaaOGaaGilaiaaykW7cqGHhis0caaI8bWaaSbaaS qaaiaadofaaeqaaOGaaGilaiaaykW7caGILbGaaGiFamaaBaaaleaa caWGtbaabeaakiaaiMcacaaISaaaaa@65CC@ (4)

где S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ подлежащее множество формы, а вертикальная черта #| S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4iaiaaiYhadaWgaaWcbaGaam4uaaqabaaaaa@3ABA@ обозначает ограничение объекта # MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4iaaaa@38B0@ на множество S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ .

Элемент T E | S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFtepvdaWgaaWcbaGaamyraaqabaGccaaI8bWaaSbaaSqaai aadofaaeqaaaaa@4674@ структуры (4) является топологией на S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ , индуцированной из E geom MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesrdaWgaaWcbaGaae4zaiaabwgacaqGVbGaaeyBaaqaba aaaa@4688@ [47, с.~49], а класс D E | S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFdeprdaWgaaWcbaGaamyraaqabaGccaaI8bWaaSbaaSqaai aadofaaeqaaaaa@4654@ является гладкой структурой на S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ , порожденной атласом, состоящим из карт срезки (slice charts) [39, с.~101]. Поле g | S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4zaiaaiYhadaWgaaWcbaGaam4uaaqabaaaaa@3AFD@ определяет риманову метрику на S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ (в рамках классической дифференциальной геометрии ей соответствует первая основная фундаментальная форма) как обратный образ (pullback) [39, с.~320] g | S := ι S g MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4zaiaaiYhadaWgaaWcbaGaam4uaaqabaGccaaI 6aGaaGypaiabeM7aPnaaDaaaleaacaWGtbaabaGaey4fIOcaaOGaaC 4zaaaa@412F@ физической метрики относительно канонической инъекции ι S :SE MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyUdK2aaSbaaSqaaiaadofaaeqaaOGaaGOoaiaa ykW7caWGtbWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiu aacqWFJgcScaWGfbaaaa@4A34@ , определяемой формулой xx. В свою очередь, | S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIeTaaGiFamaaBaaaleaacaWGtbaabeaaaaa@3B93@ есть связность Леви-Чивита [48, с.~122--123] на S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ , порожденная полем g | S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4zaiaaiYhadaWgaaWcbaGaam4uaaqabaaaaa@3AFD@ . Наконец, поле e | S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaOyzaiaaiYhadaWgaaWcbaGaam4uaaqabaaaaa@3AFE@ является формой объема на S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ , индуцированной формой объема e MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaOyzaaaa@38F4@ физического пространства. Определение последнего поля довольно деликатно и зависит от размерности n=dimB MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2daciGGKbGaaiyAaiaac2gatuuDJXwA KzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbacfaGae8xaWleaaa@48FE@ . Если n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ , подобно физическому пространству, то многообразие (S, T E | S , D E | S ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadofacaaISaGaaGPaVprr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae83eXt1aaSbaaSqaaiaadw eaaeqaaOGaaGiFamaaBaaaleaacaWGtbaabeaakiaaiYcacaaMc8Ua e83aXt0aaSbaaSqaaiaadweaaeqaaOGaaGiFamaaBaaaleaacaWGtb aabeaakiaaiMcaaaa@520E@ всегда ориентируемо, и его форма объема определяется через обратный образ e | S = ι S e MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaOyzaiaaiYhadaWgaaWcbaGaam4uaaqabaGccaaI 9aGaeqyUdK2aa0baaSqaaiaadofaaeaacqGHxiIkaaGccaGILbaaaa@406D@ [39, с.~383]. В случае же, когда n<3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaaiYdacaaIZaaaaa@3A79@ , обратный образ 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4maaaa@38C0@ -формы на n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ -мерное подмногообразие дает нуль-форму. Чтобы иметь возможность определить форму объема по римановой метрике [39, с.~389], необходимо наложить ограничение на тело B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ , потребовав, чтобы оно было ориентируемым многообразием. Тогда все его формы будут также ориентируемы. В дальнейшем это ограничение неявно подразумевается[2].

Даже если размерность тела совпадает с размерностью физического пространства, то есть, когда n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ , евклидова форма S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ может не совпадать со всем физическим пространством E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ . Это означает, что соответствующая конфигурация ϰ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+aaa@43FE@ не является обратимым отображением, поскольку не для каждого элемента xE имеется прообраз в B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ . Чтобы исправить ситуацию, достаточно воспользоваться отображением ϰ ^ :BS MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaecaaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy 0Hgip5wzaGqbaiab=b=a5dGaayPadaGaaGOoaiaaykW7tuuDJXwAKz KCHTgD1jharCqr1ngBPrgigjxyRrxDYbacgaGae4xaWlKaeyOKH4Qa e8NeXpfaaa@5751@ , определенным равенством ϰ ^ (p):=ϰ(p) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaecaaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy 0Hgip5wzaGqbaiab=b=a5dGaayPadaGaaGikamrr1ngBPrMrYf2A0v NCaeXbfv3ySLgzGyKCHTgD1jhaiyaacqGFPapCcaaIPaGaaGOoaiaa i2dacqWFWpq+caaIOaGae4xkWdNaaGykaaaa@5A63@ , т. е. ϰ ^ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaecaaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy 0Hgip5wzaGqbaiab=b=a5dGaayPadaaaaa@44C0@ получено из ϰ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+aaa@43FE@ сужением области прибытия на образ отображения. Действительно, это отображение является биекцией. Более того, оно является диффеоморфизмом.

Пусть ϰ R ,ϰ:BE MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+daWgaaWcbaGaamOuaaqabaGccaaISaGaaGPaVlab=b =a5laaiQdacaaMc8+efv3ySLgzgjxyRrxDYbqehuuDJXwAKbIrYf2A 0vNCaGGbaiab+fa8cjabgkziUkab=btifbaa@5B8E@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ произвольные конфигурации, образами которых являются евклидовы формы S R = ϰ R (B) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaGccaaI9aGae8h8dK=aaS baaSqaaiaadkfaaeqaaOGaaGikamrr1ngBPrMrYf2A0vNCaeXbfv3y SLgzGyKCHTgD1jhaiyaacqGFbaVqcaaIPaaaaa@5699@ и S=ϰ(B) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=ucaaI9aGae8h8dKVaaGikamrr1ngBPrMrYf2A0vNCae Xbfv3ySLgzGyKCHTgD1jhaiyaacqGFbaVqcaaIPaaaaa@547F@ . Определим отображения ϰ ^ R :B S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaecaaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy 0Hgip5wzaGqbaiab=b=a5dGaayPadaWaaSbaaSqaaiaadkfaaeqaaO GaaGOoaiaaykW7tuuDJXwAKzKCHTgD1jharCqr1ngBPrgigjxyRrxD YbacgaGae4xaWlKaeyOKH4Qae8NeXp1aaSbaaSqaaiaadkfaaeqaaa aa@5961@ и ϰ ^ :BS MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaecaaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy 0Hgip5wzaGqbaiab=b=a5dGaayPadaGaaGOoaiaaykW7tuuDJXwAKz KCHTgD1jharCqr1ngBPrgigjxyRrxDYbacgaGae4xaWlKaeyOKH4Qa e8NeXpfaaa@5751@ . Тогда композиция γ:= ϰ ^ ϰ ^ R 1 : S R S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdCMaaGOoaiaai2dadaqiaaqaamrr1ngBPrwt HrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae8h8dKpacaGLcmaacq WIyiYBdaqiaaqaaiab=b=a5dGaayPadaWaa0baaSqaaiaadkfaaeaa cqGHsislcaaIXaaaaOGaaGOoaiaaykW7cqWFse=udaWgaaWcbaGaam OuaaqabaGccqGHsgIRcqWFse=uaaa@5814@ характеризует переход от формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ к форме S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ и в этой связи называется деформацией. Соотношения между телом и его формами, конфигурациями и деформациями иллюстрируются на рис. 1.

 

Рис. 1.1. Конфигурации и деформации

Fig. 1.1. Configurations and deformations

 

1.3. Геометрическая структура над телом

Выбор привилегированной формы, геометрия которой в общем случае неевклидова, означает, что тело, будучи носителем этой формы, становится геометрическим пространством. Термин <<геометрический>> подчеркивает, что рассматриваются многообразия, на которых определены правило параллельного переноса и возможность измерять длины, т. е. все то, что позволяет использовать геометрический язык. Альтернативный термин MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ <<пространство аффинной связности с метрикой>> MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ в настоящей статье не употребляется в силу его громоздкости. Таким образом, структура (3) пополняется новыми элементами [26]:

B geom =(B, T B , D B ,G,,μ), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cnaaBaaaleaacaqGNbGaaeyzaiaab+gacaqGTbaabe aakiaai2dacaaIOaGaamOqaiaaiYcacaaMc8+efv3ySLgznfgDOfda rCqr1ngBPrginfgDObYtUvgaiyaacqGFtepvdaWgaaWcbaGaamOqaa qabaGccaaISaGaaGPaVlab+nq8enaaBaaaleaacaWGcbaabeaakiaa iYcacaaMc8Uaae4raiaaiYcacaaMc8Uaey4bIeTaaGilaiaaykW7cq aH8oqBcaaIPaGaaGilaaaa@6A78@ (5)

где G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ риманова метрика, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ аффинная связность, а μ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiVd0gaaa@39B9@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ форма объема на B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOqaaaa@38CA@ . Структура (5) является абстрактным представлением формы, свободной от напряжений; ее элементы G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ и μ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiVd0gaaa@39B9@ зависят от физической природы несовместности деформаций. В частности, эти поля могут быть неравноправны: одни из них могут быть определены по другим либо по дополнительным физическим полям. Таблица 1.1 иллюстрирует эту ситуацию. Она содержит примеры геометрических пространств, обычно используемых в геометрической механике континуума [20; 21].

 

Таблица 1.1. Геометрические пространства над B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@

Table 1.1. Geometric spaces over B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@

Пространство

Структура

Базисн. поля

Произв. поля

Римана

(B, D B ,G,,dV) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadkeacaaISaGaaGPaVprr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae83aXt0aaSbaaSqaaiaadk eaaeqaaOGaaGilaiaaykW7caqGhbGaaGilaiaaykW7cqGHhis0caaI SaGaaGPaVlaadsgacaWGwbGaaGykaaaa@538B@

G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@

MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ , dV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamizaiaadAfaaaa@39C7@

Вайценбока

(B, D B ,G,,μ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadkeacaaISaGaaGPaVprr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae83aXt0aaSbaaSqaaiaadk eaaeqaaOGaaGilaiaaykW7caqGhbGaaGilaiaaykW7cqGHhis0caaI SaGaaGPaVlabeY7aTjaaiMcaaaa@537D@

H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaaaa@38CE@ , μ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiVd0gaaa@39B9@

G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@

Вейля

(B, D B ,G,,μ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadkeacaaISaGaaGPaVprr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae83aXt0aaSbaaSqaaiaadk eaaeqaaOGaaGilaiaaykW7caqGhbGaaGilaiaaykW7cqGHhis0caaI SaGaaGPaVlabeY7aTjaaiMcaaaa@537D@

G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ , ν MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyVd4gaaa@39BB@ , μ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiVd0gaaa@39B9@

MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@

 

В табл. 1.1 столбец <<Базисные поля>> содержит поля, которые предписаны, исходя из тех или иных физических соображений. Эти поля не зависят от структуры геометрии. Последний столбец <<Производные поля>> содержит поля, которые получаются из базисных полей и, возможно, геометрических свойств гладких многообразий. Поясним соответствие между базисными и производными полями более подробно.

Если пространство риманово, то риманова метрика G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ является базисным полем. Другие поля из структуры (5) выражаются в терминах метрики: аффинная связность MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ является связностью Леви-Чивита, а форма объема μ=dV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiVd0MaaGypaiaadsgacaWGwbaaaa@3C44@ определяется равенством dV= detG d p 1 d p n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamizaiaadAfacaaI9aWaaOaaaeaaciGGKbGaaiyz aiaacshacaqGhbaaleqaaOGaaGjcVlaadsgatuuDJXwAKzKCHTgD1j haryqr1ngBPrgigjxyRrxDYbacfaGae8xkWd3aaWbaaSqabeaacaaI XaaaaOGaey4jIKTaeS47IWKaey4jIKTaamizaiab=Lc8WnaaCaaale qabaGaamOBaaaaaaa@57E4@ . В случае пространства Вайценбока заданы поле H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaaaa@38CE@ обратимых линейных преобразований и форма объема μ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiVd0gaaa@39B9@ . Тогда аффинная связность и метрика, являясь производными полями, порождаются полем H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaaaa@38CE@ . Наконец, аффинная связность пространства Вейля определяется метрикой и 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaaaa@38BE@ -формой ν MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyVd4gaaa@39BB@ .

Геометрия пространств аффинной связности характеризуется тензорными полями кручения T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ubaa@44A0@ , кривизны MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Xrisbaa@43D8@ и неметричности Q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rbaa@449A@ . Их соответствия каждому пространству показаны в табл. 1.2, где символ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeSigI8gaaa@393D@ указывает на то, что соответствующее поле всюду равно нулю, а MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOiGClaaa@3988@ означает, что поле принимает ненулевые значения.

 

Таблица 1.2. Соответствие между геометриями и тензорными полями кручения, кривизны и неметричности

Table 1.2. Correspondence between geometries and tensorial fields of torsion, curvature and nonmetricity

Геометрия

Кручение ( T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ubaa@44A0@ )

Кривизна ( MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Xrisbaa@43D8@ )

Неметричность ( Q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rbaa@449A@ )

Римана

MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeSigI8gaaa@393D@

MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOiGClaaa@3988@

MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeSigI8gaaa@393D@

Вайценбока

MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOiGClaaa@3988@

MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeSigI8gaaa@393D@

MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeSigI8gaaa@393D@

Вейля

MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeSigI8gaaa@393D@

MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOiGClaaa@3988@

MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOiGClaaa@3988@

 

1.4. Тело и неевклидовы формы

Рассмотрим более подробно выбор геометрии на теле B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ . Очевидным представляется способ, когда геометрия индуцируется из физического пространства E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ по некоторой конфигурации ϰ:BE MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+caaI6aGaaGPaVprr1ngBPrMrYf2A0vNCaeXbfv3ySL gzGyKCHTgD1jhaiyaacqGFbaVqcqGHsgIRcqWFWesraaa@55CF@ посредством обратных образов из евклидовой формы (4). В этом случае

G= ϰ ^ g | S ,= ϰ ^ | S ,μ= ϰ ^ e | S , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raiaai2dadaqiaaqaamrr1ngBPrwtHrhAXaqe guuDJXwAKbstHrhAG8KBLbacfaGae8h8dKpacaGLcmaadaahaaWcbe qaaiabgEHiQaaakiaahEgacaaI8bWaaSbaaSqaaiaadofaaeqaaOGa aGilaiaaywW7cqGHhis0caaI9aWaaecaaeaacqWFWpq+aiaawkWaam aaCaaaleqabaGaey4fIOcaaOGaey4bIeTaaGiFamaaBaaaleaacaWG tbaabeaakiaaiYcacaaMf8UaeqiVd0MaaGypamaaHaaabaGae8h8dK pacaGLcmaadaahaaWcbeqaaiabgEHiQaaakiaakwgacaaI8bWaaSba aSqaaiaadofaaeqaaOGaaGilaaaa@63D4@

где

G p (u,v):=g( T p ϰ(u), T p ϰ(v)), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaBaaaleaatuuDJXwAKzKCHTgD1jharyqr 1ngBPrgigjxyRrxDYbacfaGae8xkWdhabeaakiaaiIcacaqG1bGaaG ilaiaaykW7caqG2bGaaGykaiaaiQdacaaI9aGaaC4zaiaaiIcacaWG ubWaaSbaaSqaaiab=Lc8WbqabaWefv3ySLgznfgDOfdarCqr1ngBPr ginfgDObYtUvgaiyaakiab+b=a5laaiIcacaqG1bGaaGykaiaaiYca caaMc8UaamivamaaBaaaleaacqWFPapCaeqaaOGae4h8dKVaaGikai aabAhacaaIPaGaaGykaiaaiYcaaaa@6BA1@

u v:= ϰ ^ {( | S ) ϰ ^ (u) ϰ ^ (v)}, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaSbaaSqaaiaabwhaaeqaaOGaaeODaiaa iQdacaaI9aWaaecaaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0H gip5wzaGqbaiab=b=a5dGaayPadaWaaWbaaSqabeaacqGHxiIkaaGc caaI7bGaaGikaiabgEGirlaaiYhadaWgaaWcbaGaam4uaaqabaGcca aIPaWaaSbaaSqaamaaHaaabaGae8h8dKpacaGLcmaadaWgaaqaaiab gEHiQaqabaGaaGikaiaabwhacaaIPaaabeaakmaaHaaabaGae8h8dK pacaGLcmaadaWgaaWcbaGaey4fIOcabeaakiaaiIcacaqG2bGaaGyk aiaai2hacaaISaaaaa@606B@

μ p ( v 1 ,, v n ):=e | S ( T p ϰ( v 1 ),, T p ϰ( v n )). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiVd02aaSbaaSqaamrr1ngBPrMrYf2A0vNCaeHb fv3ySLgzGyKCHTgD1jhaiuaacqWFPapCaeqaaOGaaGikaiaabAhada WgaaWcbaGaaGymaaqabaGccaaISaGaaGPaVlablAciljaaiYcacaaM c8UaaeODamaaBaaaleaacaWGUbaabeaakiaaiMcacaaI6aGaaGypai aakwgacaaI8bWaaSbaaSqaaiaadofaaeqaaOGaaGikaiaadsfadaWg aaWcbaGae8xkWdhabeaatuuDJXwAK1uy0HwmaeXbfv3ySLgzG0uy0H gip5wzaGGbaOGae4h8dKVaaGikaiaabAhadaWgaaWcbaGaaGymaaqa baGccaaIPaGaaGilaiaaykW7cqWIMaYscaaISaGaaGPaVlaadsfada WgaaWcbaGae8xkWdhabeaakiab+b=a5laaiIcacaqG2bWaaSbaaSqa aiaad6gaaeqaaOGaaGykaiaaiMcacaaIUaaaaa@79A0@

Здесь pB MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Lc8WjabgIGiolab=fa8cbaa@480E@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ произвольная точка тела, u,v, v 1 ,, v n T p B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyDaiaaiYcacaaMc8UaaeODaiaaiYcacaaMc8Ua aeODamaaBaaaleaacaaIXaaabeaakiaaiYcacaaMc8UaeSOjGSKaaG ilaiaaykW7caqG2bWaaSbaaSqaaiaad6gaaeqaaOGaeyicI4Saamiv amaaBaaaleaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYb acfaGae8xkWdhabeaakiab=fa8cbaa@5940@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ произвольные касательные векторы [40, с.~22], null MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ касательное отображение [40, с.~28] (используется отождествление T ϰ(p) EV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=b=a5laaiIcatuuDJXwAKzKCHTgD1j harCqr1ngBPrgigjxyRrxDYbacgaGae4xkWdNaaGykaaqabaGccqWF WesrcqGHfjcqcqWFveVvaaa@5771@ по каноническому изоморфизму [39, с.~59]), а ϰ ^ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaecaaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy 0Hgip5wzaGqbaiab=b=a5dGaayPadaWaaWbaaSqabeaacqGHxiIkaa aaaa@45DC@ , ϰ ^ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaecaaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy 0Hgip5wzaGqbaiab=b=a5dGaayPadaWaaSbaaSqaaiabgEHiQaqaba aaaa@45DB@ являются, соответственно, обратным и прямым образами векторных полей [16, с.~67].

Рассмотренный выбор геометрии на теле соответствует случаю классической механики сплошной среды. В ней тело не играет никакой иной роли, кроме как множества меток, а геометрия на нем фиксирована и совпадает с геометрией любой из его евклидовых форм. Определим теперь неевклидову форму [25; 26]

S=(S, T S , D S , G 0 , 0 , μ 0 ), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaiaai2dacaaIOaGaam4uaiaaiYcacaaMc8+e fv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFtepvda WgaaWcbaGaam4uaaqabaGccaaISaGaaGPaVlab=nq8enaaBaaaleaa caWGtbaabeaakiaaiYcacaaMc8Uaae4ramaaBaaaleaacaaIWaaabe aakiaaiYcacaaMc8Uaey4bIe9aaSbaaSqaaiaaicdaaeqaaOGaaGil aiaaykW7cqaH8oqBdaWgaaWcbaGaaGimaaqabaGccaaIPaGaaGilaa aa@5DF0@ (6)

где T S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFtepvdaWgaaWcbaGaam4uaaqabaaaaa@446E@ и D S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFdeprdaWgaaWcbaGaam4uaaqabaaaaa@444E@ являются, соответственно, топологией и гладкой структурой на континуальном множестве S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ , а G 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaBaaaleaacaaIWaaabeaaaaa@39B3@ , 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaSbaaSqaaiaaicdaaeqaaaaa@3A6F@ , μ 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiVd02aaSbaaSqaaiaaicdaaeqaaaaa@3A9F@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ произвольными римановой метрикой, аффинной связностью и формой объема на n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ -мерном многообразии (S, T S , D S ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadofacaaISaGaaGPaVprr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae83eXt1aaSbaaSqaaiaado faaeqaaOGaaGilaiaaykW7cqWFdeprdaWgaaWcbaGaam4uaaqabaGc caaIPaaaaa@4E02@ . Хотя, с формальной точки зрения, множество S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ , топология T S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFtepvdaWgaaWcbaGaam4uaaqabaaaaa@446E@ и гладкая структура D S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFdeprdaWgaaWcbaGaam4uaaqabaaaaa@444E@ могут быть произвольными, следует иметь в виду, что каждая неевклидова форма может быть получена лишь по евклидовой (ведь только такая форма доступна наблюдениям). По этой причине ограничим общность рассуждений требованием, чтобы (см. формулу (4))

SE, T S = T E | S ,è D S = D E | S . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaiabgkOimprr1ngBPrwtHrhAXaqeguuDJXwA KbstHrhAG8KBLbacfaGae8hmHuKaaGilaiaaywW7cqWFtepvdaWgaa WcbaGaam4uaaqabaGccaaI9aGae83eXt1aaSbaaSqaaiaadweaaeqa aOGaaGiFamaaBaaaleaacaWGtbaabeaakiaaiYcacaaMf8Uaaei6ai aaywW7cqWFdeprdaWgaaWcbaGaam4uaaqabaGccaaI9aGae83aXt0a aSbaaSqaaiaadweaaeqaaOGaaGiFamaaBaaaleaacaWGtbaabeaaki aai6caaaa@5EBB@

Геометрическая структура формы (6) может не иметь ничего общего с геометрией физического пространства. Для единообразия можно предположить, что любая такая форма является образом вложения ϰ R :BR MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+daWgaaWcbaGaamOuaaqabaGccaaI6aGaaGPaVprr1n gBPrMrYf2A0vNCaeXbfv3ySLgzGyKCHTgD1jhaiyaacqGFbaVqcqGH sgIRcaWGsbaaaa@5698@ , которое мы будем называть обобщенной конфигурацией, тела B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ в некоторое неевклидово пространство R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOuaaaa@38DA@ , скажем, Римана. В этом случае тело может быть снабжено соответствующей геометрией по формулам

G= ϰ ^ R G 0 ,= ϰ ^ R 0 ,μ= ϰ ^ R μ 0 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raiaai2dadaqiaaqaamrr1ngBPrwtHrhAXaqe guuDJXwAKbstHrhAG8KBLbacfaGae8h8dKpacaGLcmaadaqhaaWcba GaamOuaaqaaiabgEHiQaaakiaabEeadaWgaaWcbaGaaGimaaqabaGc caaISaGaaGzbVlabgEGirlaai2dadaqiaaqaaiab=b=a5dGaayPada Waa0baaSqaaiaadkfaaeaacqGHxiIkaaGccqGHhis0daWgaaWcbaGa aGimaaqabaGccaaISaGaaGzbVlabeY7aTjaai2dadaqiaaqaaiab=b =a5dGaayPadaWaa0baaSqaaiaadkfaaeaacqGHxiIkaaGccqaH8oqB daWgaaWcbaGaaGimaaqabaGccaaIUaaaaa@638E@

Геометрия формы S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ полностью определяется физической причиной несовместности, поэтому одно и то же многообразие меток может обладать разными геометриями. Как отмечалось ранее, тело становится геометрическим пространством и в рамках теоретических рассуждений само может быть рассмотрено как неевклидова форма [13; 14; 23].

Пусть ϰ R :BR MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+daWgaaWcbaGaamOuaaqabaGccaaI6aGaaGPaVprr1n gBPrMrYf2A0vNCaeXbfv3ySLgzGyKCHTgD1jhaiyaacqGFbaVqcqGH sgIRcaWGsbaaaa@5698@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ обобщенная конфигурация и пусть ϰ:BE MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+caaI6aGaaGPaVprr1ngBPrMrYf2A0vNCaeXbfv3ySL gzGyKCHTgD1jhaiyaacqGFbaVqcqGHsgIRcqWFWesraaa@55CF@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ обычная конфигурация. Отображение λ= ϰ ^ ϰ ^ R 1 :SS MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWMaaGypamaaHaaabaWefv3ySLgznfgDOfda ryqr1ngBPrginfgDObYtUvgaiuaacqWFWpq+aiaawkWaaiablIHiVn aaHaaabaGae8h8dKpacaGLcmaadaqhaaWcbaGaamOuaaqaaiabgkHi TiaaigdaaaGccaaI6aGaaGPaVlaadofacqGHsgIRcqWFse=uaaa@554D@ из неевклидовой формы S= ϰ R (B) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaiaai2datuuDJXwAK1uy0HwmaeHbfv3ySLgz G0uy0Hgip5wzaGqbaiab=b=a5paaBaaaleaacaWGsbaabeaakiaaiI catuuDJXwAKzKCHTgD1jharCqr1ngBPrgigjxyRrxDYbacgaGae4xa WlKaaGykaaaa@5489@ в евклидову форму S=ϰ(B) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=ucaaI9aGae8h8dKVaaGikamrr1ngBPrMrYf2A0vNCae Xbfv3ySLgzGyKCHTgD1jhaiyaacqGFbaVqcaaIPaaaaa@547F@ на уровне гладких структур[3] неотличимо от классической деформации. По этой причине будем называть λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWgaaa@39B7@ обобщенной деформацией. Вместе с тем следует иметь в виду, что геометрические структуры над областью отправления и областью прибытия λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWgaaa@39B7@ отличны друг от друга в целом. Это отличие проявляет себя в действиях обратного и прямого образов λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaWbaaSqabeaacqGHxiIkaaaaaa@3AD3@ и λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaSbaaSqaaiabgEHiQaqabaaaaa@3AD2@ . Действительно, первое отображение переводит элементы евклидовой структуры из формы S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ на форму S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ , а второе, наоборот, переводит элементы неевклидовой структуры из формы S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ на форму S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ . Конечно, ограничиваясь сценой гладких многообразий, эти отображения всего лишь преобразуют поля.

В частном случае, когда геометрия формы S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ совпадает с евклидовой (или, если n<3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaaiYdacaaIZaaaaa@3A79@ , MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ то с геометрией, индуцированной евклидовой геометрией объемлющего пространства), обобщенная деформация сводится к классической деформации. Следует лишь отождествить S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ с некоторым подмножеством евклидова пространства, а λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWgaaa@39B7@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ с искажением соответствующей евклидовой формы в другую форму S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ . В общем же случае, когда форма S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ неевклидова, можно вложить ее в евклидово пространство размерности большей, чем 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4maaaa@38C0@ . Здесь под вложением понимается отображение в такое пространство, в котором геометрия S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uaaaa@38DB@ будет совпадать с геометрией, индуцированной из объемлющего пространства на образ вложения.

Изложенные геометрические идеи могут быть образно интерпретированы в случае размерности n=2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIYaaaaa@3A79@ , которая соответствует криволинейной упругой мембране. Здесь возможны две точки зрения. Первая из них отвечает могущественному трехмерному наблюдателю в евклидовом пространстве: процесс деформации наблюдается как растяжение и изгиб мембраны в объемлющем пространстве. Вторая точка зрения более ограничительна. Следует отождествить себя с наблюдателем, пребывающем в двумерном мире с неевклидовой геометрией, образованной криволинейной формой мембраны. Геометрически это означает, что такой наблюдатель чувствует лишь внутреннюю геометрию поверхности, ассоциированной с мембраной. Именно второй подход позволяет дать описание <<чистой>> деформации, отбросив те <<фиктивные>> деформации, которые не влияют на состояние тела.

1.5. Пример неевклидовой формы

Проиллюстрируем геометрические идеи на частном примере, основанном на решении задачи об осесимметричной конечной деформации гиперупругой мембраны [49]. Рисунок 2 содержит некоторые результаты численных расчетов с использованием этого решения. Тело B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ изображено в центре нижней части рисунка как открытый диск на евклидовой плоскости с нанесенной на нем координатной сеткой. Этим иллюстрируется, что тело имеет лишь структуру гладкого многообразия и что в рассматриваемом случае эта структура совместна с евклидовой. Физическое пространство, изображенное в правой верхней части рисунка, является двумерным евклидовым многообразием. Оно содержит две плоские евклидовы формы S 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaaGymaaqabaaaaa@444F@ и S 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaaGOmaaqabaaaaa@4450@ , которые самонапряжены (как и любые другие). Эти формы ощущаются двумерным физическим наблюдателем.

Неевклидова отсчетная форма S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ тела, в которой все представительные объемы находятся в натуральном (т. е. в свободном от напряжений) состоянии, изображена в виде полусферы в левой верхней части рисунка. На нее нанесена сферическая сетка, и форма в целом помещена на сферу, представляющую сферическую (риманову) геометрию пространства, содержащего неевклидову форму. Такой образ может ощущаться трехмерным наблюдателем.

Изменение цветовых оттенков в плоскостях форм S 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaaGymaaqabaaaaa@444F@ и S 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaaGOmaaqabaaaaa@4450@ показывает распределение накопленной упругой энергии, связанной с частными вложениями отсчетной формы в физическую плоскость. Все, что может увидеть двумерный физический наблюдатель, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ это деформация γ: S 1 S 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdCMaaGOoaiaaykW7tuuDJXwAK1uy0HwmaeHb fv3ySLgzG0uy0Hgip5wzaGqbaiab=jr8tnaaBaaaleaacaaIXaaabe aakiabgkziUkab=jr8tnaaBaaaleaacaaIYaaabeaaaaa@4CFF@ одной самонапряженной формы в другую. Для иллюстрации самонапряженной природы форм на плоскостях, перпендикулярных к ним, построены графики с распределениями главных напряжений Коши, T 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaacaaIXaaabeaaaaa@39C3@ , T 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaacaaIYaaabeaaaaa@39C4@ и накопленной упругой энергии W MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4vaaaa@38DF@ .

 

Рис. 1.2. Вложения неевклидовой отсчетной формы в плоское физическое пространство

Fig. 1.2. Embeddings of non-Euclidean reference shape into planar physical space

 

Рисунок 2 основан на вычислениях, проведенных в соответствии со статьей [49]. Мембрана, отсчетная форма которой представляет собой полусферу, растягивается и уплощается. Плотность упругой энергии мембраны полагается равной

W=W(φ),W= λ 2 + μ 2 + 1 λ 2 μ 2 3 +α λ 2 μ 2 + 1 λ 2 + 1 μ 2 3 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4vaiaai2dacaWGxbGaaGikaiabeA8aQjaaiMca caaISaGaaGzbVlaadEfacaaI9aWaaeWaaeaacqaH7oaBdaahaaWcbe qaaiaaikdaaaGccqGHRaWkcqaH8oqBdaahaaWcbeqaaiaaikdaaaGc cqGHRaWkdaWcaaqaaiaaigdaaeaacqaH7oaBdaahaaWcbeqaaiaaik daaaGccqaH8oqBdaahaaWcbeqaaiaaikdaaaaaaOGaeyOeI0IaaG4m aaGaayjkaiaawMcaaiabgUcaRiabeg7aHnaabmaabaGaeq4UdW2aaW baaSqabeaacaaIYaaaaOGaeqiVd02aaWbaaSqabeaacaaIYaaaaOGa ey4kaSYaaSaaaeaacaaIXaaabaGaeq4UdW2aaWbaaSqabeaacaaIYa aaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiabeY7aTnaaCaaaleqa baGaaGOmaaaaaaGccqGHsislcaaIZaaacaGLOaGaayzkaaGaaGilaa aa@6651@

что соответствует материалу Муни. Здесь функции φλ(φ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOXdOMaeSOPHeMaeq4UdWMaaGikaiabeA8aQjaa iMcaaaa@404F@ , φμ(φ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOXdOMaeSOPHeMaeqiVd0MaaGikaiabeA8aQjaa iMcaaaa@4051@ являются главными растяжениями, а α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySdegaaa@39A2@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ материальной константой. Независимая переменная φ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOXdOgaaa@39C0@ соответствует азимутальному углу. Результаты напряжений представлены выражениями (соответственно, радиальная и окружная компоненты напряжений)

T 1 = λ μ 1 λ 3 μ 3 (1+α μ 2 ), T 2 = μ λ 1 λ 3 μ 3 (1+α λ 2 ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaacaaIXaaabeaakiaai2dadaqa daqaamaalaaabaGaeq4UdWgabaGaeqiVd0gaaiabgkHiTmaalaaaba GaaGymaaqaaiabeU7aSnaaCaaaleqabaGaaG4maaaakiabeY7aTnaa CaaaleqabaGaaG4maaaaaaaakiaawIcacaGLPaaacaaIOaGaaGymai abgUcaRiabeg7aHjabeY7aTnaaCaaaleqabaGaaGOmaaaakiaaiMca caaISaGaaGzbVlaadsfadaWgaaWcbaGaaGOmaaqabaGccaaI9aWaae WaaeaadaWcaaqaaiabeY7aTbqaaiabeU7aSbaacqGHsisldaWcaaqa aiaaigdaaeaacqaH7oaBdaahaaWcbeqaaiaaiodaaaGccqaH8oqBda ahaaWcbeqaaiaaiodaaaaaaaGccaGLOaGaayzkaaGaaGikaiaaigda cqGHRaWkcqaHXoqycqaH7oaBdaahaaWcbeqaaiaaikdaaaGccaaIPa GaaGOlaaaa@66CE@

Замечание 1. Функции φλ(φ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOXdOMaeSOPHeMaeq4UdWMaaGikaiabeA8aQjaa iMcaaaa@404F@ и φμ(φ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOXdOMaeSOPHeMaeqiVd0MaaGikaiabeA8aQjaa iMcaaaa@4051@ являются решениями системы двух нелинейных дифференциальных уравнений первого порядка:

λ ' = λ 2 ( μ 2 λ 2 )( λ 2 μ 2 +α)+λ(λμcosφ)( λ 4 μ 2 3α μ 2 ( λ 4 μ 2 +1)) μsinφ( λ 4 μ 2 +3)(1+α μ 2 ) , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaWbaaSqabeaacaWGNaaaaOGaaGypamaa laaabaGaeq4UdW2aaWbaaSqabeaacaaIYaaaaOGaaGikaiabeY7aTn aaCaaaleqabaGaaGOmaaaakiabgkHiTiabeU7aSnaaCaaaleqabaGa aGOmaaaakiaaiMcacaaIOaGaeq4UdW2aaWbaaSqabeaacaaIYaaaaO GaeqiVd02aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaeqySdeMaaGyk aiabgUcaRiabeU7aSjaaiIcacqaH7oaBcqGHsislcqaH8oqBciGGJb Gaai4BaiaacohacqaHgpGAcaaIPaGaaGikaiabeU7aSnaaCaaaleqa baGaaGinaaaakiabeY7aTnaaCaaaleqabaGaaGOmaaaakiabgkHiTi aaiodacqGHsislcqaHXoqycqaH8oqBdaahaaWcbeqaaiaaikdaaaGc caaIOaGaeq4UdW2aaWbaaSqabeaacaaI0aaaaOGaeqiVd02aaWbaaS qabeaacaaIYaaaaOGaey4kaSIaaGymaiaaiMcacaaIPaaabaGaeqiV d0Maci4CaiaacMgacaGGUbGaeqOXdOMaaGikaiabeU7aSnaaCaaale qabaGaaGinaaaakiabeY7aTnaaCaaaleqabaGaaGOmaaaakiabgUca RiaaiodacaaIPaGaaGikaiaaigdacqGHRaWkcqaHXoqycqaH8oqBda ahaaWcbeqaaiaaikdaaaGccaaIPaaaaiaaiYcaaaa@885D@

μ ' = λμcosφ sinφ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiVd02aaWbaaSqabeaacaWGNaaaaOGaaGypamaa laaabaGaeq4UdWMaeyOeI0IaeqiVd0Maci4yaiaac+gacaGGZbGaeq OXdOgabaGaci4CaiaacMgacaGGUbGaeqOXdOgaaaaa@48EF@

с начальными условиями λ(0)= λ 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWMaaGikaiaaicdacaaIPaGaaGypaiabeU7a SnaaBaaaleaacaaIWaaabeaaaaa@3F37@ и μ(0)= μ 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiVd0MaaGikaiaaicdacaaIPaGaaGypaiabeY7a TnaaBaaaleaacaaIWaaabeaaaaa@3F3B@ .

2. Синтезирование неевклидовой отсчетной формы

2.1. Гипотеза о локальной разгрузке

Натуральное состояние.

Перейдем теперь к реализации идеи неевклидовой отсчетной формы. Для этого, в первую очередь, следует уточнить, что понимается под представительным объемом и его натуральным состоянием.

Пусть S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ некоторая форма n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ -мерного тела B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ ( n=1,2,3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIXaGaaGilaiaaykW7caaIYaGa aGilaiaaykW7caaIZaaaaa@4073@ ), наблюдаемая в эксперименте. Ее точки будем обозначать прописными символами X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwaaa@4372@ , Y MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFyeFwaaa@4374@ и т. д. Будем полагать, что при n<3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaaiYdacaaIZaaaaa@3A79@ на форме S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ задана параметризация (например, для формы, топологически эквивалентной сфере, такая параметризация может быть задана с помощью азимутального и полярного углов). Предполагается, что материал тела простой и гиперупругий [44], т. е. отклик тела на деформацию γ: S R S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdCMaaGOoaiaaykW7tuuDJXwAK1uy0HwmaeHb fv3ySLgzG0uy0Hgip5wzaGqbaiab=jr8tnaaBaaaleaacaWGsbaabe aakiabgkziUkab=jr8tbaa@4C33@ в точке X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcqGHiiIZcqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@47D4@ определяется значением плотности упругой энергии

w=W(X, F X ; G R ,G). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4Daiaai2dacaWGxbGaaGikamrr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae83fXJLaaGilaiaaykW7ca WHgbWaaSbaaSqaaiab=Dr8ybqabaGccaaI7aGaaGPaVlaahEeadaWg aaWcbaGaamOuaaqabaGccaaISaGaaGPaVlaahEeacaaIPaGaaGOlaa aa@5497@ (1)

Здесь F X = T X γHom( T X S R ; T γ(X) S) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCOramaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccaaI9aGaamivamaaBa aaleaacqWFxepwaeqaaOGaeq4SdCMaeyicI4Saaeisaiaab+gacaqG TbGaaGikaiaadsfadaWgaaWcbaGae83fXJfabeaakiab=jr8tnaaBa aaleaacaWGsbaabeaakiaaiUdacaaMc8UaamivamaaBaaaleaacqaH ZoWzcaaIOaGae83fXJLaaGykaaqabaGccqWFse=ucaaIPaaaaa@5F76@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ касательное отображение [40, с.~28], которое, в соответствии с классическим аналогом, будем называть градиентом деформации[4] в точке X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwaaa@4372@ , а G R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4ramaaBaaaleaacaWGsbaabeaaaaa@39D6@ и G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4raaaa@38D3@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ метрические тензоры на формах S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ и S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ , индуцированные метрикой физического пространства g MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4zaaaa@38F3@ на них. Представление (1) плотности упругой энергии является наиболее общим и предполагает независимую параметризацию отсчетной и актуальной форм. Возможны следующие частные случаи. 1. Параметризации воспроизводят координаты физического пространства. В этом случае G R =G=g MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4ramaaBaaaleaacaWGsbaabeaakiaai2dacaWH hbGaaGypaiaahEgaaaa@3D2E@ , а координатное представление отображения Tγ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivaiabeo7aNbaa@3A83@ в точке задается некоторой числовой матрицей общего вида. 2. Параметризации согласованы таким образом, что координатное представление Tγ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivaiabeo7aNbaa@3A83@ в любой точке совпадает с единичной матрицей. В последнем случае локальные базисы координатных представлений называются векторными базисами [50]. В дальнейшем зависимости от G R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4ramaaBaaaleaacaWGsbaabeaaaaa@39D6@ и G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4raaaa@38D3@ не будут указаны явно.

Соотношение (1) задает плотность упругой энергии, т. е. энергию, отнесенную к единице объема формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ . Однако само понятие единицы объема требует уточнений. Несмотря на то что с точки зрения математического формализма здесь можно говорить об инфинитезимальном элементе объема, физически мы не можем допустить возможность выбора сколь угодно малого элемента. Для выхода из этого противоречия мы принимаем гипотезу локального термостатического состояния [51], согласно которой представительный объем достаточно мал для того, чтобы считать его инфинитезимальным с точки зрения макроскопического описания, но в то же время достаточно велик для того, чтобы полагать его находящимся в состоянии термостатического равновесия. Принимая эту гипотезу, мы можем локально интерпретировать деформацию как линейное преобразование между соответствующими касательными слоями отсчетной и актуальной форм, то есть как касательное отображение T X γ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccqaHZoWzaaa@4628@ .

Под натуральным состоянием понимается некоторое привилегированное физическое состояние представительного объема. Свободное от напряжений состояние, когда тензор напряжений Коши равен нулю в рассматриваемой точке, может служить примером. С формальной точки зрения предполагается, что натуральное состояние характеризуется некоторым тензором S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4uaaaa@38DF@ второго ранга.

Скажем, что деформация γ ( X 0 ) : S R S ( X 0 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaWbaaSqabeaacaaIOaWefv3ySLgznfgD Ofdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwdaWgaaqaaiaaic daaeqaaiaaiMcaaaGccaaI6aGaaGPaVlab=jr8tnaaBaaaleaacaWG sbaabeaakiabgkziUkab=jr8tnaaCaaaleqabaGaaGikaiab=Dr8yn aaBaaabaGaaGimaaqabaGaaGykaaaaaaa@54E1@ преобразует представительный объем, окружающий точку X 0 S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwdaWgaaWcbaGaaGimaaqabaGccqGHiiIZcqWFse=uda WgaaWcbaGaamOuaaqabaaaaa@48C4@ , в натуральное (или единообразное) состояние, если

W(X,F) F F= T Y γ ( X 0 ) | Y= X 0 =S. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaqGaaeaadaWcaaqaaiabgkGi2kaadEfacaaIOaWe fv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwca aISaGaaGPaVlaahAeacaaIPaaabaGaeyOaIyRaaCOraaaaaiaawIa7 amaaBaaaleaacaWHgbGaaGypaiaadsfadaWgaaqaaiab=Hr8zbqaba Gaeq4SdC2aaWbaaeqabaGaaGikaiab=Dr8ynaaBaaabaGaaGimaaqa baGaaGykaaaacaaI8bWaaSbaaeaacqWFyeFwcaaI9aGae83fXJ1aaS baaeaacaaIWaaabeaaaeqaaaqabaGccaaI9aGaaC4uaiaai6caaaa@618F@

Используя уточненное понятие натурального состояния, теперь можно сформулировать гипотезу, на которой основана классическая механика деформируемого твердого тела: существует деформация γ 0 : S R S 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaSbaaSqaaiaaicdaaeqaaOGaaGOoaiaa ykW7tuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaiab=j r8tnaaBaaaleaacaWGsbaabeaakiabgkziUkab=jr8tnaaBaaaleaa caaIWaaabeaaaaa@4E09@ такая, что представительный объем, окружающий каждую точку X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcqGHiiIZcqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@47D4@ , преобразуется в натуральное состояние, представленное тензором S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4uaaaa@38DF@ (одного для всех точек), то есть,

X S R : W(X,F) F F= T X γ 0 =S. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyiaIiYefv3ySLgznfgDOfdaryqr1ngBPrginfgD ObYtUvgaiuaacqWFxepwcqGHiiIZcqWFse=udaWgaaWcbaGaamOuaa qabaGccaaI6aGaaGjbVpaaeiaabaWaaSaaaeaacqGHciITcaWGxbGa aGikaiab=Dr8yjaaiYcacaaMc8UaaCOraiaaiMcaaeaacqGHciITca WHgbaaaaGaayjcSdWaaSbaaSqaaiaahAeacaaI9aGaamivamaaBaaa baGae83fXJfabeaacqaHZoWzdaWgaaqaaiaaicdaaeqaaaqabaGcca aI9aGaaC4uaiaai6caaaa@60FE@

Таким образом, евклидова форма S 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaaGimaaqabaaaaa@444E@ состоит из представительных объемов, каждый из которых находится в натуральном состоянии. Это MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ глобальная натуральная форма.

Формулировка гипотезы о локальной разгрузке.

Заменим классическую гипотезу о существовании глобального натурального состояния гипотезой, которую будем называть гипотезой локальной разгрузки [24; 26]. Пусть S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4uaaaa@38DF@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ тензор второго ранга, определяющий натуральное состояние представительного объема. Будем полагать, что существует семейство { γ (X) } X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4Eaiabeo7aNnaaCaaaleqabaGaaGikamrr1ngB PrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae83fXJLaaGykaa aakiaai2hadaWgaaWcbaGae83fXJLaeyicI4Sae8NeXp1aaSbaaeaa caWGsbaabeaaaeqaaaaa@4F29@ деформаций γ (X) : S R S (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaWbaaSqabeaacaaIOaWefv3ySLgznfgD Ofdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwcaaIPaaaaOGaaG OoaiaaykW7cqWFse=udaWgaaWcbaGaamOuaaqabaGccqGHsgIRcqWF se=udaahaaWcbeqaaiaaiIcacqWFxepwcaaIPaaaaaaa@532B@ , каждая из которых, γ (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaWbaaSqabeaacaaIOaWefv3ySLgznfgD Ofdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwcaaIPaaaaaaa@46AB@ , преобразует представительный объем, окружающий точку X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwaaa@4372@ , в натуральное состояние, то есть

X S R : W(X,F) F F= T Y γ (X) | Y=X =S. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyiaIiYefv3ySLgznfgDOfdaryqr1ngBPrginfgD ObYtUvgaiuaacqWFxepwcqGHiiIZcqWFse=udaWgaaWcbaGaamOuaa qabaGccaaI6aGaaGjbVpaaeiaabaWaaSaaaeaacqGHciITcaWGxbGa aGikaiab=Dr8yjaaiYcacaaMc8UaaCOraiaaiMcaaeaacqGHciITca WHgbaaaaGaayjcSdWaaSbaaSqaaiaahAeacaaI9aGaamivamaaBaaa baGae8hgXNfabeaacqaHZoWzdaahaaqabeaacaaIOaGae83fXJLaaG ykaaaacaaI8bWaaSbaaeaacqWFyeFwcaaI9aGae83fXJfabeaaaeqa aOGaaGypaiaahofacaaIUaaaaa@694B@

С экспериментальной точки зрения принятая гипотеза представляется вполне естественной: можно осуществить деформирование тестового образца таким образом, чтобы инфинитезимальная окрестность любой его фиксированной точки оказалась свободной от напряжений. Разумеется, для каждой точки нужно подобрать свою деформацию.

Гипотеза локальной разгрузки иллюстрируется на рис. 3 для некоторых трех точек X 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwdaWgaaWcbaGaaGymaaqabaaaaa@4459@ , X 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwdaWgaaWcbaGaaGOmaaqabaaaaa@445A@ и X 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwdaWgaaWcbaGaaG4maaqabaaaaa@445B@ из формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ . Предполагается, что тело является трехмерным многообразием. Ячейки каждого региона изображают представительные объемы; более темные соответствуют представительным объемам в натуральном состоянии. Отображения ϰ ( X i ) :B S ( X i ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+daahaaWcbeqaaiaaiIcacqWFxepwdaWgaaqaaiaadM gaaeqaaiaaiMcaaaGccaaI6aGaaGPaVprr1ngBPrMrYf2A0vNCaeXb fv3ySLgzGyKCHTgD1jhaiyaacqGFbaVqcqGHsgIRcqWFse=udaahaa WcbeqaaiaaiIcacqWFxepwdaWgaaqaaiaadMgaaeqaaiaaiMcaaaaa aa@5FA5@ , и γ ( X i ) : S R S ( X i ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaWbaaSqabeaacaaIOaWefv3ySLgznfgD Ofdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwdaWgaaqaaiaadM gaaeqaaiaaiMcaaaGccaaI6aGaaGPaVlab=jr8tnaaBaaaleaacaWG sbaabeaakiabgkziUkab=jr8tnaaCaaaleqabaGaaGikaiab=Dr8yn aaBaaabaGaamyAaaqabaGaaGykaaaaaaa@5549@ , i=1,2,3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyAaiaai2dacaaIXaGaaGilaiaaykW7caaIYaGa aGilaiaaykW7caaIZaaaaa@406E@ , являются, соответственно, конфигурациями и деформациями.

 

Рис. 2.1. Семейство локально-натуральных форм

Fig. 2.1. Family of locally natural shapes

 

Для каждой точки X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcqGHiiIZcqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@47D4@ деформация γ (X) : S R S (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaWbaaSqabeaacaaIOaWefv3ySLgznfgD Ofdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwcaaIPaaaaOGaaG OoaiaaykW7cqWFse=udaWgaaWcbaGaamOuaaqabaGccqGHsgIRcqWF se=udaahaaWcbeqaaiaaiIcacqWFxepwcaaIPaaaaaaa@532B@ определяет касательное отображение F (X) =T γ (X) :T S R T S (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCOramaaCaaaleqabaGaaGikamrr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae83fXJLaaGykaaaakiaai2 dacaWGubGaeq4SdC2aaWbaaSqabeaacaaIOaGae83fXJLaaGykaaaa kiaaiQdacaaMc8Uaamivaiab=jr8tnaaBaaaleaacaWGsbaabeaaki abgkziUkaadsfacqWFse=udaahaaWcbeqaaiaaiIcacqWFxepwcaaI Paaaaaaa@5ACD@ . Если Y S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFyeFwcqGHiiIZcqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@47D6@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ некоторая точка, которая может не совпадать с X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwaaa@4372@ , то обратимое линейное отображение

F (X) | Y = T Y γ (X) Hom( T Y S R ; T γ (X) (Y) S (X) ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCOramaaCaaaleqabaGaaGikamrr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae83fXJLaaGykaaaakiaaiY hadaWgaaWcbaGae8hgXNfabeaakiaai2dacaWGubWaaSbaaSqaaiab =Hr8zbqabaGccqaHZoWzdaahaaWcbeqaaiaaiIcacqWFxepwcaaIPa aaaOGaeyicI4Saaeisaiaab+gacaqGTbGaaGikaiaadsfadaWgaaWc baGae8hgXNfabeaakiab=jr8tnaaBaaaleaacaWGsbaabeaakiaaiU dacaaMc8UaamivamaaBaaaleaacqaHZoWzdaahaaqabeaacaaIOaGa e83fXJLaaGykaaaacaaIOaGae8hgXNLaaGykaaqabaGccqWFse=uda ahaaWcbeqaaiaaiIcacqWFxepwcaaIPaaaaOGaaGykaaaa@6E73@

преобразует представительный объем, окружающий эту точку, в некоторое состояние. Заметим, что последнее не обязательно является натуральным. Вместе с тем, если Y=X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFyeFwcaaI9aGae83fXJfaaa@4620@ , то отображение F (X) | Y=X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCOramaaCaaaleqabaGaaGikamrr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae83fXJLaaGykaaaakiaaiY hadaWgaaWcbaGae8hgXNLaaGypaiab=Dr8ybqabaaaaa@4BA2@ преобразует представительный объем T X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccqWFse=udaWgaaWcba GaamOuaaqabaaaaa@475F@ в соответствующий объем, содержащийся в форме S (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaahaaWcbeqaaiaaiIcacqWFxepwcaaIPaaaaaaa@46DF@ , который на этот раз будет в натуральном состоянии.

Поле локальных деформаций.

Предположим, что тело B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ материально единообразно, то есть состоит из представительных объемов с одинаковыми физическими свойствами [13]. Последнее означает, что, будучи извлеченным из состава тела и приведенным в натуральное состояние, каждый такой представительный объем даст один и тот же отклик на одну и ту же деформацию. Общий для всех образ представительного объема в натуральном состоянии, следуя [18], назовем архетипом.

Для математической формализации архетипа ассоциируем с ним фиксированное семейство ( c A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahogadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EB8@ попарно ортогональных векторов единичной длины. Здесь n=dimB MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2daciGGKbGaaiyAaiaac2gatuuDJXwA KzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbacfaGae8xaWleaaa@48FE@ . С физической точки зрения элементы этого семейства определяют кристаллографические направления решетки идеального кристалла. Тогда прообраз семейства ( c A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahogadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EB8@ в произвольно выделенном представительном объеме, находящемся в составе наблюдаемой формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ , будет некоторым семейством ( z A | X ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahQhadaWgaaWcbaGaamyqaaqabaGccaaI 8bWaaSbaaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLb acfaGae83fXJfabeaakiaaiMcadaqhaaWcbaGaamyqaiaai2dacaaI XaaabaGaamOBaaaaaaa@4B7A@ , определяющим направления искаженной решетки кристалла. Здесь X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwaaa@4372@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ метка рассматриваемого представительного объема.

Преобразование ( z A | X ) A=1 n ( c A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahQhadaWgaaWcbaGaamyqaaqabaGccaaI 8bWaaSbaaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLb acfaGae83fXJfabeaakiaaiMcadaqhaaWcbaGaamyqaiaai2dacaaI XaaabaGaamOBaaaakiablAAiHjaaiIcacaWHJbWaaSbaaSqaaiaadg eaaeqaaOGaaGykamaaDaaaleaacaWGbbGaaGypaiaaigdaaeaacaWG Ubaaaaaa@53F2@ определяет деформацию представительного объема, переводящую его из текущего состояния в натуральное. Поскольку каждое из семейств ( c A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahogadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EB8@ , ( z A | X ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahQhadaWgaaWcbaGaamyqaaqabaGccaaI 8bWaaSbaaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLb acfaGae83fXJfabeaakiaaiMcadaqhaaWcbaGaamyqaiaai2dacaaI XaaabaGaamOBaaaaaaa@4B7A@ находится в векторном пространстве V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFveVvaaa@436E@ , то эту деформацию можно распространить на однозначно определенное линейное отображение H X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaaaaa@446F@ , переводящее линейную оболочку, порождаемую репером ( z A | X ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahQhadaWgaaWcbaGaamyqaaqabaGccaaI 8bWaaSbaaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLb acfaGae83fXJfabeaakiaaiMcadaqhaaWcbaGaamyqaiaai2dacaaI XaaabaGaamOBaaaaaaa@4B7A@ , в линейную оболочку, порождаемую репером ( c A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahogadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EB8@ . Первая линейная оболочка есть всего лишь касательное пространство T X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccqWFse=udaWgaaWcba GaamOuaaqabaaaaa@475F@ . Для второй линейной оболочки будем использовать обозначение U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFueFvaaa@436C@ . Если через g U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4zamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=rr8vbqabaaaaa@4488@ обозначить сужение евклидовой метрики g MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4zaaaa@38F3@ на U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFueFvaaa@436C@ , то окончательно под архетипом будем понимать евклидово векторное пространство (U, g U ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8hfXxLaaGilaiaaykW7caWHNbWaaSbaaSqaaiab=r r8vbqabaGccaaIPaaaaa@4A17@ , где

UVè g U :=g | U . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFueFvcqGHckcZcqWFveVvcaaMf8Uaaei6aiaaywW7caWHNb WaaSbaaSqaaiab=rr8vbqabaGccaaI6aGaaGypaiaahEgacaaI8bWa aSbaaSqaaiab=rr8vbqabaGccaaIUaaaaa@5523@

Соответствующее линейное отображение H X Hom( T X S R ;U) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccqGHiiIZcaqGibGaae 4Baiaab2gacaaIOaGaamivamaaBaaaleaacqWFxepwaeqaaOGae8Ne Xp1aaSbaaSqaaiaadkfaaeqaaOGaaG4oaiaaykW7cqWFueFvcaaIPa aaaa@541A@ , в свою очередь, будем называть локальной деформацией.

Гипотезу локальной разгрузки в рамках свойства материального единообразия следует дополнить предположением, что для каждой точки X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcqGHiiIZcqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@47D4@ евклидово пространство ( T γ (X) (X) S (X) ,g | S (X) ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadsfadaWgaaWcbaGaeq4SdC2aaWbaaeqa baGaaGikamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfa Gae83fXJLaaGykaaaacaaIOaGae83fXJLaaGykaaqabaGccqWFse=u daahaaWcbeqaaiaaiIcacqWFxepwcaaIPaaaaOGaaGilaiaaykW7ca WHNbGaaGiFamaaBaaaleaacqWFse=udaahaaqabeaacaaIOaGae83f XJLaaGykaaaaaeqaaOGaaGykaaaa@5B6E@ канонически изометрично[5] пространству (U, g U ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8hfXxLaaGilaiaaykW7caWHNbWaaSbaaSqaaiab=r r8vbqabaGccaaIPaaaaa@4A17@ . Производя, в соответствии с этим предположением, для каждой точки X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcqGHiiIZcqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@47D4@ отождествление T γ (X) (X) S (X) U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaacqaHZoWzdaahaaqabeaacaaI OaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxe pwcaaIPaaaaiaaiIcacqWFxepwcaaIPaaabeaakiab=jr8tnaaCaaa leqabaGaaGikaiab=Dr8yjaaiMcaaaGccqGHHjIUcqWFueFvaaa@53FD@ , приходим к линейному отображению

H X := F (X) | Y=X Hom( T X S R ;U), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccaaI6aGaaGypaiaahA eadaahaaWcbeqaaiaaiIcacqWFxepwcaaIPaaaaOGaaGiFamaaBaaa leaacqWFyeFwcaaI9aGae83fXJfabeaakiabgIGiolaabIeacaqGVb GaaeyBaiaaiIcacaWGubWaaSbaaSqaaiab=Dr8ybqabaGccqWFse=u daWgaaWcbaGaamOuaaqabaGccaaI7aGaaGPaVlab=rr8vjaaiMcaca aISaaaaa@607A@ (2)

которое представляет локальную деформацию. По построению, обратимое линейное отображение H X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaaaaa@446F@ преобразует представительный объем, окружающий точку X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwaaa@4372@ , в натуральное состояние.

В свою очередь, по отображениям H X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaaaaa@446F@ синтезируем глобальное поле следующим образом. Совместно с касательным расслоением T S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8NeXp1aaSbaaSqaaiaadkfaaeqaaaaa@4544@ определим тривиальное векторное расслоение над S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ , типовым слоем которого будет U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFueFvaaa@436C@ , и обозначим это расслоение символом T( S R ,U) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivaiaaiIcatuuDJXwAK1uy0HwmaeHbfv3ySLgz G0uy0Hgip5wzaGqbaiab=jr8tnaaBaaaleaacaWGsbaabeaakiaaiY cacaaMc8Uae8hfXxLaaGykaaaa@4AD3@ . Таким образом, по построению T( S R ,U):= S R ×U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivaiaaiIcatuuDJXwAK1uy0HwmaeHbfv3ySLgz G0uy0Hgip5wzaGqbaiab=jr8tnaaBaaaleaacaWGsbaabeaakiaaiY cacaaMc8Uae8hfXxLaaGykaiaaiQdacaaI9aGae8NeXp1aaSbaaSqa aiaadkfaaeqaaOGaey41aqRae8hfXxfaaa@533C@ . Теперь определим новое отображение:

H:T S R T( S R ,U),H(X,v):=(X, H X v). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaiaaiQdacaaMc8Uaamivamrr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae8NeXp1aaSbaaSqaaiaadk faaeqaaOGaeyOKH4QaamivaiaaiIcacqWFse=udaWgaaWcbaGaamOu aaqabaGccaaISaGaaGPaVlab=rr8vjaaiMcacaaISaGaaGzbVlaahI eacaaIOaGae83fXJLaaGilaiaaykW7caWH2bGaaGykaiaaiQdacaaI 9aGaaGikaiab=Dr8yjaaiYcacaaMc8UaaCisamaaBaaaleaacqWFxe pwaeqaaOGaaCODaiaaiMcacaaIUaaaaa@68F9@ (3)

Предполагается, что полученное отображение (3) гладко как отображение между многообразиями. В таком случае, согласно положениям общей теории векторных расслоений, оно является гомоморфизмом векторных расслоений [39, с.~261] над S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ . Соответствующее отображение между базами этих расслоений является тождественным отображением. Будем называть отображение H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ полем локальных деформаций [26].

Если семейство { γ (X) } X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4Eaiabeo7aNnaaCaaaleqabaGaaGikamrr1ngB PrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae83fXJLaaGykaa aakiaai2hadaWgaaWcbaGae83fXJLaeyicI4Sae8NeXp1aaSbaaeaa caWGsbaabeaaaeqaaaaa@4F29@ можно выбрать таким образом, что оно состоит из одного элемента γ 0 : S R S 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaSbaaSqaaiaaicdaaeqaaOGaaGOoaiaa ykW7tuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaiab=j r8tnaaBaaaleaacaWGsbaabeaakiabgkziUkab=jr8tnaaBaaaleaa caaIWaaabeaaaaa@4E09@ , то образ γ 0 ( S R ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaSbaaSqaaiaaicdaaeqaaOGaaGikamrr 1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae8NeXp1aaS baaSqaaiaadkfaaeqaaOGaaGykaaaa@4871@ является глобальной натуральной формой тела B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ . В этом случае отображение H X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaaaaa@446F@ может быть отождествлено с касательным отображением T X γ 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccqaHZoWzdaWgaaWcba GaaGimaaqabaaaaa@470E@ для каждой точки X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcqGHiiIZcqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@47D4@ и локальные деформации называются совместными. В противном случае, когда семейство деформаций не сводится к одному элементу, локальные деформации называются несовместными, то есть линейное отображение H X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaaaaa@446F@ не может быть выражено как касательное отображение к некоторой деформации, единой для всех X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwaaa@4372@ [13].

Замечание 2. Тот факт, что расслоение T( S R ,U) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivaiaaiIcatuuDJXwAK1uy0HwmaeHbfv3ySLgz G0uy0Hgip5wzaGqbaiab=jr8tnaaBaaaleaacaWGsbaabeaakiaaiY cacaaMc8Uae8hfXxLaaGykaaaa@4AD3@ было выбрано тривиальным (т. е. тотальное пространство MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbcKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C75@ цилиндр), а отображение H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ было определено формулой (3), нисколько не умаляет общность дальнейших рассуждений, поскольку в них фигурируют лишь локальные деформации в точке.

Представления поля локальных деформаций.

Поле локальных деформаций H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ имеет ряд альтернативных представлений. Действительно, во-первых, отображение H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ , будучи синтезированным по линейным отображениям H X Hom( T X S R ;U) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccqGHiiIZcaqGibGaae 4Baiaab2gacaaIOaGaamivamaaBaaaleaacqWFxepwaeqaaOGae8Ne Xp1aaSbaaSqaaiaadkfaaeqaaOGaaG4oaiaaykW7cqWFueFvcaaIPa aaaa@541A@ , индуцирует семейство { H X } X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4EaiaahIeadaWgaaWcbaWefv3ySLgznfgDOfda ryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwaeqaaOGaaGyFamaaBa aaleaacqWFxepwcqGHiiIZcqWFse=udaWgaaqaaiaadkfaaeqaaaqa baaaaa@4CED@ линейных трансформаций. Предположим, в частности, что n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ . Тогда U=V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFueFvcaaI9aGae8xfXBfaaa@4614@ и T X S R V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccqWFse=udaWgaaWcba GaamOuaaqabaGccqGHfjcqcqWFveVvaaa@4A7D@ (канонический изоморфизм). По этой причине H X End(V) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccqGHiiIZcaqGfbGaae OBaiaabsgacaaIOaGae8xfXBLaaGykaaaa@4BE3@ , и семейство { H X } X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4EaiaahIeadaWgaaWcbaWefv3ySLgznfgDOfda ryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwaeqaaOGaaGyFamaaBa aaleaacqWFxepwcqGHiiIZcqWFse=udaWgaaqaaiaadkfaaeqaaaqa baaaaa@4CED@ , в свою очередь, сводится к гладкому полю H: S R End(V) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaiaaiQdacaaMc8+efv3ySLgznfgDOfdaryqr 1ngBPrginfgDObYtUvgaiuaacqWFse=udaWgaaWcbaGaamOuaaqaba GccqGHsgIRcaqGfbGaaeOBaiaabsgacaaIOaGae8xfXBLaaGykaaaa @4F68@ линейных преобразований. В таком виде H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ используется в классической теории дефектов [52].

Пусть теперь ( X A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae83fXJ1aaWbaaSqabeaacaWGbbaaaOGaaGykamaaDa aaleaacaWGbbGaaGypaiaaigdaaeaacaWGUbaaaaaa@493C@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ локальные координаты на многообразии S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ и пусть ( e A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahwgadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EBA@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ некоторый базис U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFueFvaaa@436C@ . Тогда линейное отображение H X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaaaaa@446F@ имеет следующее диадное представление:

H X = H B A e A d X B | X , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccaaI9aGaamisamaaDa aaleaacaWGcbaabaGaamyqaaaakiaahwgadaWgaaWcbaGaamyqaaqa baGccqGHxkcXcaWGKbGae83fXJ1aaWbaaSqabeaacaWGcbaaaOGaaG iFamaaBaaaleaacqWFxepwaeqaaOGaaGilaaaa@5367@ (4)

где (d X A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadsgatuuDJXwAK1uy0HwmaeHbfv3ySLgz G0uy0Hgip5wzaGqbaiab=Dr8ynaaCaaaleqabaGaamyqaaaakiaaiM cadaqhaaWcbaGaamyqaiaai2dacaaIXaaabaGaamOBaaaaaaa@4A25@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ поле координатных кореперов. Последнее представление может быть записано в следующей краткой форме:

H X = e A H X A , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccaaI9aGaaCyzamaaBa aaleaacaWGbbaabeaakiabgEPielaadIeadaqhaaWcbaGae83fXJfa baGaamyqaaaakiaaiYcaaaa@4D98@

в которой

H X A = H B A d X B | X ,A=1,,n, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamisamaaDaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqaaiaadgeaaaGccaaI9aGaam isamaaDaaaleaacaWGcbaabaGaamyqaaaakiaadsgacqWFxepwdaah aaWcbeqaaiaadkeaaaGccaaI8bWaaSbaaSqaaiab=Dr8ybqabaGcca aISaGaaGzbVlaadgeacaaI9aGaaGymaiaaiYcacaaMc8UaeSOjGSKa aGilaiaaykW7caWGUbGaaGilaaaa@5B5A@

MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ совокупность 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaaaa@38BE@ -форм. Это означает, что вместо отображения (3) можно рассматривать n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ гладких полей 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaaaa@38BE@ -форм H A MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamisamaaCaaaleqabaGaamyqaaaaaaa@39C3@ , A=1,,n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyqaiaai2dacaaIXaGaaGilaiaaykW7cqWIMaYs caaISaGaaGPaVlaad6gaaaa@40E2@ . Если базис ( e A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahwgadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EBA@ фиксирован (например, соответствует положению кристаллографических осей в натуральном состоянии), то эти поля позволяют однозначно восстановить отображение H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ .

Замечание об условиях совместности.

С использованием дифференциальных форм совместность локальных деформаций сводится к следующему условию: должны существовать n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ гладких скалярных функций γ A : S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaWbaaSqabeaacaWGbbaaaOGaaGOoaiaa ykW7tuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaiab=j r8tnaaBaaaleaacaWGsbaabeaakiabgkziUorr1ngBPrwtHrhAYaqe huuDJXwAKbstHrhAGq1DVbacgaGae4xhHifaaa@560E@ , A=1,,n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyqaiaai2dacaaIXaGaaGilaiaaykW7cqWIMaYs caaISaGaaGPaVlaad6gaaaa@40E2@ , таких, что

H A =d γ A ,A=1,,n. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamisamaaCaaaleqabaGaamyqaaaakiaai2dacaWG KbGaeq4SdC2aaWbaaSqabeaacaWGbbaaaOGaaGilaiaaywW7caWGbb GaaGypaiaaigdacaaISaGaaGPaVlablAciljaaiYcacaaMc8UaamOB aiaai6caaaa@49FC@

Здесь d MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamizaaaa@38EC@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ внешний дифференциал [39, с.~365]. Таким образом, необходимым (а в случае, когда форма S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ односвязна) и достаточным условием совместности деформаций являются равенства

d H A =0,A=1,,n, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamizaiaadIeadaahaaWcbeqaaiaadgeaaaGccaaI 9aGaaGimaiaaiYcacaaMf8Uaamyqaiaai2dacaaIXaGaaGilaiaayk W7cqWIMaYscaaISaGaaGPaVlaad6gacaaISaaaaa@4810@ (5)

которым, в случае n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ , соответствует классическое условие

rotH=0. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOCaiaab+gacaqG0bGaaGjcVlaahIeacaaI9aGa aCimaiaai6caaaa@3F7B@

В общем случае, когда отсчетная форма неодносвязна (например, является полым шаром), достаточные условия могут быть также сформулированы, но равенств (5) уже недостаточно. Необходимо привлекать методы алгебраической топологии, что сделано, например, в работе [53].

2.2.Материальная метрика и связность

Стирание евклидовой геометрии из формы.

Пусть поле (3) локальных деформаций известно. Тогда вместо использования континуального семейства { S (X) } X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4Eamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8NeXp1aaWbaaSqabeaacaaIOaGae83fXJLaaGykaa aakiaai2hadaWgaaWcbaGae83fXJLaeyicI4Sae8NeXp1aaSbaaeaa caWGsbaabeaaaeqaaaaa@4F5D@ локально единообразных форм можно синтезировать новую глобально единообразную форму с неевклидовой геометрией. Для этого будем исходить из формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ и первым шагом <<сотрем>> с нее геометрию. Полученное многообразие будет обозначаться через M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ , то есть M R =( S R , T E | S R , D E | S R ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaakiaai2dacaaI OaGaam4uamaaBaaaleaacaWGsbaabeaakiaaiYcacaaMc8+efv3ySL gznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFtepvdaWgaaWc baGaamyraaqabaGccaaI8bWaaSbaaSqaaiaadofadaWgaaqaaiaadk faaeqaaaqabaGccaaISaGaaGPaVlab=nq8enaaBaaaleaacaWGfbaa beaakiaaiYhadaWgaaWcbaGaam4uamaaBaaabaGaamOuaaqabaaabe aakiaaiMcaaaa@57B1@ . В таком случае точки формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ становятся всего лишь точками многообразия и по этой причине, чтобы отличать многообразие M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ от геометрического пространства S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ , будем обозначать точки M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ символами вида X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ , Y MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamywaaaa@38E1@ и Z MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOwaaaa@38E2@ . Отличие структуры S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ от M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ можно подчеркнуть в явном виде, определив отображение (каноническую инъекцию) ι M R : M R E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyUdK2aaSbaaSqaaiaad2eadaWgaaqaaiaadkfa aeqaaaqabaGccaaI6aGaaGPaVlaad2eadaWgaaWcbaGaamOuaaqaba Wefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaakiab=nAi Wkab=btifbaa@4C7E@ , XX MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaiablAAiHnrr1ngBPrwtHrhAXaqeguuDJXwA KbstHrhAG8KBLbacfaGae83fXJfaaa@4608@ . Здесь X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ обозначает точку из M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ , а символ X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwaaa@4372@ , обозначающий элемент S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ , представляет точно такую же точку, но рассматриваемую в пространстве E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ .

После <<стирания>> геометрии с формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ касательные пространства к ней также изменяют свои атрибуты. Действительно, если T X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccqWFse=udaWgaaWcba GaamOuaaqabaaaaa@475F@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ касательное пространство к S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ , то оно автоматически снабжается скалярным произведением, индуцированным из евклидова векторного пространства V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFveVvaaa@436E@ , и, таким образом, рассматривается как подпространство последнего. Вместе с тем касательное пространство к M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ не имеет никакой дополнительной структуры. Чтобы подчеркнуть это, тензорные поля на M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ обозначаются как P MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiuaaaa@38D6@ , Q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyuaaaa@38D7@ , и R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaaaa@38D8@ . В частности, поле (3) обозначается символом H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaaaa@38CE@ . В явном виде отображение H:T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaiaaiQdacaaMc8Uaamivaiaad2eadaWgaaWc baGaamOuaaqabaGccqGHsgIRaaa@3FC2@ T( S R ,U) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOKH4QaamivaiaaiIcatuuDJXwAK1uy0HwmaeHb fv3ySLgzG0uy0Hgip5wzaGqbaiab=jr8tnaaBaaaleaacaWGsbaabe aakiaaiYcacaaMc8Uae8hfXxLaaGykaaaa@4CC0@ определяется равенством

H X = H X p X , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaBaaaleaacaWGybaabeaakiaai2dacaWH ibWaaSbaaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLb acfaGae83fXJfabeaakiablIHiVjaadchadaWgaaWcbaGaamiwaaqa baGccaaISaaaaa@4B16@

где X= ι M R (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcaaI9aGaeqyUdK2aaSbaaSqaaiaad2eadaWgaaqaai aadkfaaeqaaaqabaGccaaIOaGaamiwaiaaiMcaaaa@4A2A@ , а p X : T X M R V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiCamaaBaaaleaacaWGybaabeaakiaaiQdacaaM c8UaamivamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0H gip5wzaGqbaiab=Dr8ybqabaGccaWGnbWaaSbaaSqaaiaadkfaaeqa aOGaeyOKH4Qae8xfXBfaaa@4E85@ изоморфизм на свой образ, такой, что

u= u A A | X u= p X (u)= u A e A | X . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyDaiaai2dacaWG1bWaaWbaaSqabeaacaWGbbaa aOGaeyOaIy7aaSbaaSqaaiaadgeaaeqaaOGaaGiFamaaBaaaleaaca WGybaabeaakiablAAiHjaahwhacaaI9aGaamiCamaaBaaaleaacaWG ybaabeaakiaaiIcacaqG1bGaaGykaiaai2dacaWG1bWaaWbaaSqabe aacaWGbbaaaOGaaCyzamaaBaaaleaacaWGbbaabeaakiaaiYhadaWg aaWcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacq WFxepwaeqaaOGaaGOlaaaa@5A22@

Таким образом, H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaaaa@38CE@ играет точно такую же роль, как его евклидов аналог, поле локальных деформаций H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ , но определен на многообразии, очищенном от какой-либо геометрии.

Замечание 3. С общетеоретической точки зрения нет необходимости извлекать подлежащее многообразие из-под некоторой формы, поскольку общее для всех форм многообразие B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ было определено заранее. Вместе с тем, особенно в частных задачах, можно явно описать лишь формы, поскольку только они наблюдаемы. Процедура <<стирания>> геометрии с формы есть фактически способ восстановить многообразие B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ и затем построить на нем геометрию. Таким образом, в действительности речь по-прежнему идет о теле.

Материальная метрика.

Все, что имеется на данный момент, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ поле H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaaaa@38CE@ локальных деформаций. Оно трансформирует каждый представительный объем в натуральное состояние, где измерения проводятся посредством метрического тензора g U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4zamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=rr8vbqabaaaaa@4488@ . Построим обратный образ этой метрики на многообразие M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ и, таким образом, снабдим инфинитезимальные волокна в M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ мерами, которые они имеют в натуральном состоянии. В явном виде определим тензорное поле GSec T M R T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raiabgIGiolaabofacaqGLbGaae4yamaabmaa baGaamivamaaCaaaleqabaGaey4fIOcaaOGaamytamaaBaaaleaaca WGsbaabeaakiabgEPielaadsfadaahaaWcbeqaaiabgEHiQaaakiaa d2eadaWgaaWcbaGaamOuaaqabaaakiaawIcacaGLPaaaaaa@4843@ равенством [13]:

X M R u,v T X M R : G X (u,v):= g U H X [u], H X [v] . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyiaIiIaamiwaiabgIGiolaad2eadaWgaaWcbaGa amOuaaqabaGccaaMi8UaeyiaIiIaaeyDaiaaiYcacaaMc8UaaeODai abgIGiolaadsfadaWgaaWcbaGaamiwaaqabaGccaWGnbWaaSbaaSqa aiaadkfaaeqaaOGaaGOoaiaaysW7caqGhbWaaSbaaSqaaiaadIfaae qaaOGaaGikaiaabwhacaaISaGaaGPaVlaabAhacaaIPaGaaGOoaiaa i2dacaWHNbWaaSbaaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHr hAG8KBLbacfaGae8hfXxfabeaakmaabmaabaGaaeisamaaBaaaleaa caWGybaabeaakiaaiUfacaqG1bGaaGyxaiaaiYcacaaMc8Uaaeisam aaBaaaleaacaWGybaabeaakiaaiUfacaqG2bGaaGyxaaGaayjkaiaa wMcaaiaai6caaaa@6E9B@ (6)

Будем называть риманову метрику G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ материальной метрикой. В координатном репере, компоненты G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ имеют вид G AB = g CD H A C H B D MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaBaaaleaacaWGbbGaamOqaaqabaGccaaI 9aGaam4zamaaBaaaleaacaWGdbGaamiraaqabaGccaqGibWaa0baaS qaaiaadgeaaeaacaWGdbaaaOGaaeisamaaDaaaleaacaWGcbaabaGa amiraaaaaaa@4322@ , где H B A MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaDaaaleaacaWGcbaabaGaamyqaaaaaaa@3A88@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ коэффициенты разложения (4), имеющего в нынешних обозначениях вид H X = H B A e A d X B | X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaBaaaleaacaWGybaabeaakiaai2dacaqG ibWaa0baaSqaaiaadkeaaeaacaWGbbaaaOGaaCyzamaaBaaaleaaca WGbbaabeaakiabgEPielaadsgacaWGybWaaWbaaSqabeaacaWGcbaa aOGaaGiFamaaBaaaleaacaWGybaabeaaaaa@45FD@ , а g CD = e C e D MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4zamaaBaaaleaacaWGdbGaamiraaqabaGccaaI 9aGaaCyzamaaBaaaleaacaWGdbaabeaakiaaygW7cqGHflY1caaMb8 UaaCyzamaaBaaaleaacaWGebaabeaaaaa@44AA@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ компоненты метрики g U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4zamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=rr8vbqabaaaaa@4488@ .

К определению (6) материальной метрики можно подойти иным способом [23]. Рассмотрим семейство { γ (X) } X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4Eaiabeo7aNnaaCaaaleqabaGaaGikamrr1ngB PrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae83fXJLaaGykaa aakiaai2hadaWgaaWcbaGae83fXJLaeyicI4Sae8NeXp1aaSbaaeaa caWGsbaabeaaaeqaaaaa@4F29@ . По нему можно определить новое семейство { γ (X) } X M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4Eaiabeo7aNnaaCaaaleqabaGaaGikaiaadIfa caaIPaaaaOGaaGyFamaaBaaaleaacaWGybGaeyicI4SaamytamaaBa aabaGaamOuaaqabaaabeaaaaa@4286@ , где отображения γ (X) : M R E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaWbaaSqabeaacaaIOaGaamiwaiaaiMca aaGccaaI6aGaaGPaVlaad2eadaWgaaWcbaGaamOuaaqabaGccqGHsg IRtuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaiab=bti fbaa@4CE3@ связаны с деформациями γ (X) : S R E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaWbaaSqabeaacaaIOaWefv3ySLgznfgD Ofdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwcaaIPaaaaOGaaG OoaiaaykW7cqWFse=udaWgaaWcbaGaamOuaaqabaGccqGHsgIRcqWF Wesraaa@4EF4@ соотношениями γ (X) = γ (X) ι M R ; S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaWbaaSqabeaacaaIOaGaamiwaiaaiMca aaGccaaI9aGaeq4SdC2aaWbaaSqabeaacaaIOaWefv3ySLgznfgDOf daryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwcaaIPaaaaOGaeSig I8MaeqyUdK2aaSbaaSqaaiaad2eadaWgaaqaaiaadkfaaeqaaiaaiU dacaaMc8Uae8NeXp1aaSbaaeaacaWGsbaabeaaaeqaaaaa@559E@ . Здесь ι M R ; S R : M R S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyUdK2aaSbaaSqaaiaad2eadaWgaaqaaiaadkfa aeqaaiaaiUdacaaMc8+efv3ySLgznfgDOfdaryqr1ngBPrginfgDOb YtUvgaiuaacqWFse=udaWgaaqaaiaadkfaaeqaaaqabaGccaaI6aGa aGPaVlaad2eadaWgaaWcbaGaamOuaaqabaGccqWFJgcScqWFse=uda WgaaWcbaGaamOuaaqabaaaaa@5364@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ каноническая инъекция, которая отображает точку X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ в ту же самую точку X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwaaa@4372@ , но в пространстве с геометрией.

Каждое отображение γ (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaWbaaSqabeaacaaIOaGaamiwaiaaiMca aaaaaa@3C19@ определяет метрику G (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaCaaaleqabaGaaGikaiaadIfacaaIPaaa aaaa@3B3C@ как обратный образ, G (X) :=( γ (X) ) g MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaCaaaleqabaGaaGikaiaadIfacaaIPaaa aOGaaGOoaiaai2dacaaIOaGaeq4SdC2aaWbaaSqabeaacaaIOaGaam iwaiaaiMcaaaGccaaIPaWaaWbaaSqabeaacqGHxiIkaaGccaWHNbaa aa@446C@ :

Y M R u,v T Y M R : G (X) | Y (u,v):= T Y γ (X) [u] T Y γ (X) [v]. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyiaIiIaamywaiabgIGiolaad2eadaWgaaWcbaGa amOuaaqabaGccaaMi8UaeyiaIiIaaeyDaiaaiYcacaaMc8UaaeODai abgIGiolaadsfadaWgaaWcbaGaamywaaqabaGccaWGnbWaaSbaaSqa aiaadkfaaeqaaOGaaGOoaiaaysW7caqGhbWaaWbaaSqabeaacaaIOa GaamiwaiaaiMcaaaGccaaI8bWaaSbaaSqaaiaadMfaaeqaaOGaaGik aiaabwhacaaISaGaaGPaVlaabAhacaaIPaGaaGOoaiaai2dacaWGub WaaSbaaSqaaiaadMfaaeqaaOGaeq4SdC2aaWbaaSqabeaacaaIOaGa amiwaiaaiMcaaaGccaaIBbGaaeyDaiaai2facqGHflY1caWGubWaaS baaSqaaiaadMfaaeqaaOGaeq4SdC2aaWbaaSqabeaacaaIOaGaamiw aiaaiMcaaaGccaaIBbGaaeODaiaai2facaaIUaaaaa@6C6C@

Тензорное поле G (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaCaaaleqabaGaaGikaiaadIfacaaIPaaa aaaa@3B3C@ задает метрическую структуру на M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ , индуцированную из физического пространства. Теперь пусть G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ тензорное поле второго ранга, такое, что G:X G (X) | X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raiaaiQdacaaMc8UaamiwaiablAAiHjaabEea daahaaWcbeqaaiaaiIcacaWGybGaaGykaaaakiaaiYhadaWgaaWcba Gaamiwaaqabaaaaa@4304@ . Это поле и есть в точности (6).

Таким образом, материальная метрика может быть синтезирована по семейству { G (X) } X M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4EaiaabEeadaahaaWcbeqaaiaaiIcacaWGybGa aGykaaaakiaai2hadaWgaaWcbaGaamiwaiabgIGiolaad2eadaWgaa qaaiaadkfaaeqaaaqabaaaaa@41A9@ . Образно говоря, метрическая структура над M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ может быть получена путем перебора элементов семейства { S (X) } X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4Eamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8NeXp1aaWbaaSqabeaacaaIOaGae83fXJLaaGykaa aakiaai2hadaWgaaWcbaGae83fXJLaeyicI4Sae8NeXp1aaSbaaeaa caWGsbaabeaaaeqaaaaa@4F5D@ и извлечения соответствующей метрической структуры (скалярного произведения) из каждого. Иными словами, различные метрические структуры синтезируются в одну глобальную, неевклидову, структуру.

Материальная связность.

Синтезировав материальную метрику, приходим к структуре ( M R ,G) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaad2eadaWgaaWcbaGaamOuaaqabaGccaaI SaGaaGPaVlaabEeacaaIPaaaaa@3E52@ риманова многообразия, которое почти является искомой глобально единообразной формой тела. Чтобы завершить построение этой формы, осталось добавить еще одну составляющую MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ аффинную связность. Действительно, чтобы записать уравнения баланса на M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ , равно как и отразить конкретную физическую природу несовместности, необходимо установить некоторое правило параллельного перенесения, которое как раз и определяется аффинной связностью. По определению, аффинная связность MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ это отображение :Vec( M R )×Vec( M R )Vec( M R ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIeTaaGOoaiaaykW7caqGwbGaaeyzaiaaboga caaIOaGaamytamaaBaaaleaacaWGsbaabeaakiaaiMcacqGHxdaTca qGwbGaaeyzaiaabogacaaIOaGaamytamaaBaaaleaacaWGsbaabeaa kiaaiMcacqGHsgIRcaqGwbGaaeyzaiaabogacaaIOaGaamytamaaBa aaleaacaWGsbaabeaakiaaiMcaaaa@519D@ , (u,v) u v MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaabwhacaaISaGaaGPaVlaabAhacaaIPaGa eSOPHeMaey4bIe9aaSbaaSqaaiaabwhaaeqaaOGaaeODaaaa@4300@ , которое удовлетворяет следующим аксиомам [54, 55]:

u,v,wVec( M R ): u+v w= u w+ v w MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyiaIiIaaeyDaiaaiYcacaaMc8UaaeODaiaaiYca caaMc8Uaae4DaiabgIGiolaabAfacaqGLbGaae4yaiaaiIcacaWGnb WaaSbaaSqaaiaadkfaaeqaaOGaaGykaiaaiQdacaaMe8Uaey4bIe9a aSbaaSqaaiaabwhacqGHRaWkcaqG2baabeaakiaabEhacaaI9aGaey 4bIe9aaSbaaSqaaiaabwhaaeqaaOGaae4DaiabgUcaRiabgEGirpaa BaaaleaacaqG2baabeaakiaabEhaaaa@588F@ ;

u,vVec( M R )f C ( M R ): fu v=f u v MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyiaIiIaaeyDaiaaiYcacaaMc8UaaeODaiabgIGi olaabAfacaqGLbGaae4yaiaaiIcacaWGnbWaaSbaaSqaaiaadkfaae qaaOGaaGykaiaaykW7cqGHaiIicaWGMbGaeyicI4Saam4qamaaCaaa leqabaGaeyOhIukaaOGaaGikaiaad2eadaWgaaWcbaGaamOuaaqaba GccaaIPaGaaGOoaiaaysW7cqGHhis0daWgaaWcbaGaamOzaiaabwha aeqaaOGaaeODaiaai2dacaWGMbGaey4bIe9aaSbaaSqaaiaabwhaae qaaOGaaeODaaaa@5B3A@ ;

u,v,wVec( M R ): u (v+w)= u v+ u w MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyiaIiIaaeyDaiaaiYcacaaMc8UaaeODaiaaiYca caaMc8Uaae4DaiabgIGiolaabAfacaqGLbGaae4yaiaaiIcacaWGnb WaaSbaaSqaaiaadkfaaeqaaOGaaGykaiaaiQdacaaMe8Uaey4bIe9a aSbaaSqaaiaadwhaaeqaaOGaaGikaiaabAhacqGHRaWkcaqG3bGaaG ykaiaai2dacqGHhis0daWgaaWcbaGaaeyDaaqabaGccaqG2bGaey4k aSIaey4bIe9aaSbaaSqaaiaabwhaaeqaaOGaae4Daaaa@59F4@ ;

u,vVec( M R )f C ( M R ): u (fv)=f u v+(uf)v MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyiaIiIaaeyDaiaaiYcacaaMc8UaaeODaiabgIGi olaabAfacaqGLbGaae4yaiaaiIcacaWGnbWaaSbaaSqaaiaadkfaae qaaOGaaGykaiaaykW7cqGHaiIicaWGMbGaeyicI4Saam4qamaaCaaa leqabaGaeyOhIukaaOGaaGikaiaad2eadaWgaaWcbaGaamOuaaqaba GccaaIPaGaaGOoaiaaysW7cqGHhis0daWgaaWcbaGaaeyDaaqabaGc caaIOaGaamOzaiaabAhacaaIPaGaaGypaiaadAgacqGHhis0daWgaa WcbaGaaeyDaaqabaGccaqG2bGaey4kaSIaaGikaiaabwhacaWGMbGa aGykaiaabAhaaaa@61C2@ .

Здесь символ uf MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyDaiaadAgaaaa@39E6@ обозначает действие векторного поля u MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyDaaaa@38FB@ , рассматриваемого как дифференцирование, на скалярное поле f MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzaaaa@38EE@ .

Пусть ( e A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaabwgadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EB4@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ локальный репер касательного расслоения T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivaiaad2eadaWgaaWcbaGaamOuaaqabaaaaa@3AB1@ , то есть совокупность гладких векторных полей e A :UT M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyzamaaBaaaleaacaWGbbaabeaakiaaiQdacaaM c8UaamyvaiabgkziUkaadsfacaWGnbWaaSbaaSqaaiaadkfaaeqaaa aa@41AB@ , A=1,,n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyqaiaai2dacaaIXaGaaGilaiaaykW7cqWIMaYs caaISaGaaGPaVlaad6gaaaa@40E2@ , заданных на открытом множестве U M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyvaiabgkOimlaad2eadaWgaaWcbaGaamOuaaqa baaaaa@3CAE@ , такая, что в каждой точке XU MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaiabgIGiolaadwfaaaa@3B3E@ упорядоченный набор ( e A | X ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaabwgadaWgaaWcbaGaamyqaaqabaGccaaI 8bWaaSbaaSqaaiaadIfaaeqaaOGaaGykamaaDaaaleaacaWGbbGaaG ypaiaaigdaaeaacaWGUbaaaaaa@40CD@ является базисом касательного пространства T X M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaacaWGybaabeaakiaad2eadaWg aaWcbaGaamOuaaqabaaaaa@3BC4@ . Для любых A,B=1,,n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyqaiaaiYcacaaMc8UaamOqaiaai2dacaaIXaGa aGilaiaaykW7cqWIMaYscaaISaGaaGPaVlaad6gaaaa@43EA@ , ковариантная производная e A e B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaSbaaSqaaiaabwgadaWgaaqaaiaadgea aeqaaaqabaGccaqGLbWaaSbaaSqaaiaadkeaaeqaaaaa@3D69@ является векторным полем, заданным на U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyvaaaa@38DD@ . По этой причине ее тоже можно разложить по локальному реперу ( e A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaabwgadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EB4@ : e A e B = Γ A C B e C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaSbaaSqaaiaabwgadaWgaaqaaiaadgea aeqaaaqabaGccaqGLbWaaSbaaSqaaiaadkeaaeqaaOGaaGypaiabfo 5ahnaaDaaaleaacaWGbbaabaGaam4qaaaakmaaBaaaleaacaWGcbaa beaakiaabwgadaWgaaWcbaGaam4qaaqabaaaaa@4440@ . Коэффициенты разложения MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ n 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBamaaCaaaleqabaGaaG4maaaaaaa@39E0@ скалярных полей Γ A C B :U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaadgeaaeaacaWGdbaaaOWa aSbaaSqaaiaadkeaaeqaaOGaaGOoaiaaykW7caWGvbGaeyOKH46efv 3ySLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFDeIuaaa@4BFB@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ называются коэффициентами связности. Если они известны, то для произвольных векторных полей u= u A e A MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyDaiaai2dacaqG1bWaaWbaaSqabeaacaWGbbaa aOGaaeyzamaaBaaaleaacaWGbbaabeaaaaa@3D91@ и v= v B e B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeODaiaai2dacaqG2bWaaWbaaSqabeaacaWGcbaa aOGaaeyzamaaBaaaleaacaWGcbaabeaaaaa@3D95@ , используя аксиомы связности, можно получить выражение

u v= u A e A ( v C )+ v B Γ A C B e C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaSbaaSqaaiaabwhaaeqaaOGaaeODaiaa i2dacaqG1bWaaWbaaSqabeaacaWGbbaaaOWaaeWaaeaacaqGLbWaaS baaSqaaiaadgeaaeqaaOGaaGikaiaabAhadaahaaWcbeqaaiaadoea aaGccaaIPaGaey4kaSIaaeODamaaCaaaleqabaGaamOqaaaakiabfo 5ahnaaDaaaleaacaWGbbaabaGaam4qaaaakmaaBaaaleaacaWGcbaa beaaaOGaayjkaiaawMcaaiaabwgadaWgaaWcbaGaam4qaaqabaaaaa@4E15@

для ковариантной производной.

Для дальнейших рассуждений необходимо иметь общий закон преобразования коэффициентов связности. Пусть ( e A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaabwgadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EB4@ и ( ϑ A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabeg9aknaaCaaaleqabaGaamyqaaaakiaa iMcadaqhaaWcbaGaamyqaiaai2dacaaIXaaabaGaamOBaaaaaaa@3F75@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ локальный репер и дуальный к нему корепер, определенные на открытом множестве U M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyvaiabgkOimlaad2eadaWgaaWcbaGaamOuaaqa baaaaa@3CAE@ , которое одновременно является координатной областью некоторой карты. Предположим, что на том же множестве U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyvaaaa@38DD@ определены другие локальный репер ( e ˜ A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikamaaGaaabaGaaeyzaaGaay5adaWaaSbaaSqa aiaadgeaaeqaaOGaaGykamaaDaaaleaacaWGbbGaaGypaiaaigdaae aacaWGUbaaaaaa@3F76@ и корепер ( ϑ ˜ A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikamaaGaaabaGaeqy0dOeacaGLdmaadaahaaWc beqaaiaadgeaaaGccaaIPaWaa0baaSqaaiaadgeacaaI9aGaaGymaa qaaiaad6gaaaaaaa@4037@ , связанные с предыдущими репером и корепером гладким полем невырожденных n×n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiabgEna0kaad6gaaaa@3C00@ -матриц Ω=[ Ω B A ]:UGL(n;) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuyQdCLaaGypaiaaiUfacqqHPoWvdaqhaaWcbaGa amOqaaqaaiaadgeaaaGccaaIDbGaaGOoaiaaykW7caWGvbGaeyOKH4 Qaae4raiaabYeacaaIOaGaamOBaiaaiUdacaaMc8+efv3ySLgznfgD Ojdaryqr1ngBPrginfgDObcv39gaiuaacqWFDeIucaaIPaaaaa@5585@ :

e ˜ A = Ω A B e B , ϑ ˜ A = B A ϑ B , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacaqGLbaacaGLdmaadaWgaaWcbaGaamyq aaqabaGccaaI9aGaeuyQdC1aa0baaSqaaiaadgeaaeaacaWGcbaaaO GaaeyzamaaBaaaleaacaWGcbaabeaakiaaiYcacaaMf8+aaacaaeaa cqaHrpGsaiaawoWaamaaCaaaleqabaGaamyqaaaakiaai2darqqr1n gBPrgifHhDYfgaiuaacqWFNeIsdaqhaaWcbaGaamOqaaqaaiaadgea aaGccqaHrpGsdaahaaWcbeqaaiaadkeaaaGccaaISaaaaa@51EC@

где =[ B A ]= Ω 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaarqqr1ngBPrgifHhDYfgaiuaapaGae83jHOKaaGypaiaa iUfacqWFNeIsdaqhaaWcbaGaamOqaaqaaiaadgeaaaGccaaIDbGaaG ypaiabfM6axnaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@474E@ . Обозначим соответствующие коэффициенты связности через Γ A C B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaadgeaaeaacaWGdbaaaOWa aSbaaSqaaiaadkeaaeqaaaaa@3C23@ и Γ ˜ A C B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqqHtoWraiaawoWaamaaDaaaleaacaWG bbaabaGaam4qaaaakmaaBaaaleaacaWGcbaabeaaaaa@3CE5@ , то есть

e A e B = Γ A C B e C ,è e ˜ A e ˜ B = Γ ˜ A C B e ˜ C . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaSbaaSqaaiaabwgadaWgaaqaaiaadgea aeqaaaqabaGccaqGLbWaaSbaaSqaaiaadkeaaeqaaOGaaGypaiabfo 5ahnaaDaaaleaacaWGbbaabaGaam4qaaaakmaaBaaaleaacaWGcbaa beaakiaabwgadaWgaaWcbaGaam4qaaqabaGccaaISaGaaGzbVlaabI oacaaMf8Uaey4bIe9aaSbaaSqaamaaGaaabaGaaeyzaaGaay5adaWa aSbaaeaacaWGbbaabeaaaeqaaOWaaacaaeaacaqGLbaacaGLdmaada WgaaWcbaGaamOqaaqabaGccaaI9aWaaacaaeaacqqHtoWraiaawoWa amaaDaaaleaacaWGbbaabaGaam4qaaaakmaaBaaaleaacaWGcbaabe aakmaaGaaabaGaaeyzaaGaay5adaWaaSbaaSqaaiaadoeaaeqaaOGa aGOlaaaa@598E@

Если ( X A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabgkGi2oaaBaaaleaacaWGybWaaWbaaeqa baGaamyqaaaaaeqaaOGaaGykamaaDaaaleaacaWGbbGaaGypaiaaig daaeaacaWGUbaaaaaa@4031@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ координатный репер на U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyvaaaa@38DD@ , то e A = Ψ A B X B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyzamaaBaaaleaacaWGbbaabeaakiaai2dacqqH OoqwdaqhaaWcbaGaamyqaaqaaiaadkeaaaGccqGHciITdaWgaaWcba GaamiwamaaCaaabeqaaiaadkeaaaaabeaaaaa@4159@ , где Ψ=[ Ψ B A ]:UGL(n;) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuiQdKLaaGypaiaaiUfacqqHOoqwdaqhaaWcbaGa amOqaaqaaiaadgeaaaGccaaIDbGaaGOoaiaaykW7caWGvbGaeyOKH4 Qaae4raiaabYeacaaIOaGaamOBaiaaiUdacaaMc8+efv3ySLgznfgD Ojdaryqr1ngBPrginfgDObcv39gaiuaacqWFDeIucaaIPaaaaa@5587@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ однозначно определенное гладкое поле n×n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiabgEna0kaad6gaaaa@3C00@ матриц. Тогда справедливо следующее соотношение [23]:

Γ ˜ B A C = Γ Q P R P A Ω B Q Ω C R + Ψ Q R P A Ω B Q X R Ω C P . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqqHtoWraiaawoWaamaaDaaaleaacaWG cbaabaGaamyqaaaakmaaBaaaleaacaWGdbaabeaakiaai2dacqqHto WrdaqhaaWcbaGaamyuaaqaaiaadcfaaaGcdaWgaaWcbaGaamOuaaqa baGccaaMi8EeeuuDJXwAKbsr4rNCHbacfaGae83jHO0aa0baaSqaai aadcfaaeaacaWGbbaaaOGaaGjcVlabfM6axnaaDaaaleaacaWGcbaa baGaamyuaaaakiaayIW7cqqHPoWvdaqhaaWcbaGaam4qaaqaaiaadk faaaGccqGHRaWkcqqHOoqwdaqhaaWcbaGaamyuaaqaaiaadkfaaaGc caaMi8Uae83jHO0aa0baaSqaaiaadcfaaeaacaWGbbaaaOGaaGjcVl abfM6axnaaDaaaleaacaWGcbaabaGaamyuaaaakiaayIW7cqGHciIT daWgaaWcbaGaamiwamaaCaaabeqaaiaadkfaaaaabeaakiabfM6axn aaDaaaleaacaWGdbaabaGaamiuaaaakiaai6caaaa@6BEA@ (7)

Формула (7) определяет закон преобразования коэффициентов связности в том случае, когда неголономный (т. е. некоординатный) репер заменяется на другой неголономный репер. Если же исходный репер ( e A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaabwgadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EB4@ является голономным, то есть Ψ B A = δ B A MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuiQdK1aa0baaSqaaiaadkeaaeaacaWGbbaaaOGa aGypaiabes7aKnaaDaaaleaacaWGcbaabaGaamyqaaaaaaa@3F7C@ , то соотношение (7) упрощается:

Γ ˜ B A C = Γ Q P R P A Ω B Q Ω C R + P A Ω B Q X Q Ω C P . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqqHtoWraiaawoWaamaaDaaaleaacaWG cbaabaGaamyqaaaakmaaBaaaleaacaWGdbaabeaakiaai2dacqqHto WrdaqhaaWcbaGaamyuaaqaaiaadcfaaaGcdaWgaaWcbaGaamOuaaqa baGccaaMi8EeeuuDJXwAKbsr4rNCHbacfaGae83jHO0aa0baaSqaai aadcfaaeaacaWGbbaaaOGaaGjcVlabfM6axnaaDaaaleaacaWGcbaa baGaamyuaaaakiaayIW7cqqHPoWvdaqhaaWcbaGaam4qaaqaaiaadk faaaGccqGHRaWkcqWFNeIsdaqhaaWcbaGaamiuaaqaaiaadgeaaaGc caaMi8UaeuyQdC1aa0baaSqaaiaadkeaaeaacaWGrbaaaOGaaGjcVl abgkGi2oaaBaaaleaacaWGybWaaWbaaeqabaGaamyuaaaaaeqaaOGa euyQdC1aa0baaSqaaiaadoeaaeaacaWGqbaaaOGaaGOlaaaa@66E5@ (8)

Наконец, когда оба локальных репера ( e A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaabwgadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EB4@ и ( e ˜ A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikamaaGaaabaGaaeyzaaGaay5adaWaaSbaaSqa aiaadgeaaeqaaOGaaGykamaaDaaaleaacaWGbbGaaGypaiaaigdaae aacaWGUbaaaaaa@3F76@ голономны, то тогда речь идет о преобразовании координат. В таком случае Ω B A = X A X ˜ B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuyQdC1aa0baaSqaaiaadkeaaeaacaWGbbaaaOGa aGypamaalaaabaGaeyOaIyRaamiwamaaCaaaleqabaGaamyqaaaaaO qaaiabgkGi2oaaGaaabaGaamiwaaGaay5adaWaaWbaaSqabeaacaWG cbaaaaaaaaa@4365@ , где координаты ( X A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadIfadaahaaWcbeqaaiaadgeaaaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EAA@ порождают репер ( e A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaabwgadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EB4@ , а координаты ( X ˜ A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikamaaGaaabaGaamiwaaGaay5adaWaaWbaaSqa beaacaWGbbaaaOGaaGykamaaDaaaleaacaWGbbGaaGypaiaaigdaae aacaWGUbaaaaaa@3F6C@ порождают репер ( e ˜ A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikamaaGaaabaGaaeyzaaGaay5adaWaaSbaaSqa aiaadgeaaeqaaOGaaGykamaaDaaaleaacaWGbbGaaGypaiaaigdaae aacaWGUbaaaaaa@3F76@ . Выражение (8) сводится к соотношению

Γ ˜ B A C = X ˜ A X P X Q X ˜ B X R X ˜ C Γ Q P R + X ˜ A X P 2 X P X ˜ B X ˜ C . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqqHtoWraiaawoWaamaaDaaaleaacaWG cbaabaGaamyqaaaakmaaBaaaleaacaWGdbaabeaakiaai2dadaWcaa qaaiabgkGi2oaaGaaabaGaamiwaaGaay5adaWaaWbaaSqabeaacaWG bbaaaaGcbaGaeyOaIyRaamiwamaaCaaaleqabaGaamiuaaaaaaGcda WcaaqaaiabgkGi2kaadIfadaahaaWcbeqaaiaadgfaaaaakeaacqGH ciITdaaiaaqaaiaadIfaaiaawoWaamaaCaaaleqabaGaamOqaaaaaa GcdaWcaaqaaiabgkGi2kaadIfadaahaaWcbeqaaiaadkfaaaaakeaa cqGHciITdaaiaaqaaiaadIfaaiaawoWaamaaCaaaleqabaGaam4qaa aaaaGccqqHtoWrdaqhaaWcbaGaamyuaaqaaiaadcfaaaGcdaWgaaWc baGaamOuaaqabaGccqGHRaWkdaWcaaqaaiabgkGi2oaaGaaabaGaam iwaaGaay5adaWaaWbaaSqabeaacaWGbbaaaaGcbaGaeyOaIyRaamiw amaaCaaaleqabaGaamiuaaaaaaGcdaWcaaqaaiabgkGi2oaaCaaale qabaGaaGOmaaaakiaadIfadaahaaWcbeqaaiaadcfaaaaakeaacqGH ciITdaaiaaqaaiaadIfaaiaawoWaamaaCaaaleqabaGaamOqaaaaki abgkGi2oaaGaaabaGaamiwaaGaay5adaWaaWbaaSqabeaacaWGdbaa aaaakiaai6caaaa@6D8A@ (9)

Даже при n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ коэффициенты связности Γ B A C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaadkeaaeaacaWGbbaaaOWa aSbaaSqaaiaadoeaaeqaaaaa@3C23@ , равные нулю в декартовых координатах, могут быть отличны от нуля в криволинейных координатах. Вместе с тем в обоих случаях связность одна и та же MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ евклидова. По этой причине коэффициенты связности не подходят в качестве индикаторов неевклидовой природы пространства. Для этой цели используются тензорные поля кручения T:Vec( M R )×Vec( M R )Vec( M R ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ujaaiQdacaaMc8UaaeOvaiaabwgacaqGJbGaaGikai aad2eadaWgaaWcbaGaamOuaaqabaGccaaIPaGaey41aqRaaeOvaiaa bwgacaqGJbGaaGikaiaad2eadaWgaaWcbaGaamOuaaqabaGccaaIPa GaeyOKH4QaaeOvaiaabwgacaqGJbGaaGikaiaad2eadaWgaaWcbaGa amOuaaqabaGccaaIPaaaaa@5CB4@ , кривизны :Vec( M R )×Vec( M R )×Vec( M R )Vec( M R ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=XrisjaaiQdacaaMc8UaaeOvaiaabwgacaqGJbGaaGikai aad2eadaWgaaWcbaGaamOuaaqabaGccaaIPaGaey41aqRaaeOvaiaa bwgacaqGJbGaaGikaiaad2eadaWgaaWcbaGaamOuaaqabaGccaaIPa Gaey41aqRaaeOvaiaabwgacaqGJbGaaGikaiaad2eadaWgaaWcbaGa amOuaaqabaGccaaIPaGaeyOKH4QaaeOvaiaabwgacaqGJbGaaGikai aad2eadaWgaaWcbaGaamOuaaqabaGccaaIPaaaaa@63EE@ и неметричности Q:Vec( M R )×Vec( M R )×Vec( M R ) C ( M R ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rjaaiQdacaaMc8UaaeOvaiaabwgacaqGJbGaaGikai aad2eadaWgaaWcbaGaamOuaaqabaGccaaIPaGaey41aqRaaeOvaiaa bwgacaqGJbGaaGikaiaad2eadaWgaaWcbaGaamOuaaqabaGccaaIPa Gaey41aqRaaeOvaiaabwgacaqGJbGaaGikaiaad2eadaWgaaWcbaGa amOuaaqabaGccaaIPaGaeyOKH4Qaam4qamaaCaaaleqabaGaeyOhIu kaaOGaaGikaiaad2eadaWgaaWcbaGaamOuaaqabaGccaaIPaaaaa@6479@ , определяемые, соответственно, формулами [48; 54; 56]

T(u,v):= u v v u[u,v]; MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ujaaiIcacaqG1bGaaGilaiaaykW7caqG2bGaaGykai aaiQdacaaI9aGaey4bIe9aaSbaaSqaaiaabwhaaeqaaOGaaeODaiab gkHiTiabgEGirpaaBaaaleaacaqG2baabeaakiaabwhacqGHsislca aIBbGaaeyDaiaaiYcacaaMc8UaaeODaiaai2facaaI7aaaaa@5BB9@ (10)

(u,v)w:=[ u , v ]w [u,v] w; MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=XrisjaaiIcacaqG1bGaaGilaiaaykW7caqG2bGaaGykai aabEhacaaI6aGaaGypaiaaiUfacqGHhis0daWgaaWcbaGaaeyDaaqa baGccaaISaGaaGPaVlabgEGirpaaBaaaleaacaqG2baabeaakiaai2 facaqG3bGaeyOeI0Iaey4bIe9aaSbaaSqaaiaaiUfacaqG1bGaaGil aiaaykW7caqG2bGaaGyxaaqabaGccaqG3bGaaG4oaaaa@60CA@ (11)

Q(u,v,w):=g( u v,w)+g(v, u w)u[g(v,w)]. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rjaaiIcacaqG1bGaaGilaiaaykW7caqG2bGaaGilai aaykW7caqG3bGaaGykaiaaiQdacaaI9aGaae4zaiaaiIcacqGHhis0 daWgaaWcbaGaaeyDaaqabaGccaqG2bGaaGilaiaaykW7caqG3bGaaG ykaiabgUcaRiaabEgacaaIOaGaaeODaiaaiYcacaaMc8Uaey4bIe9a aSbaaSqaaiaabwhaaeqaaOGaae4DaiaaiMcacqGHsislcaqG1bGaaG 4waiaabEgacaaIOaGaaeODaiaaiYcacaaMc8Uaae4DaiaaiMcacaaI DbGaaGOlaaaa@6D33@ (12)

Здесь[6] [u,v] MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4waiaabwhacaaISaGaaGPaVlaabAhacaaIDbaa aa@3E01@ и [ u , v ] MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4waiabgEGirpaaBaaaleaacaqG1baabeaakiaa iYcacaaMc8Uaey4bIe9aaSbaaSqaaiaabAhaaeqaaOGaaGyxaaaa@4179@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ коммутаторы,

[u,v]=uvvu,[ u , v ]= u v v u , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4waiaabwhacaaISaGaaGPaVlaabAhacaaIDbGa aGypaiaabwhacqWIyiYBcaqG2bGaeyOeI0IaaeODaiablIHiVjaabw hacaaISaGaaGzbVlaaiUfacqGHhis0daWgaaWcbaGaaeyDaaqabaGc caaISaGaaGPaVlabgEGirpaaBaaaleaacaqG2baabeaakiaai2faca aI9aGaey4bIe9aaSbaaSqaaiaabwhaaeqaaOGaeSigI8Maey4bIe9a aSbaaSqaaiaabAhaaeqaaOGaeyOeI0Iaey4bIe9aaSbaaSqaaiaabA haaeqaaOGaeSigI8Maey4bIe9aaSbaaSqaaiaabwhaaeqaaOGaaGil aaaa@6175@

а u[g(v,w)] MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyDaiaaiUfacaqGNbGaaGikaiaabAhacaaISaGa aGPaVlaabEhacaaIPaGaaGyxaaaa@414A@ есть действие векторного поля u MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyDaaaa@38FB@ на скалярное поле g(v,w) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4zaiaaiIcacaqG2bGaaGilaiaaykW7caqG3bGa aGykaaaa@3E86@ . Евклидова геометрия характеризуется следующими значениями: T=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ujaai2dacaqGWaaaaa@461A@ , =0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Xrisjaai2dacaqGWaaaaa@4552@ и Q=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rjaai2dacaqGWaaaaa@4614@ .

Таким образом, приходим к структуре S R =( M R ,G,) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaakiaai2dacaaI OaGaamytamaaBaaaleaacaWGsbaabeaakiaaiYcacaaMc8Uaae4rai aaiYcacaaMc8Uaey4bIeTaaGykaaaa@44C5@ , которая является неевклидовой глобально единообразной формой. Заметим, что в общем случае S R S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaGccqGHGjsUcaWGtbWaaS baaSqaaiaadkfaaeqaaaaa@4817@ , поскольку g | S R G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4zaiaaiYhadaWgaaWcbaGaam4uamaaBaaabaGa amOuaaqabaaabeaakiabgcMi5kaabEeaaaa@3E90@ и | S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIeTaaGiFamaaBaaaleaacaWGtbWaaSbaaeaa caWGsbaabeaaaeqaaOGaeyiyIKRaey4bIenaaa@3FE2@ .

Обсудим более детально связь между неевклидовой формой S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ и непрерывным семейством { S (X) } X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4Eamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8NeXp1aaWbaaSqabeaacaaIOaGae83fXJLaaGykaa aakiaai2hadaWgaaWcbaGae83fXJLaeyicI4Sae8NeXp1aaSbaaeaa caWGsbaabeaaaeqaaaaa@4F5D@ локально единообразных форм, каждая из которых, S (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaahaaWcbeqaaiaaiIcacqWFxepwcaaIPaaaaaaa@46DF@ , является образом <<разгрузочной>> деформации γ (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaWbaaSqabeaacaaIOaWefv3ySLgznfgD Ofdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwcaaIPaaaaaaa@46AB@ . Последнему семейству отвечает поле H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ локальных деформаций, значение которого в каждой точке X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcqGHiiIZcqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@47D4@ является линейным отображением H X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaaaaa@446F@ , преобразующим представительный объем, окружающий точку X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwaaa@4372@ , в единообразное состояние.

С геометрической точки зрения несовместность локальных деформаций означает, что связи, наложенные евклидовой геометрией на формы тела, являются слишком обременительными. В этом месте представляется удобным апеллировать к следующей образной интерпретации. Представительные объемы в единообразном состоянии MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ <<элементарные параллелепипеды>>, как их часто изображают в монографиях по теории упругости, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ являются частями различных представителей S (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaahaaWcbeqaaiaaiIcacqWFxepwcaaIPaaaaaaa@46DF@ семейства форм. Мысленно отделим каждый такой параллелепипед от остальной формы, что приводит к континуальной совокупности представительных объемов. Следуя Можену, будем называть такую совокупность <<кристаллической отсчетной>> [15]. Элементы кристаллической отсчетной геометрически несовместны: попытка собрать из них непрерывное тело приводит к несогласованным индивидуальным искажениям, и, таким образом, снова получается некоторая неединообразная евклидова форма. Эту проблему можно решить, лишь освободив связи, наложенные евклидовой структурой на формы. Тогда кристаллическая отсчетная в своем объединении (без предварительных искажений) даст непрерывную область, вложенную в пространство с более общей геометрией, чем евклидова. Возвращаясь к семейству форм, получаем, что в этом случае локальная деформация может быть представлена в виде градиента вложения неевклидовой формы в евклидово физическое пространство.

Замечание 4. Следует отметить, что при работе с глобальной неевклидовой формой неявно используется исходное семейство локально единообразных форм { S (X) } X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4Eamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8NeXp1aaWbaaSqabeaacaaIOaGae83fXJLaaGykaa aakiaai2hadaWgaaWcbaGae83fXJLaeyicI4Sae8NeXp1aaSbaaeaa caWGsbaabeaaaeqaaaaa@4F5D@ . Действительно, например, при вычислении отклика в точке неевклидовой формы этот отклик на самом деле определяется относительно соответствующей формы из локально единообразного семейства форм. Далее, значения материальной метрики и связности также определяются по значениям соответствующих полей на определенной форме из семейства { S (X) } X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4Eamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8NeXp1aaWbaaSqabeaacaaIOaGae83fXJLaaGykaa aakiaai2hadaWgaaWcbaGae83fXJLaeyicI4Sae8NeXp1aaSbaaeaa caWGsbaabeaaaeqaaaaa@4F5D@ . Все, что дает неевклидова форма, MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbcKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C75@ это возможность представить семейство форм в виде единой области.

2.3. Частные случаи материальной связности

Связность Леви-Чивита.

Рассмотрим примеры материальных связностей. Первый пример MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ связность Леви-Чивита L MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaWbaaSqabeaacaWGmbaaaaaa@3A87@ [57], индуцируемая на многообразии M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ материальной метрикой (6). В этом случае коэффициенты связности в координатном репере ( A ) i=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabgkGi2oaaBaaaleaacaWGbbaabeaakiaa iMcadaqhaaWcbaGaamyAaiaai2dacaaIXaaabaGaamOBaaaaaaa@3F5A@ представлены выражениями

Γ A C B = G CD 2 A G DB + B G AD D G AB , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaadgeaaeaacaWGdbaaaOWa aSbaaSqaaiaadkeaaeqaaOGaaGypamaalaaabaGaae4ramaaCaaale qabaGaam4qaiaadseaaaaakeaacaaIYaaaamaabmaabaGaeyOaIy7a aSbaaSqaaiaadgeaaeqaaOGaae4ramaaBaaaleaacaWGebGaamOqaa qabaGccqGHRaWkcqGHciITdaWgaaWcbaGaamOqaaqabaGccaqGhbWa aSbaaSqaaiaadgeacaWGebaabeaakiabgkHiTiabgkGi2oaaBaaale aacaWGebaabeaakiaabEeadaWgaaWcbaGaamyqaiaadkeaaeqaaaGc caGLOaGaayzkaaGaaGilaaaa@5336@ (13)

в которых [ G AB ]=[ G AB ] 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4waiaabEeadaahaaWcbeqaaiaadgeacaWGcbaa aOGaaGyxaiaai2dacaaIBbGaae4ramaaBaaaleaacaWGbbGaamOqaa qabaGccaaIDbWaaWbaaSqabeaacqGHsislcaaIXaaaaaaa@4352@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ матрица, обратная к матрице метрических коэффициентов G AB =G( A , B ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaBaaaleaacaWGbbGaamOqaaqabaGccaaI 9aGaae4raiaaiIcacqGHciITdaWgaaWcbaGaamyqaaqabaGccaaISa GaaGPaVlabgkGi2oaaBaaaleaacaWGcbaabeaakiaaiMcaaaa@448C@ .

Связность L MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaWbaaSqabeaacaWGmbaaaaaa@3A87@ наделяет многообразие M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ геометрией, которая полностью характеризуется тензором кривизны Римана MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Xrisbaa@43D8@ (11), поскольку кручение (10) и неметричность (12) связности Леви-Чивита равны нулю. Компоненты MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Xrisbaa@43D8@ в координатном репере связаны с коэффициентами связности формулой

A D B C = A Γ B D C B Γ A D C + Γ B E C Γ A D E Γ A E C Γ B D E . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=XrisnaaDaaaleaacaWGbbaabaGaamiraaaakmaaBaaale aacaWGcbaabeaakmaaBaaaleaacaWGdbaabeaakiaai2dacqGHciIT daWgaaWcbaGaamyqaaqabaGccqqHtoWrdaqhaaWcbaGaamOqaaqaai aadseaaaGcdaWgaaWcbaGaam4qaaqabaGccqGHsislcqGHciITdaWg aaWcbaGaamOqaaqabaGccqqHtoWrdaqhaaWcbaGaamyqaaqaaiaads eaaaGcdaWgaaWcbaGaam4qaaqabaGccqGHRaWkcqqHtoWrdaqhaaWc baGaamOqaaqaaiaadweaaaGcdaWgaaWcbaGaam4qaaqabaGccqqHto WrdaqhaaWcbaGaamyqaaqaaiaadseaaaGcdaWgaaWcbaGaamyraaqa baGccqGHsislcqqHtoWrdaqhaaWcbaGaamyqaaqaaiaadweaaaGcda WgaaWcbaGaam4qaaqabaGccqqHtoWrdaqhaaWcbaGaamOqaaqaaiaa dseaaaGcdaWgaaWcbaGaamyraaqabaGccaaIUaaaaa@69AA@ (14)

Тензорное поле MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Xrisbaa@43D8@ , в свою очередь, определяет тензор кривизны Риччи RicSec T M R T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbGaeyicI4Saae4uaiaabwga caqGJbWaaeWaaeaacaWGubWaaWbaaSqabeaacqGHxiIkaaGccaWGnb WaaSbaaSqaaiaadkfaaeqaaOGaey4LIqSaamivamaaCaaaleqabaGa ey4fIOcaaOGaamytamaaBaaaleaacaWGsbaabeaaaOGaayjkaiaawM caaaaa@4A20@ , компоненты которого в координатном репере получаются при помощи свертки компонент кривизны Римана:

Ric AB = A C C B . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbWaaSbaaSqaaiaadgeacaWG cbaabeaakiaai2datuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRr xDYbacfaGae8hhHi1aa0baaSqaaiaadgeaaeaacaWGdbaaaOWaaSba aSqaaiaadoeaaeqaaOWaaSbaaSqaaiaadkeaaeqaaOGaaGOlaaaa@4D81@

Являясь тензорным полем второго ранга на n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ -мерном многообразии, тензор Ric MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbaaaa@3AAA@ имеет в качестве одного из главных инвариантов след

Scal= G AB Ric AB , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4uaiaabogacaqGHbGaaeiBaiaai2dacaqGhbWa aWbaaSqabeaacaWGbbGaamOqaaaakiaabkfacaqGPbGaae4yamaaBa aaleaacaWGbbGaamOqaaqabaGccaaISaaaaa@4407@

называемый скалярной кривизной.

Хотя в общем случае определение главных инвариантов тензора четвертого ранга является довольно сложной задачей, для малых размерностей, то есть, когда n3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBamrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhA Gq1DVbacfaGae8xFQqOaaG4maaaa@4506@ , можно полностью охарактеризовать геометрию отсчетной формы либо с помощью тензора Риччи ( n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ ), либо посредством скалярной кривизны ( n=2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIYaaaaa@3A79@ ). Действительно, определим следующую операцию, называемую произведением Кулкарни MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFtaca aa@3C72@ Номидзу [48, с.~213]:

hk(w,x,y,z):=h(w,z)k(x,y)+h(x,y)k(w,z)h(w,y)k(x,z)h(x,z)k(w,y), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiAaiaabUgacaaIOaGaae4DaiaaiYcacaaMc8Ua aeiEaiaaiYcacaaMc8UaaeyEaiaaiYcacaaMc8UaaeOEaiaaiMcaca aI6aGaaGypaiaabIgacaaIOaGaae4DaiaaiYcacaaMc8UaaeOEaiaa iMcacaqGRbGaaGikaiaabIhacaaISaGaaGPaVlaabMhacaaIPaGaey 4kaSIaaeiAaiaaiIcacaqG4bGaaGilaiaaykW7caqG5bGaaGykaiaa bUgacaaIOaGaae4DaiaaiYcacaaMc8UaaeOEaiaaiMcacqGHsislca qGObGaaGikaiaabEhacaaISaGaaGPaVlaabMhacaaIPaGaae4Aaiaa iIcacaqG4bGaaGilaiaaykW7caqG6bGaaGykaiabgkHiTiaabIgaca aIOaGaaeiEaiaaiYcacaaMc8UaaeOEaiaaiMcacaqGRbGaaGikaiaa bEhacaaISaGaaGPaVlaabMhacaaIPaGaaGilaaaa@7F3B@

действующую над симметричными тензорами h,kSec T M R T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiAaiaaiYcacaaMc8Uaae4AaiabgIGiolaabofa caqGLbGaae4yamaabmaabaGaamivamaaCaaaleqabaGaey4fIOcaaO GaamytamaaBaaaleaacaWGsbaabeaakiabgEPielaadsfadaahaaWc beqaaiabgEHiQaaakiaad2eadaWgaaWcbaGaamOuaaqabaaakiaawI cacaGLPaaaaaa@4B93@ . Тогда для n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ будем иметь[7] (см. [48, следствие 7.26])

# =RicG Scal 4 GG, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=XrisnaaCaaaleqabaGaai4iaaaakiaai2dacaqGsbGaae yAaiaabogacaqGhbGaeyOeI0YaaSaaaeaacaqGtbGaae4yaiaabgga caqGSbaabaGaaGinaaaacaqGhbGaae4raiaaiYcaaaa@5082@

а для n=2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIYaaaaa@3A79@ , согласно [48, следствие 7.27], MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@

# = Scal 4 GG. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=XrisnaaCaaaleqabaGaai4iaaaakiaai2dadaWcaaqaai aabofacaqGJbGaaeyyaiaabYgaaeaacaaI0aaaaiaabEeacaqGhbGa aGOlaaaa@4C26@

Заметим, что для n=1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIXaaaaa@3A78@ справедливо равенство 1 1 1 1 =0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=XrisnaaBaaaleaacaaIXaaabeaakmaaBaaaleaacaaIXa aabeaakmaaDaaaleaacaaIXaaabaGaaGymaaaakiaai2dacaaIWaaa aa@48E8@ .

Таким образом, если n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ , то =0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Xrisjaai2dacaqGWaaaaa@4552@ тогда и только тогда, когда Ric=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbGaaGypaiaabcdaaaa@3C24@ , а если n=2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIYaaaaa@3A79@ , то =0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Xrisjaai2dacaqGWaaaaa@4552@ тогда и только тогда, когда Scal=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4uaiaabogacaqGHbGaaeiBaiaai2dacaaIWaaa aa@3D13@ . Это означает, что в соответствии с размерностью многообразия кривизна Риччи или скалярная кривизна полностью характеризуют несовместность локальных деформаций. Следует отметить, что в одномерном случае локальные деформации всегда совместны, поскольку кривизна всегда равна нулю.

Связность Вайценбока.

Поскольку для каждой точки X M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaiabgIGiolaad2eadaWgaaWcbaGaamOuaaqa baaaaa@3C39@ линейное отображение H X : T X M R U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaBaaaleaacaWGybaabeaakiaaiQdacaaM c8UaamivamaaBaaaleaacaWGybaabeaakiaad2eadaWgaaWcbaGaam OuaaqabaGccqGHsgIRtuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgi p5wzaGqbaiab=rr8vbaa@4D51@ обратимо, можно определить отображение H 1 :T( S R ,U)T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaa iQdacaaMc8UaamivaiaaiIcatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0 uy0Hgip5wzaGqbaiab=jr8tnaaBaaaleaacaWGsbaabeaakiaaiYca caaMc8Uae8hfXxLaaGykaiabgkziUkaadsfacaWGnbWaaSbaaSqaai aadkfaaeqaaaaa@5467@ , синтезированное из обратных отображений H X 1 :U T X M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaDaaaleaacaWGybaabaGaeyOeI0IaaGym aaaakiaaiQdacaaMc8+efv3ySLgznfgDOfdaryqr1ngBPrginfgDOb YtUvgaiuaacqWFueFvcqGHsgIRcaWGubWaaSbaaSqaaiaadIfaaeqa aOGaamytamaaBaaaleaacaWGsbaabeaaaaa@4EF0@ . Далее, напомним, что архетип U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFueFvaaa@436C@ снабжен скалярным произведением, индуцированным евклидовой метрикой. Тогда можно выбрать некоторый ортонормированный базис ( c A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahogadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EB8@ в U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFueFvaaa@436C@ , что позволяет определить специальное семейство ( z A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaabQhadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EC9@ векторных полей z A Vec( M R ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOEamaaBaaaleaacaWGbbaabeaakiabgIGiolaa bAfacaqGLbGaae4yaiaaiIcacaWGnbWaaSbaaSqaaiaadkfaaeqaaO GaaGykaaaa@416B@ на M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ :

z A := H 1 [ c A ],A=1,,n. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOEamaaBaaaleaacaWGbbaabeaakiaaiQdacaaI 9aGaaeisamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaaiUfacaWHJb WaaSbaaSqaaiaadgeaaeqaaOGaaGyxaiaaiYcacaaMf8Uaamyqaiaa i2dacaaIXaGaaGilaiaaykW7cqWIMaYscaaISaGaaGPaVlaad6gaca aIUaaaaa@4DC0@

Являясь образами базисных векторов относительно невырожденного линейного отображения, векторы ( z A | X ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaabQhadaWgaaWcbaGaamyqaaqabaGccaaI 8bWaaSbaaSqaaiaadIfaaeqaaOGaaGykamaaDaaaleaacaWGbbGaaG ypaiaaigdaaeaacaWGUbaaaaaa@40E2@ образуют базис в каждой точке X M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaiabgIGiolaad2eadaWgaaWcbaGaamOuaaqa baaaaa@3C39@ . Таким образом, семейство ( z A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaabQhadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EC9@ является репером (следуя Картану [58], этот репер обычно называют подвижным репером). Согласно диадному разложению H 1 =[ H 1 ] B A A c B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaa i2dacaaIBbGaaeisamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaai2 fadaqhaaWcbaGaamOqaaqaaiaadgeaaaGccqGHciITdaWgaaWcbaGa amyqaaqabaGccqGHxkcXcaWHJbWaaWbaaSqabeaacaWGcbaaaaaa@47F9@ , где ( A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabgkGi2oaaBaaaleaacaWGbbaabeaakiaa iMcadaqhaaWcbaGaamyqaiaai2dacaaIXaaabaGaamOBaaaaaaa@3F32@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ координатный репер, приходим к эквивалентному представлению элементов подвижного репера: z A =[ H 1 ] A B B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOEamaaBaaaleaacaWGbbaabeaakiaai2dacaaI BbGaaeisamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaai2fadaqhaa WcbaGaamyqaaqaaiaadkeaaaGccqGHciITdaWgaaWcbaGaamOqaaqa baaaaa@4356@ .

Многообразие M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ снабжается такой связностью W MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaWbaaSqabeaacaWGxbaaaaaa@3A92@ , что

z A W z B =0,A,B{1,,n}. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aa0baaSqaaiaabQhadaWgaaqaaiaadgea aeqaaaqaaiaadEfaaaGccaqG6bWaaSbaaSqaaiaadkeaaeqaaOGaaG ypaiaabcdacaaISaGaaGzbVlaadgeacaaISaGaaGPaVlaadkeacqGH iiIZcaaI7bGaaGymaiaaiYcacaaMc8UaeSOjGSKaaGilaiaaykW7ca WGUbGaaGyFaiaai6caaaa@51A0@

Будем называть ее связностью Вайценбока [59]. По построению, коэффициенты связности Вайценбока относительно репера ( z A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaabQhadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EC9@ равны нулю. Вместе с тем, если Γ B A C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaadkeaaeaacaWGbbaaaOWa aSbaaSqaaiaadoeaaeqaaaaa@3C23@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ коэффициенты связности W MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaWbaaSqabeaacaWGxbaaaaaa@3A92@ в координатном репере, то для них справедливы равенства

Γ B A C = H C D X B [ H 1 ] D A =[ H 1 ] D A X B H C D , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaadkeaaeaacaWGbbaaaOWa aSbaaSqaaiaadoeaaeqaaOGaaGypaiabgkHiTiaabIeadaqhaaWcba Gaam4qaaqaaiaadseaaaGccqGHciITdaWgaaWcbaGaamiwamaaCaaa beqaaiaadkeaaaaabeaakiaaiUfacaqGibWaaWbaaSqabeaacqGHsi slcaaIXaaaaOGaaGyxamaaDaaaleaacaWGebaabaGaamyqaaaakiaa i2dacaaIBbGaaeisamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaai2 fadaqhaaWcbaGaamiraaqaaiaadgeaaaGccqGHciITdaWgaaWcbaGa amiwamaaCaaabeqaaiaadkeaaaaabeaakiaabIeadaqhaaWcbaGaam 4qaaqaaiaadseaaaGccaaISaaaaa@57C0@ (15)

являющиеся следствиями равенств (8), в которых нужно положить Γ ˜ B A C =0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqqHtoWraiaawoWaamaaDaaaleaacaWG cbaabaGaamyqaaaakmaaBaaaleaacaWGdbaabeaakiaai2dacaaIWa aaaa@3E70@ и =[ H B A ] MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaarqqr1ngBPrgifHhDYfgaiuaapaGae83jHOKaaGypaiaa iUfacaqGibWaa0baaSqaaiaadkeaaeaacaWGbbaaaOGaaGyxaaaa@42D8@ .

Геометрия, устанавливаемая на многообразии M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ связностью W MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaWbaaSqabeaacaWGxbaaaaaa@3A92@ , полностью характеризуется тензором кручения T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ubaa@44A0@ (10), поскольку как тензор кривизны Римана (11), так и неметричность (12) связности W MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaWbaaSqabeaacaWGxbaaaaaa@3A92@ равны нулю. В координатном репере компоненты кручения имеют следующий вид:

T B A C = Γ B A C Γ C A B . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8unaaDaaaleaacaWGcbaabaGaamyqaaaakmaaBaaale aacaWGdbaabeaakiaai2dacqqHtoWrdaqhaaWcbaGaamOqaaqaaiaa dgeaaaGcdaWgaaWcbaGaam4qaaqabaGccqGHsislcqqHtoWrdaqhaa WcbaGaam4qaaqaaiaadgeaaaGcdaWgaaWcbaGaamOqaaqabaGccaaI Uaaaaa@5222@ (16)

Принимая во внимание выражения (15), приходим к альтернативному представлению компонент кручения:

T B A C =[ H 1 ] D A X B H C D [ H 1 ] D A X C H B D . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8unaaDaaaleaacaWGcbaabaGaamyqaaaakmaaBaaale aacaWGdbaabeaakiaai2dacaaIBbGaaeisamaaCaaaleqabaGaeyOe I0IaaGymaaaakiaai2fadaqhaaWcbaGaamiraaqaaiaadgeaaaGccq GHciITdaWgaaWcbaGaamiwamaaCaaabeqaaiaadkeaaaaabeaakiaa bIeadaqhaaWcbaGaam4qaaqaaiaadseaaaGccqGHsislcaaIBbGaae isamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaai2fadaqhaaWcbaGa amiraaqaaiaadgeaaaGccqGHciITdaWgaaWcbaGaamiwamaaCaaabe qaaiaadoeaaaaabeaakiaabIeadaqhaaWcbaGaamOqaaqaaiaadsea aaGccaaIUaaaaa@6230@

В случае n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ часто бывает удобно (особенно, в континуальной теории дефектов) заменить тензор кручения на тензорное поле αSec T M R T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySdeMaeyicI4Saae4uaiaabwgacaqGJbWaaeWa aeaacaWGubGaamytamaaBaaaleaacaWGsbaabeaakiabgEPielaads facaWGnbWaaSbaaSqaaiaadkfaaeqaaaGccaGLOaGaayzkaaaaaa@46CC@ , α= α AB X A X B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySdeMaaGypaiabeg7aHnaaCaaaleqabaGaamyq aiaadkeaaaGccqGHciITdaWgaaWcbaGaamiwamaaCaaabeqaaiaadg eaaaaabeaakiabgEPielabgkGi2oaaBaaaleaacaWGybWaaWbaaeqa baGaamOqaaaaaeqaaaaa@468E@ с компонентами [44]

α AB = 1 2 ε ACD T C B D . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySde2aaWbaaSqabeaacaWGbbGaamOqaaaakiaa i2dacqGHsisldaWcaaqaaiaaigdaaeaacaaIYaaaaiabew7aLnaaCa aaleqabaGaamyqaiaadoeacaWGebaaamrr1ngBPrMrYf2A0vNCaeHb fv3ySLgzGyKCHTgD1jhaiuaakiab=nb8unaaDaaaleaacaWGdbaaba GaamOqaaaakmaaBaaaleaacaWGebaabeaakiaai6caaaa@52F0@ (17)

Это поле принято называть тензором плотности дислокаций. Здесь ε MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTdugaaa@39AA@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ тензор Леви-Чивита с компонентами ε ABC = e ABC detG MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTdu2aaWbaaSqabeaacaWGbbGaamOqaiaadoea aaGccaaI9aWaaSaaaeaacaWGLbWaaWbaaSqabeaacaWGbbGaamOqai aadoeaaaaakeaadaGcaaqaaiGacsgacaGGLbGaaiiDaiaabEeaaSqa baaaaaaa@4433@ , в которых detG MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaciizaiaacwgacaGG0bGaae4raaaa@3B98@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ определитель материальной метрики G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ , а e ABC MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyzamaaCaaaleqabaGaamyqaiaadkeacaWGdbaa aaaa@3B6F@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ альтернатор[8]. Тензорное поле α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySdegaaa@39A2@ может быть эквивалентно выбрано в качестве меры несовместности, поскольку T=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ujaai2dacaqGWaaaaa@461A@ тогда и только тогда, когда α=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySdeMaaGypaiaabcdaaaa@3B1C@ .

Связность Вейля.

Рассмотрим последний частный случай материальной связности. Пусть ν MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyVd4MaeyicI4maaa@3B3F@ Ω 1 M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyicI4SaeuyQdC1aaWbaaSqabeaacaaIXaaaaOWa aeWaaeaacaWGnbWaaSbaaSqaaiaadkfaaeqaaaGccaGLOaGaayzkaa aaaa@3F6F@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ произвольная 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaaaa@38BE@ -форма. Определим связность Wl MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaWbaaSqabeaacaWGxbGaamiBaaaaaaa@3B83@ условиями:

Wl MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaWbaaSqabeaacaWGxbGaamiBaaaaaaa@3B83@ симметрична, то есть T=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ujaai2dacaqGWaaaaa@461A@ ,

u Wl G=ν(u)G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aa0baaSqaaiaabwhaaeaacaWGxbGaamiB aaaakiaabEeacaaI9aGaeqyVd4MaaGikaiaabwhacaaIPaGaae4raa aa@42F5@ .

В частности, из условия (b) следует, что тензор неметричности (12) равен Q=νG MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rjaai2dacqGHsislcqaH9oGBcqGHxkcXcaqGhbaaaa@4AD9@ . Связность Wl MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaWbaaSqabeaacaWGxbGaamiBaaaaaaa@3B83@ называется связностью Вейля, а 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaaaa@38BE@ -форма ν MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyVd4gaaa@39BB@ называется 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaaaa@38BE@ -формой Вейля [60].

В координатном репере компоненты Q ABC MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rnaaBaaaleaacaWGbbGaamOqaiaadoeaaeqaaaaa@471B@ тензора неметричности Q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rbaa@449A@ представлены равенствами Q ABC = ν A G BC MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rnaaBaaaleaacaWGbbGaamOqaiaadoeaaeqaaOGaaG ypaiabgkHiTiabe27aUnaaBaaaleaacaWGbbaabeaakiaabEeadaWg aaWcbaGaamOqaiaadoeaaeqaaaaa@4E12@ . Поэтому коэффициенты связности Вейля имеют вид:

Γ B A C = G AD 2 ( B G DC + C G DB D G BC ) 1 2 ( ν B δ C A + ν C δ B A ν D G BC G AD ), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaadkeaaeaacaWGbbaaaOWa aSbaaSqaaiaadoeaaeqaaOGaaGypamaalaaabaGaae4ramaaCaaale qabaGaamyqaiaadseaaaaakeaacaaIYaaaaiaaiIcacqGHciITdaWg aaWcbaGaamOqaaqabaGccaqGhbWaaSbaaSqaaiaadseacaWGdbaabe aakiabgUcaRiabgkGi2oaaBaaaleaacaWGdbaabeaakiaabEeadaWg aaWcbaGaamiraiaadkeaaeqaaOGaeyOeI0IaeyOaIy7aaSbaaSqaai aadseaaeqaaOGaae4ramaaBaaaleaacaWGcbGaam4qaaqabaGccaaI PaGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacaaIOaGaeqyVd4 2aaSbaaSqaaiaadkeaaeqaaOGaeqiTdq2aa0baaSqaaiaadoeaaeaa caWGbbaaaOGaey4kaSIaeqyVd42aaSbaaSqaaiaadoeaaeqaaOGaeq iTdq2aa0baaSqaaiaadkeaaeaacaWGbbaaaOGaeyOeI0IaeqyVd42a aSbaaSqaaiaadseaaeqaaOGaae4ramaaBaaaleaacaWGcbGaam4qaa qabaGccaqGhbWaaWbaaSqabeaacaWGbbGaamiraaaakiaaiMcacaaI Saaaaa@6CD2@ (18)

из которого следует, что связность Вейля полностью определена материальной метрикой G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ и 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaaaa@38BE@ -формой ν MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyVd4gaaa@39BB@ .

В общем случае кривизна связности Вейля отлична от нуля и в силу дополнительных слагаемых в (18), определяющих отклонение связности Вейля от связности Леви-Чивита, соответствующий тензор Риччи несимметричен. По этой причине, тензор Риччи может быть однозначно разложен в виде суммы нетривиальных симметричной и антисимметричной частей Ric (sym) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbWaaWbaaSqabeaacaaIOaGa ae4CaiaabMhacaqGTbGaaGykaaaaaaa@3F1E@ и Ric (asym) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbWaaWbaaSqabeaacaaIOaGa aeyyaiaabohacaqG5bGaaeyBaiaaiMcaaaaaaa@4002@ :

Ric= Ric (sym) + Ric (asym) . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbGaaGypaiaabkfacaqGPbGa ae4yamaaCaaaleqabaGaaGikaiaabohacaqG5bGaaeyBaiaaiMcaaa GccqGHRaWkcaqGsbGaaeyAaiaabogadaahaaWcbeqaaiaaiIcacaqG HbGaae4CaiaabMhacaqGTbGaaGykaaaakiaai6caaaa@4C39@

Первое слагаемое разложения определяет скалярную кривизну Scal MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4uaiaabogacaqGHbGaaeiBaaaa@3B92@ связности Вейля, а второе слагаемое связано с 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaaaa@38BE@ -формой Вейля соотношением [60]

Ric (asym) =dν, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbWaaWbaaSqabeaacaaIOaGa aeyyaiaabohacaqG5bGaaeyBaiaaiMcaaaGccaaI9aGaamizaiabe2 7aUjaaiYcaaaa@442A@

где dν MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamizaiabe27aUbaa@3AA4@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ внешний дифференциал ковекторного поля ( 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaaaa@38BE@ -формы) ν MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyVd4gaaa@39BB@ , который в координатном представлении имеет вид

dν= A<B ( A ν B B ν A )d X A d X B . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamizaiabe27aUjaai2dadaaeqbqabSqaaiaadgea caaI8aGaamOqaaqab0GaeyyeIuoakiaaiIcacqGHciITdaWgaaWcba GaamyqaaqabaGccqaH9oGBdaWgaaWcbaGaamOqaaqabaGccqGHsisl cqGHciITdaWgaaWcbaGaamOqaaqabaGccqaH9oGBdaWgaaWcbaGaam yqaaqabaGccaaIPaGaamizaiaadIfadaahaaWcbeqaaiaadgeaaaGc cqGHNis2caWGKbGaamiwamaaCaaaleqabaGaamOqaaaakiaai6caaa a@5459@

Если форма Вейля является полным дифференциалом, т. е., когда ν=df MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyVd4MaaGypaiaadsgacaWGMbaaaa@3C56@ для некоторой скалярной функции f C ( M R ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzaiabgIGiolaadoeadaahaaWcbeqaaiabg6Hi LcaakiaaiIcacaWGnbWaaSbaaSqaaiaadkfaaeqaaOGaaGykaaaa@4026@ , то Ric (asym) =ddf=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbWaaWbaaSqabeaacaaIOaGa aeyyaiaabohacaqG5bGaaeyBaiaaiMcaaaGccaaI9aGaamizaiaads gacaWGMbGaaGypaiaabcdaaaa@450A@ и тензор Риччи вновь оказывается симметричным. Обратное утверждение справедливо в следующей форме: если кривизна пространства Вейля равна нулю, то тогда dν=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamizaiabe27aUjaai2dacaqGWaaaaa@3C1E@ и, следовательно, в случае односвязного многообразия M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ форма ν MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyVd4gaaa@39BB@ является полным дифференциалом.

2.4.Замена отсчетной формы

Неевклидова связность на форме S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ , которая изначально предполагалась частью евклидова пространства, задавалась в два этапа. На первом этапе геометрия, индуцированная из физического пространства, <<стиралась>>. При этом форма превращалась в гладкое многообразие M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ без какой-либо связности, определенной на нем. На втором этапе на подготовленном таким образом многообразии задавалась новая связность с помощью поля невырожденных линейных преобразований, роль которых выполняли локальные деформации H X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaaaaa@446F@ . Для доказательства корректности определения связности требуется показать, что результат не зависит от выбора конкретной формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ , то есть инварианты новой связности не должны зависеть от гладкого преобразования формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ в некоторую иную форму S R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaqhaaWcbaGaamOuaaqaaiaadEcaaaaaaa@4518@ [26].

Пусть S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ и S R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaqhaaWcbaGaamOuaaqaaiaadEcaaaaaaa@4518@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ произвольные евклидовы формы и пусть γ: S R ' S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdCMaaGOoaiaaykW7tuuDJXwAK1uy0HwmaeHb fv3ySLgzG0uy0Hgip5wzaGqbaiab=jr8tnaaDaaaleaacaWGsbaaba Gaam4jaaaakiabgkziUkab=jr8tnaaBaaaleaacaWGsbaabeaaaaa@4DE3@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ деформация, трансформирующая одну форму в другую. Обозначим через M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ и M R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaDaaaleaacaWGsbaabaGaam4jaaaaaaa@3A85@ соответствующие подлежащие многообразия. Предположим, что поля H:T S R T( S R ,U) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaiaaiQdacaaMc8Uaamivamrr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae8NeXp1aaSbaaSqaaiaadk faaeqaaOGaeyOKH4QaamivaiaaiIcacqWFse=udaWgaaWcbaGaamOu aaqabaGccaaISaGaaGPaVlab=rr8vjaaiMcaaaa@53A1@ и H ' :T S R ' T( S R ,U) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaCaaaleqabaGaam4jaaaakiaaiQdacaaM c8Uaamivamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfa Gae8NeXp1aa0baaSqaaiaadkfaaeaacaWGNaaaaOGaeyOKH4Qaamiv aiaaiIcacqWFse=udaWgaaWcbaGaamOuaaqabaGccaaISaGaaGPaVl ab=rr8vjaaiMcaaaa@5531@ локальных деформаций связаны равенством: H ' =HF MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaCaaaleqabaGaam4jaaaakiaai2dacaWH ibGaaCOraaaa@3C1E@ , где F=Tγ:T S R ' T S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCOraiaai2dacaWGubGaeq4SdCMaaGOoaiaaykW7 caWGubWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacq WFse=udaqhaaWcbaGaamOuaaqaaiaadEcaaaGccqGHsgIRcaWGubGa e8NeXp1aaSbaaSqaaiaadkfaaeqaaaaa@5204@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ градиент деформации. С физической точки зрения это предположение означает, что разгрузка произвольного представительного объема из формы S R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaqhaaWcbaGaamOuaaqaaiaadEcaaaaaaa@4518@ в натуральное состояние производится таким образом, что этот объем вначале преобразуется в соответствующий представительный объем из S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ , а затем разгружается в натуральное состояние посредством H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ . Далее будет показано, что в этом случае геометрии, построенные на соответствующих формах, совпадают.

Чтобы установить этот факт, рассмотрим преобразование тензорных полей, представляющих геометрии, под действием деформации как отображения γ: M R ' M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdCMaaGOoaiaaykW7caWGnbWaa0baaSqaaiaa dkfaaeaacaWGNaaaaOGaeyOKH4QaamytamaaBaaaleaacaWGsbaabe aaaaa@4247@ между многообразиями. Это преобразование производится в терминах операции обратного образа [39, с.~320], которая сохраняет геометрическую структуру. Если установить, что обратный образ тензорных полей, представляющих геометрию многообразия M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ , на многообразии M R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaDaaaleaacaWGsbaabaGaam4jaaaaaaa@3A85@ совпадает с соответствующими полями, определенными на M R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaDaaaleaacaWGsbaabaGaam4jaaaaaaa@3A85@ , то тогда этим будет показано, что рассматриваемые геометрии совпадают.

Пусть семейство ( X A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadIfadaahaaWcbeqaaiaadgeaaaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EAA@ представляет локальные координаты на M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ , и пусть семейство ( Y A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadMfadaahaaWcbeqaaiaadgeaaaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EAB@ соответствует локальным координатам на M R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaDaaaleaacaWGsbaabaGaam4jaaaaaaa@3A85@ . В этих координатах имеют место следующие диадные представления для градиента деформации и обратного к нему отображения:

F= F B A X A d Y B è F 1 =[ F 1 ] B A Y A d X B , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOraiaai2dacaqGgbWaa0baaSqaaiaadkeaaeaa caWGbbaaaOGaeyOaIy7aaSbaaSqaaiaadIfadaahaaqabeaacaWGbb aaaaqabaGccqGHxkcXcaWGKbGaamywamaaCaaaleqabaGaamOqaaaa kiaaywW7caqGOdGaaGzbVlaabAeadaahaaWcbeqaaiabgkHiTiaaig daaaGccaaI9aGaaG4waiaabAeadaahaaWcbeqaaiabgkHiTiaaigda aaGccaaIDbWaa0baaSqaaiaadkeaaeaacaWGbbaaaOGaeyOaIy7aaS baaSqaaiaadMfadaahaaqabeaacaWGbbaaaaqabaGccqGHxkcXcaWG KbGaamiwamaaCaaaleqabaGaamOqaaaakiaaiYcaaaa@5B62@

где F B A = X A Y B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOramaaDaaaleaacaWGcbaabaGaamyqaaaakiaa i2dadaWcaaqaaiabgkGi2kaadIfadaahaaWcbeqaaiaadgeaaaaake aacqGHciITcaWGzbWaaWbaaSqabeaacaWGcbaaaaaaaaa@41DF@ и [ F 1 ] B A = Y A X B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4waiaabAeadaahaaWcbeqaaiabgkHiTiaaigda aaGccaaIDbWaa0baaSqaaiaadkeaaeaacaWGbbaaaOGaaGypamaala aabaGaeyOaIyRaamywamaaCaaaleqabaGaamyqaaaaaOqaaiabgkGi 2kaadIfadaahaaWcbeqaaiaadkeaaaaaaaaa@458A@ . Для дальнейших рассуждений удобно иметь явные формулы для компонент обратного образа. Пусть PSec T M R T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiuaiabgIGiolaabofacaqGLbGaae4yamaabmaa baGaamivaiaad2eadaWgaaWcbaGaamOuaaqabaGccqGHxkcXcaWGub GaamytamaaBaaaleaacaWGsbaabeaaaOGaayjkaiaawMcaaaaa@4600@ , QSec T M R T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyuaiabgIGiolaabofacaqGLbGaae4yamaabmaa baGaamivamaaCaaaleqabaGaey4fIOcaaOGaamytamaaBaaaleaaca WGsbaabeaakiabgEPielaadsfadaahaaWcbeqaaiabgEHiQaaakiaa d2eadaWgaaWcbaGaamOuaaqabaaakiaawIcacaGLPaaaaaa@484D@ , R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiabgIGiodaa@3A5C@ Sec T M R T M R T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyicI4Saae4uaiaabwgacaqGJbWaaeWaaeaacaWG ubGaamytamaaBaaaleaacaWGsbaabeaakiabgEPielaadsfadaahaa WcbeqaaiabgEHiQaaakiaad2eadaWgaaWcbaGaamOuaaqabaGccqGH xkcXcaWGubWaaWbaaSqabeaacqGHxiIkaaGccaWGnbWaaSbaaSqaai aadkfaaeqaaaGccaGLOaGaayzkaaaaaa@4C3A@ и SSec T M R T M R T M R T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4uaiabgIGiolaabofacaqGLbGaae4yamaabmaa baGaamivaiaad2eadaWgaaWcbaGaamOuaaqabaGccqGHxkcXcaWGub WaaWbaaSqabeaacqGHxiIkaaGccaWGnbWaaSbaaSqaaiaadkfaaeqa aOGaey4LIqSaamivamaaCaaaleqabaGaey4fIOcaaOGaamytamaaBa aaleaacaWGsbaabeaakiabgEPielaadsfadaahaaWcbeqaaiabgEHi Qaaakiaad2eadaWgaaWcbaGaamOuaaqabaaakiaawIcacaGLPaaaaa a@52F7@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ произвольные тензорные поля. Тогда справедливы равенства:

( γ P) A B =[ F 1 ] C A [ F 1 ] D B P C D | γ() , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabeo7aNnaaCaaaleqabaGaey4fIOcaaOGa aeiuaiaaiMcadaahaaWcbeqaaiaadgeaaaGcdaahaaWcbeqaaiaadk eaaaGccaaI9aGaaG4waiaabAeadaahaaWcbeqaaiabgkHiTiaaigda aaGccaaIDbWaa0baaSqaaiaadoeaaeaacaWGbbaaaOGaaG4waiaabA eadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaaIDbWaa0baaSqaaiaa dseaaeaacaWGcbaaaOGaaeiuamaaCaaaleqabaGaam4qaaaakmaaCa aaleqabaGaamiraaaakiaaiYhadaWgaaWcbaGaeq4SdCMaaGikaiab gwSixlaaiMcaaeqaaOGaaGilaaaa@5658@ (19)

( γ Q) A B = F A C F B D Q C D | γ() , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabeo7aNnaaCaaaleqabaGaey4fIOcaaOGa aeyuaiaaiMcadaWgaaWcbaGaamyqaaqabaGcdaWgaaWcbaGaamOqaa qabaGccaaI9aGaaeOramaaDaaaleaacaWGbbaabaGaam4qaaaakiaa bAeadaqhaaWcbaGaamOqaaqaaiaadseaaaGccaqGrbWaaSbaaSqaai aadoeaaeqaaOWaaSbaaSqaaiaadseaaeqaaOGaaGiFamaaBaaaleaa cqaHZoWzcaaIOaGaeyyXICTaaGykaaqabaGccaaISaaaaa@4F00@ (20)

( γ R) B A C =[ F 1 ] D A F B E F C F R E D F | γ() , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabeo7aNnaaCaaaleqabaGaey4fIOcaaOGa aeOuaiaaiMcadaqhaaWcbaGaamOqaaqaaiaadgeaaaGcdaWgaaWcba Gaam4qaaqabaGccaaI9aGaaG4waiaabAeadaahaaWcbeqaaiabgkHi TiaaigdaaaGccaaIDbWaa0baaSqaaiaadseaaeaacaWGbbaaaOGaae OramaaDaaaleaacaWGcbaabaGaamyraaaakiaabAeadaqhaaWcbaGa am4qaaqaaiaadAeaaaGccaqGsbWaa0baaSqaaiaadweaaeaacaWGeb aaaOWaaSbaaSqaaiaadAeaaeqaaOGaaGiFamaaBaaaleaacqaHZoWz caaIOaGaeyyXICTaaGykaaqabaGccaaISaaaaa@56D9@ (21)

( γ S) B A C D =[ F 1 ] E A F B F F C G F D H S F E G H | γ() . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabeo7aNnaaCaaaleqabaGaey4fIOcaaOGa ae4uaiaaiMcadaqhaaWcbaGaamOqaaqaaiaadgeaaaGcdaWgaaWcba Gaam4qaaqabaGcdaWgaaWcbaGaamiraaqabaGccaaI9aGaaG4waiaa bAeadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaaIDbWaa0baaSqaai aadweaaeaacaWGbbaaaOGaaeOramaaDaaaleaacaWGcbaabaGaamOr aaaakiaabAeadaqhaaWcbaGaam4qaaqaaiaadEeaaaGccaqGgbWaa0 baaSqaaiaadseaaeaacaWGibaaaOGaae4uamaaDaaaleaacaWGgbaa baGaamyraaaakmaaBaaaleaacaWGhbaabeaakmaaBaaaleaacaWGib aabeaakiaaiYhadaWgaaWcbaGaeq4SdCMaaGikaiabgwSixlaaiMca aeqaaOGaaGOlaaaa@5B7B@ (22)

Все готово для следующего утверждения:

Теорема 1. Пусть S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ и S R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaqhaaWcbaGaamOuaaqaaiaadEcaaaaaaa@4518@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbcKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C75@ евклидовы формы тела B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ с подлежащими многообразиями M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ и M R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaDaaaleaacaWGsbaabaGaam4jaaaaaaa@3A85@ соответственно. Пусть γ: S R ' S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdCMaaGOoaiaaykW7tuuDJXwAK1uy0HwmaeHb fv3ySLgzG0uy0Hgip5wzaGqbaiab=jr8tnaaDaaaleaacaWGsbaaba Gaam4jaaaakiabgkziUkab=jr8tnaaBaaaleaacaWGsbaabeaaaaa@4DE3@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbcKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C75@ соответствующая деформация и пусть F=Tγ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCOraiaai2dacaWGubGaeq4SdCgaaa@3C19@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbcKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C75@ ее градиент. Если поля H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ и H ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaCaaaleqabaGaam4jaaaaaaa@39AD@ локальных деформаций, определенные на формах S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ и S R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaqhaaWcbaGaamOuaaqaaiaadEcaaaaaaa@4518@ соответственно, связаны между собой равенством H ' =HF MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaCaaaleqabaGaam4jaaaakiaai2dacaWH ibGaaCOraaaa@3C1E@ , то имеют место следующие свойства:

Материальные метрики G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ и G ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaCaaaleqabaGaam4jaaaaaaa@39A6@ , порожденные полями H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ и H ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaCaaaleqabaGaam4jaaaaaaa@39AD@ соответственно, связаны равенством G ' = γ G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaCaaaleqabaGaam4jaaaakiaai2dacqaH ZoWzdaahaaWcbeqaaiabgEHiQaaakiaabEeaaaa@3E0E@ .

В случае римановых геометрий, построенных по материальным метрикам G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ и G ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaCaaaleqabaGaam4jaaaaaaa@39A6@ , справедливы равенства ' = γ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=XrisnaaCaaaleqabaGaam4jaaaakiaai2dacqaHZoWzda ahaaWcbeqaaiabgEHiQaaakiab=Xrisbaa@4963@ между тензорами кривизны Римана, Ric ' = γ Ric MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbWaaWbaaSqabeaacaWGNaaa aOGaaGypaiabeo7aNnaaCaaaleqabaGaey4fIOcaaOGaaeOuaiaabM gacaqGJbaaaa@41C8@ между тензорами Риччи, и Scal ' =Scalγ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4uaiaabogacaqGHbGaaeiBamaaCaaaleqabaGa am4jaaaakiaai2dacaqGtbGaae4yaiaabggacaqGSbGaeSigI8Maeq 4SdCgaaa@43AC@ между скалярными кривизнами.

В случае геометрий Вайценбока, построенных по локальным деформациям H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaaaa@38CE@ и H ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaCaaaleqabaGaam4jaaaaaaa@39A7@ , имеется соотношение T ' = γ T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8unaaCaaaleqabaGaam4jaaaakiaai2dacqaHZoWzda ahaaWcbeqaaiabgEHiQaaakiab=nb8ubaa@4AF3@ между тензорами кручения. Если n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ , то справедливо дополнительное соотношение α ' = γ α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySde2aaWbaaSqabeaacaWGNaaaaOGaaGypaiab eo7aNnaaCaaaleqabaGaey4fIOcaaOGaeqySdegaaa@3FB8@ между тензорами плотности дислокаций.

Proof. Будем использовать введенные выше обозначения для локальных координат на многообразиях M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ и M R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaDaaaleaacaWGsbaabaGaam4jaaaaaaa@3A85@ . Более того, пусть ( e A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahwgadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EBA@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ некоторый базис архетипа U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFueFvaaa@436C@ . Тогда имеет место диадное представление

H= H B A e A d X B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaiaai2dacaqGibWaa0baaSqaaiaadkeaaeaa caWGbbaaaOGaaCyzamaaBaaaleaacaWGbbaabeaakiabgEPielaads gacaWGybWaaWbaaSqabeaacaWGcbaaaaaa@42D1@

для поля H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaaaa@38CE@ , что влечет равенство

H ' =HF= H B A F C B e A | γ() d Y C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaCaaaleqabaGaam4jaaaakiaai2dacaqG ibGaaeOraiaai2dacaqGibWaa0baaSqaaiaadkeaaeaacaWGbbaaaO GaaeOramaaDaaaleaacaWGdbaabaGaamOqaaaakiaahwgadaWgaaWc baGaamyqaaqabaGccaaI8bWaaSbaaSqaaiabeo7aNjaaiIcacqGHfl Y1caaIPaaabeaakiabgEPielaadsgacaWGzbWaaWbaaSqabeaacaWG dbaaaaaa@4F32@

для другого поля H ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaCaaaleqabaGaam4jaaaaaaa@39A7@ .

(a) Определение (6) материальной метрики влечет, что

G AB ' = G ' ( Y A , Y B )= g CD F A E F B F H E C H F D , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaDaaaleaacaWGbbGaamOqaaqaaiaadEca aaGccaaI9aGaae4ramaaCaaaleqabaGaam4jaaaakiaaiIcacqGHci ITdaWgaaWcbaGaamywamaaCaaabeqaaiaadgeaaaaabeaakiaaiYca caaMc8UaeyOaIy7aaSbaaSqaaiaadMfadaahaaqabeaacaWGcbaaaa qabaGccaaIPaGaaGypaiaadEgadaWgaaWcbaGaam4qaiaadseaaeqa aOGaaeOramaaDaaaleaacaWGbbaabaGaamyraaaakiaabAeadaqhaa WcbaGaamOqaaqaaiaadAeaaaGccaqGibWaa0baaSqaaiaadweaaeaa caWGdbaaaOGaaeisamaaDaaaleaacaWGgbaabaGaamiraaaakiaaiY caaaa@5698@

где g CD = e C e D MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4zamaaBaaaleaacaWGdbGaamiraaqabaGccaaI 9aGaaCyzamaaBaaaleaacaWGdbaabeaakiaaygW7cqGHflY1caaMb8 UaaCyzamaaBaaaleaacaWGebaabeaaaaa@44AA@ . Аналогично то же самое определение дает равенство

G EF = g CD H E C H F D , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaBaaaleaacaWGfbGaamOraaqabaGccaaI 9aGaam4zamaaBaaaleaacaWGdbGaamiraaqabaGccaqGibWaa0baaS qaaiaadweaaeaacaWGdbaaaOGaaeisamaaDaaaleaacaWGgbaabaGa amiraaaakiaaiYcaaaa@43F2@

поэтому

G AB ' = F A E F B F G EF | γ() . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaDaaaleaacaWGbbGaamOqaaqaaiaadEca aaGccaaI9aGaaeOramaaDaaaleaacaWGbbaabaGaamyraaaakiaabA eadaqhaaWcbaGaamOqaaqaaiaadAeaaaGccaqGhbWaaSbaaSqaaiaa dweacaWGgbaabeaakiaaiYhadaWgaaWcbaGaeq4SdCMaaGikaiabgw SixlaaiMcaaeqaaOGaaGOlaaaa@4B05@

Поскольку GSec T M R T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raiabgIGiolaabofacaqGLbGaae4yamaabmaa baGaamivamaaCaaaleqabaGaey4fIOcaaOGaamytamaaBaaaleaaca WGsbaabeaakiabgEPielaadsfadaahaaWcbeqaaiabgEHiQaaakiaa d2eadaWgaaWcbaGaamOuaaqabaaakiaawIcacaGLPaaaaaa@4843@ , выражение в правой части последнего равенства совпадает с координатным представлением (20) для обратного образа, где символ Q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyuaaaa@38D7@ нужно заменить на G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ . Приходим к искомому равенству G ' = γ G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaCaaaleqabaGaam4jaaaakiaai2dacqaH ZoWzdaahaaWcbeqaaiabgEHiQaaakiaabEeaaaa@3E0E@ .

(b) Коэффициенты (13) связности Леви-Чивита, определенные на многообразии M R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaDaaaleaacaWGsbaabaGaam4jaaaaaaa@3A85@ посредством метрики G ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaCaaaleqabaGaam4jaaaaaaa@39A6@ , равны

Γ ' I K J = G ' KL 2 Y I G JL ' + Y J G IL ' Y L G IJ ' , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aaWbaaSqabeaacaWGNaaaaOWaa0baaSqa aiaadMeaaeaacaWGlbaaaOWaaSbaaSqaaiaadQeaaeqaaOGaaGypam aalaaabaGaae4ramaaCaaaleqabaGaam4jaaaakmaaCaaaleqabaGa am4saiaadYeaaaaakeaacaaIYaaaamaabmaabaGaeyOaIy7aaSbaaS qaaiaadMfadaahaaqabeaacaWGjbaaaaqabaGccaqGhbWaa0baaSqa aiaadQeacaWGmbaabaGaam4jaaaakiabgUcaRiabgkGi2oaaBaaale aacaWGzbWaaWbaaeqabaGaamOsaaaaaeqaaOGaae4ramaaDaaaleaa caWGjbGaamitaaqaaiaadEcaaaGccqGHsislcqGHciITdaWgaaWcba GaamywamaaCaaabeqaaiaadYeaaaaabeaakiaabEeadaqhaaWcbaGa amysaiaadQeaaeaacaWGNaaaaaGccaGLOaGaayzkaaGaaGilaaaa@5A73@

где[9]

G IJ ' = F I C F J D G CD è G ' KL =[ F 1 ] S K [ F 1 ] P L G SP . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4ramaaDaaaleaacaWGjbGaamOsaaqaaiaadEca aaGccaaI9aGaaeOramaaDaaaleaacaWGjbaabaGaam4qaaaakiaabA eadaqhaaWcbaGaamOsaaqaaiaadseaaaGccaqGhbWaaSbaaSqaaiaa doeacaWGebaabeaakiaaywW7caqGOdGaaGzbVlaabEeadaahaaWcbe qaaiaadEcaaaGcdaahaaWcbeqaaiaadUeacaWGmbaaaOGaaGypaiaa iUfacaqGgbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaGyxamaaDa aaleaacaWGtbaabaGaam4saaaakiaaiUfacaqGgbWaaWbaaSqabeaa cqGHsislcaaIXaaaaOGaaGyxamaaDaaaleaacaWGqbaabaGaamitaa aakiaabEeadaahaaWcbeqaaiaadofacaWGqbaaaOGaaGOlaaaa@5CB0@

Используя свойство коммутативности смешанных частных производных, получаем

Y I F J K = 2 X K Y I Y J = Y J F I K . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOaIy7aaSbaaSqaaiaadMfadaahaaqabeaacaWG jbaaaaqabaGccaqGgbWaa0baaSqaaiaadQeaaeaacaWGlbaaaOGaaG ypamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamiwamaa CaaaleqabaGaam4saaaaaOqaaiabgkGi2kaadMfadaahaaWcbeqaai aadMeaaaGccqGHciITcaWGzbWaaWbaaSqabeaacaWGkbaaaaaakiaa i2dacqGHciITdaWgaaWcbaGaamywamaaCaaabeqaaiaadQeaaaaabe aakiaabAeadaqhaaWcbaGaamysaaqaaiaadUeaaaGccaaIUaaaaa@513B@

По этой причине

Γ ' I K J = G SP 2 [ F 1 ] S K F J C Y I G CP + [ F 1 ] S K F I C Y J G CP MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aaWbaaSqabeaacaWGNaaaaOWaa0baaSqa aiaadMeaaeaacaWGlbaaaOWaaSbaaSqaaiaadQeaaeqaaOGaaGypam aalaaabaGaae4ramaaCaaaleqabaGaam4uaiaadcfaaaaakeaacaaI YaaaamaabeaabaGaaG4waiaabAeadaahaaWcbeqaaiabgkHiTiaaig daaaGccaaIDbWaa0baaSqaaiaadofaaeaacaWGlbaaaOGaaeOramaa DaaaleaacaWGkbaabaGaam4qaaaakiabgkGi2oaaBaaaleaacaWGzb WaaWbaaeqabaGaamysaaaaaeqaaOGaae4ramaaBaaaleaacaWGdbGa amiuaaqabaGccqGHRaWkcaaIBbGaaeOramaaCaaaleqabaGaeyOeI0 IaaGymaaaakiaai2fadaqhaaWcbaGaam4uaaqaaiaadUeaaaGccaqG gbWaa0baaSqaaiaadMeaaeaacaWGdbaaaOGaeyOaIy7aaSbaaSqaai aadMfadaahaaqabeaacaWGkbaaaaqabaGccaqGhbWaaSbaaSqaaiaa doeacaWGqbaabeaakiabgkHiTaGaayjkaaaaaa@61E0@

[ F 1 ] S K [ F 1 ] P L F I C F J D Y L G CD + [ F 1 ] S K Y I F J S . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaeGaaeaacqGHsislcaaIBbGaaeOramaaCaaaleqa baGaeyOeI0IaaGymaaaakiaai2fadaqhaaWcbaGaam4uaaqaaiaadU eaaaGccaaIBbGaaeOramaaCaaaleqabaGaeyOeI0IaaGymaaaakiaa i2fadaqhaaWcbaGaamiuaaqaaiaadYeaaaGccaqGgbWaa0baaSqaai aadMeaaeaacaWGdbaaaOGaaeOramaaDaaaleaacaWGkbaabaGaamir aaaakiabgkGi2oaaBaaaleaacaWGzbWaaWbaaeqabaGaamitaaaaae qaaOGaae4ramaaBaaaleaacaWGdbGaamiraaqabaaakiaawMcaaiab gUcaRiaaiUfacaqGgbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaG yxamaaDaaaleaacaWGtbaabaGaam4saaaakiabgkGi2oaaBaaaleaa caWGzbWaaWbaaeqabaGaamysaaaaaeqaaOGaaeOramaaDaaaleaaca WGkbaabaGaam4uaaaakiaai6caaaa@5F86@

Поскольку

Y I G JK = Y I ( G JK γ)= F I O X O G CP , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOaIy7aaSbaaSqaaiaadMfadaahaaqabeaacaWG jbaaaaqabaGccaqGhbWaaSbaaSqaaiaadQeacaWGlbaabeaakiaai2 dacqGHciITdaWgaaWcbaGaamywamaaCaaabeqaaiaadMeaaaaabeaa kiaaiIcacaqGhbWaaSbaaSqaaiaadQeacaWGlbaabeaakiablIHiVj abeo7aNjaaiMcacaaI9aGaaeOramaaDaaaleaacaWGjbaabaGaam4t aaaakiabgkGi2oaaBaaaleaacaWGybWaaWbaaeqabaGaam4taaaaae qaaOGaae4ramaaBaaaleaacaWGdbGaamiuaaqabaGccaaISaaaaa@534D@

приходим к равенству

Γ ' I K J =[ F 1 ] S K F I O F J C Γ O S C | γ() + [ F 1 ] S K Y I F J S , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aaWbaaSqabeaacaWGNaaaaOWaa0baaSqa aiaadMeaaeaacaWGlbaaaOWaaSbaaSqaaiaadQeaaeqaaOGaaGypai aaiUfacaqGgbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaGyxamaa DaaaleaacaWGtbaabaGaam4saaaakiaabAeadaqhaaWcbaGaamysaa qaaiaad+eaaaGccaqGgbWaa0baaSqaaiaadQeaaeaacaWGdbaaaOGa eu4KdC0aa0baaSqaaiaad+eaaeaacaWGtbaaaOWaaSbaaSqaaiaado eaaeqaaOGaaGiFamaaBaaaleaacqaHZoWzcaaIOaGaeyyXICTaaGyk aaqabaGccqGHRaWkcaaIBbGaaeOramaaCaaaleqabaGaeyOeI0IaaG ymaaaakiaai2fadaqhaaWcbaGaam4uaaqaaiaadUeaaaGccqGHciIT daWgaaWcbaGaamywamaaCaaabeqaaiaadMeaaaaabeaakiaabAeada qhaaWcbaGaamOsaaqaaiaadofaaaGccaaISaaaaa@6252@ (23)

где скалярные поля Γ O S C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaad+eaaeaacaWGtbaaaOWa aSbaaSqaaiaadoeaaeqaaaaa@3C42@ соответствуют коэффициентам связности Леви-Чивита на M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ , порожденным материальной метрикой G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ .

Отметим сходство полученного соотношения для коэффициентов связности на M R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaDaaaleaacaWGsbaabaGaam4jaaaaaaa@3A85@ с законом преобразования (9) коэффициентов связности при замене координат. Действительно, с точностью до композиций с координатными отображениями деформацию γ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdCgaaa@39AA@ можно рассматривать как замену локальных координат в M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ . Вместе с тем в рассматриваемом случае имеется большее, чем просто преобразование координат, поскольку формы также преобразуются. В этом причина, почему прямые соотношения между <<штрихованными>> и <<нештрихованными>> полями не могут быть получены без использования преобразования обратного образа.

Подставляя полученное соотношение (23) для коэффициентов связности в координатное представление (14) тензора кривизны, получаем

' I T J K = Y I Γ ' J T K Y J Γ ' I T K + Γ ' J L K Γ ' I T L Γ ' I L K Γ ' J T L = = Y I [ F 1 ] S T F J O F K C Γ O S C + [ F 1 ] S T Y J F K S Y J [ F 1 ] S T F I O F K C Γ O S C + [ F 1 ] S T Y I F K S + + [ F 1 ] S L F J O F K C Γ O S C + [ F 1 ] S L Y I F K S [ F 1 ] B T F I A F L D Γ A B D + [ F 1 ] B T Y I F L B [ F 1 ] S L F I O F K C Γ O S C + [ F 1 ] S L Y I F K S [ F 1 ] B T F J A F L D Γ A B D + [ F 1 ] B T Y J F L B . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaqbaeaabqqaaaaabaWefv3ySLgzgjxyRrxDYbqeguuD JXwAKbIrYf2A0vNCaGqbaiab=XrisnaaCaaaleqabaGaam4jaaaakm aaDaaaleaacaWGjbaabaGaamivaaaakmaaBaaaleaacaWGkbaabeaa kmaaBaaaleaacaWGlbaabeaakiaai2dacqGHciITdaWgaaWcbaGaam ywamaaCaaabeqaaiaadMeaaaaabeaakiabfo5ahnaaCaaaleqabaGa am4jaaaakmaaDaaaleaacaWGkbaabaGaamivaaaakmaaBaaaleaaca WGlbaabeaakiabgkHiTiabgkGi2oaaBaaaleaacaWGzbWaaWbaaeqa baGaamOsaaaaaeqaaOGaeu4KdC0aaWbaaSqabeaacaWGNaaaaOWaa0 baaSqaaiaadMeaaeaacaWGubaaaOWaaSbaaSqaaiaadUeaaeqaaOGa ey4kaSIaeu4KdC0aaWbaaSqabeaacaWGNaaaaOWaa0baaSqaaiaadQ eaaeaacaWGmbaaaOWaaSbaaSqaaiaadUeaaeqaaOGaeu4KdC0aaWba aSqabeaacaWGNaaaaOWaa0baaSqaaiaadMeaaeaacaWGubaaaOWaaS baaSqaaiaadYeaaeqaaOGaeyOeI0Iaeu4KdC0aaWbaaSqabeaacaWG NaaaaOWaa0baaSqaaiaadMeaaeaacaWGmbaaaOWaaSbaaSqaaiaadU eaaeqaaOGaeu4KdC0aaWbaaSqabeaacaWGNaaaaOWaa0baaSqaaiaa dQeaaeaacaWGubaaaOWaaSbaaSqaaiaadYeaaeqaaOGaaGypaaqaai aai2dacqGHciITdaWgaaWcbaGaamywamaaCaaabeqaaiaadMeaaaaa beaakmaabmaabaGaaG4waiaabAeadaahaaWcbeqaaiabgkHiTiaaig daaaGccaaIDbWaa0baaSqaaiaadofaaeaacaWGubaaaOGaaeOramaa DaaaleaacaWGkbaabaGaam4taaaakiaabAeadaqhaaWcbaGaam4saa qaaiaadoeaaaGccqqHtoWrdaqhaaWcbaGaam4taaqaaiaadofaaaGc daWgaaWcbaGaam4qaaqabaGccqGHRaWkcaaIBbGaaeOramaaCaaale qabaGaeyOeI0IaaGymaaaakiaai2fadaqhaaWcbaGaam4uaaqaaiaa dsfaaaGccqGHciITdaWgaaWcbaGaamywamaaCaaabeqaaiaadQeaaa aabeaakiaabAeadaqhaaWcbaGaam4saaqaaiaadofaaaaakiaawIca caGLPaaacqGHsislcqGHciITdaWgaaWcbaGaamywamaaCaaabeqaai aadQeaaaaabeaakmaabmaabaGaaG4waiaabAeadaahaaWcbeqaaiab gkHiTiaaigdaaaGccaaIDbWaa0baaSqaaiaadofaaeaacaWGubaaaO GaaeOramaaDaaaleaacaWGjbaabaGaam4taaaakiaabAeadaqhaaWc baGaam4saaqaaiaadoeaaaGccqqHtoWrdaqhaaWcbaGaam4taaqaai aadofaaaGcdaWgaaWcbaGaam4qaaqabaGccqGHRaWkcaaIBbGaaeOr amaaCaaaleqabaGaeyOeI0IaaGymaaaakiaai2fadaqhaaWcbaGaam 4uaaqaaiaadsfaaaGccqGHciITdaWgaaWcbaGaamywamaaCaaabeqa aiaadMeaaaaabeaakiaabAeadaqhaaWcbaGaam4saaqaaiaadofaaa aakiaawIcacaGLPaaacqGHRaWkaeaacqGHRaWkdaqadaqaaiaaiUfa caqGgbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaGyxamaaDaaale aacaWGtbaabaGaamitaaaakiaabAeadaqhaaWcbaGaamOsaaqaaiaa d+eaaaGccaqGgbWaa0baaSqaaiaadUeaaeaacaWGdbaaaOGaeu4KdC 0aa0baaSqaaiaad+eaaeaacaWGtbaaaOWaaSbaaSqaaiaadoeaaeqa aOGaey4kaSIaaG4waiaabAeadaahaaWcbeqaaiabgkHiTiaaigdaaa GccaaIDbWaa0baaSqaaiaadofaaeaacaWGmbaaaOGaeyOaIy7aaSba aSqaaiaadMfadaahaaqabeaacaWGjbaaaaqabaGccaqGgbWaa0baaS qaaiaadUeaaeaacaWGtbaaaaGccaGLOaGaayzkaaWaaeWaaeaacaaI BbGaaeOramaaCaaaleqabaGaeyOeI0IaaGymaaaakiaai2fadaqhaa WcbaGaamOqaaqaaiaadsfaaaGccaqGgbWaa0baaSqaaiaadMeaaeaa caWGbbaaaOGaaeOramaaDaaaleaacaWGmbaabaGaamiraaaakiabfo 5ahnaaDaaaleaacaWGbbaabaGaamOqaaaakmaaBaaaleaacaWGebaa beaakiabgUcaRiaaiUfacaqGgbWaaWbaaSqabeaacqGHsislcaaIXa aaaOGaaGyxamaaDaaaleaacaWGcbaabaGaamivaaaakiabgkGi2oaa BaaaleaacaWGzbWaaWbaaeqabaGaamysaaaaaeqaaOGaaeOramaaDa aaleaacaWGmbaabaGaamOqaaaaaOGaayjkaiaawMcaaiabgkHiTaqa aiabgkHiTmaabmaabaGaaG4waiaabAeadaahaaWcbeqaaiabgkHiTi aaigdaaaGccaaIDbWaa0baaSqaaiaadofaaeaacaWGmbaaaOGaaeOr amaaDaaaleaacaWGjbaabaGaam4taaaakiaabAeadaqhaaWcbaGaam 4saaqaaiaadoeaaaGccqqHtoWrdaqhaaWcbaGaam4taaqaaiaadofa aaGcdaWgaaWcbaGaam4qaaqabaGccqGHRaWkcaaIBbGaaeOramaaCa aaleqabaGaeyOeI0IaaGymaaaakiaai2fadaqhaaWcbaGaam4uaaqa aiaadYeaaaGccqGHciITdaWgaaWcbaGaamywamaaCaaabeqaaiaadM eaaaaabeaakiaabAeadaqhaaWcbaGaam4saaqaaiaadofaaaaakiaa wIcacaGLPaaadaqadaqaaiaaiUfacaqGgbWaaWbaaSqabeaacqGHsi slcaaIXaaaaOGaaGyxamaaDaaaleaacaWGcbaabaGaamivaaaakiaa bAeadaqhaaWcbaGaamOsaaqaaiaadgeaaaGccaqGgbWaa0baaSqaai aadYeaaeaacaWGebaaaOGaeu4KdC0aa0baaSqaaiaadgeaaeaacaWG cbaaaOWaaSbaaSqaaiaadseaaeqaaOGaey4kaSIaaG4waiaabAeada ahaaWcbeqaaiabgkHiTiaaigdaaaGccaaIDbWaa0baaSqaaiaadkea aeaacaWGubaaaOGaeyOaIy7aaSbaaSqaaiaadMfadaahaaqabeaaca WGkbaaaaqabaGccaqGgbWaa0baaSqaaiaadYeaaeaacaWGcbaaaaGc caGLOaGaayzkaaGaaGOlaaaaaaa@373E@ (24)

Используя формулу

Y I [ F 1 ] L M = [ F 1 ] N M [ F 1 ] L K Y I F K N , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOaIy7aaSbaaSqaaiaadMfadaahaaqabeaacaWG jbaaaaqabaGccaaIBbGaaeOramaaCaaaleqabaGaeyOeI0IaaGymaa aakiaai2fadaqhaaWcbaGaamitaaqaaiaad2eaaaGccaaI9aGaeyOe I0IaaG4waiaabAeadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaaIDb Waa0baaSqaaiaad6eaaeaacaWGnbaaaOGaaG4waiaabAeadaahaaWc beqaaiabgkHiTiaaigdaaaGccaaIDbWaa0baaSqaaiaadYeaaeaaca WGlbaaaOGaeyOaIy7aaSbaaSqaaiaadMfadaahaaqabeaacaWGjbaa aaqabaGccaqGgbWaa0baaSqaaiaadUeaaeaacaWGobaaaOGaaGilaa aa@56CE@

которая вытекает из дифференцирования обеих частей соотношения [ F 1 ] N M | γ() F K N = δ K M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4waiaabAeadaahaaWcbeqaaiabgkHiTiaaigda aaGccaaIDbWaa0baaSqaaiaad6eaaeaacaWGnbaaaOGaaGiFamaaBa aaleaacqaHZoWzcaaIOaGaeyyXICTaaGykaaqabaGccaaMi8UaaeOr amaaDaaaleaacaWGlbaabaGaamOtaaaakiaai2dacqaH0oazdaqhaa WcbaGaam4saaqaaiaad2eaaaaaaa@4D54@ по переменным Y I MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamywamaaCaaaleqabaGaamysaaaaaaa@39DC@ , и равенство

Y I Γ O S C = F I P X P Γ O S C , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOaIy7aaSbaaSqaaiaadMfadaahaaqabeaacaWG jbaaaaqabaGccqqHtoWrdaqhaaWcbaGaam4taaqaaiaadofaaaGcda WgaaWcbaGaam4qaaqabaGccaaI9aGaaeOramaaDaaaleaacaWGjbaa baGaamiuaaaakiabgkGi2oaaBaaaleaacaWGybWaaWbaaeqabaGaam iuaaaaaeqaaOGaeu4KdC0aa0baaSqaaiaad+eaaeaacaWGtbaaaOWa aSbaaSqaaiaadoeaaeqaaOGaaGilaaaa@4B8F@

которое следует из цепного правила дифференцирования, после сокращения соответствующих слагаемых из (24) получаем, что

' I T J K =[ F 1 ] S T F I P F J O F K C P O C S | γ() . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=XrisnaaCaaaleqabaGaam4jaaaakmaaDaaaleaacaWGjb aabaGaamivaaaakmaaBaaaleaacaWGkbaabeaakmaaBaaaleaacaWG lbaabeaakiaai2dacaaIBbGaaeOramaaCaaaleqabaGaeyOeI0IaaG ymaaaakiaai2fadaqhaaWcbaGaam4uaaqaaiaadsfaaaGccaqGgbWa a0baaSqaaiaadMeaaeaacaWGqbaaaOGaaeOramaaDaaaleaacaWGkb aabaGaam4taaaakiaabAeadaqhaaWcbaGaam4saaqaaiaadoeaaaGc cqWFCeIudaWgaaWcbaGaamiuaaqabaGcdaWgaaWcbaGaam4taaqaba GcdaqhaaWcbaGaam4qaaqaaiaadofaaaGccaaI8bWaaSbaaSqaaiab eo7aNjaaiIcacqGHflY1caaIPaaabeaakiaai6caaaa@63EF@

Сравнивая полученное соотношение с формулой (22), в которой S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4uaaaa@38D9@ следует заменить на MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Xrisbaa@43D8@ , приходим к выводу, что ' = γ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=XrisnaaCaaaleqabaGaam4jaaaakiaai2dacqaHZoWzda ahaaWcbeqaaiabgEHiQaaakiab=Xrisbaa@4963@ .

Для тензора кривизны Риччи имеем равенства

Ric IK ' = ' I L L K = F I P F K C P S S C = F I P F K C Ric P C | γ() . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbWaa0baaSqaaiaadMeacaWG lbaabaGaam4jaaaakiaai2datuuDJXwAKzKCHTgD1jharyqr1ngBPr gigjxyRrxDYbacfaGae8hhHi1aaWbaaSqabeaacaWGNaaaaOWaa0ba aSqaaiaadMeaaeaacaWGmbaaaOWaaSbaaSqaaiaadYeaaeqaaOWaaS baaSqaaiaadUeaaeqaaOGaaGypaiaabAeadaqhaaWcbaGaamysaaqa aiaadcfaaaGccaqGgbWaa0baaSqaaiaadUeaaeaacaWGdbaaaOGae8 hhHi1aa0baaSqaaiaadcfaaeaacaWGtbaaaOWaaSbaaSqaaiaadofa aeqaaOWaaSbaaSqaaiaadoeaaeqaaOGaaGypaiaabAeadaqhaaWcba GaamysaaqaaiaadcfaaaGccaqGgbWaa0baaSqaaiaadUeaaeaacaWG dbaaaOGaaeOuaiaabMgacaqGJbWaaSbaaSqaaiaadcfaaeqaaOWaaS baaSqaaiaadoeaaeqaaOGaaGiFamaaBaaaleaacqaHZoWzcaaIOaGa eyyXICTaaGykaaqabaGccaaIUaaaaa@6B8F@

Таким образом, приходим к соотношению

Ric IK ' = F I P F K C Ric P C | γ() . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbWaa0baaSqaaiaadMeacaWG lbaabaGaam4jaaaakiaai2dacaqGgbWaa0baaSqaaiaadMeaaeaaca WGqbaaaOGaaeOramaaDaaaleaacaWGlbaabaGaam4qaaaakiaabkfa caqGPbGaae4yamaaBaaaleaacaWGqbaabeaakmaaBaaaleaacaWGdb aabeaakiaaiYhadaWgaaWcbaGaeq4SdCMaaGikaiabgwSixlaaiMca aeqaaOGaaGOlaaaa@4F27@

Поскольку RicSec T M R T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbGaeyicI4Saae4uaiaabwga caqGJbWaaeWaaeaacaWGubWaaWbaaSqabeaacqGHxiIkaaGccaWGnb WaaSbaaSqaaiaadkfaaeqaaOGaey4LIqSaamivamaaCaaaleqabaGa ey4fIOcaaOGaamytamaaBaaaleaacaWGsbaabeaaaOGaayjkaiaawM caaaaa@4A20@ , выражение в правой части полученного равенства совпадает с координатным представлением (20) обратного образа, в котором тензорное поле Q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyuaaaa@38D7@ нужно заменить на Ric MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbaaaa@3AAA@ . Тем самым равенство Ric ' = γ Ric MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaiaabMgacaqGJbWaaWbaaSqabeaacaWGNaaa aOGaaGypaiabeo7aNnaaCaaaleqabaGaey4fIOcaaOGaaeOuaiaabM gacaqGJbaaaa@41C8@ установлено.

Наконец, получим соотношение между скалярными кривизнами:

Scal ' = G ' AB Ric AB ' =[ F 1 ] S A [ F 1 ] P B G SP F A L F B M Ric L M | γ() = G LM Ric L M | γ() =Scalγ, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4uaiaabogacaqGHbGaaeiBamaaCaaaleqabaGa am4jaaaakiaai2dacaqGhbWaaWbaaSqabeaacaWGNaaaaOWaaWbaaS qabeaacaWGbbGaamOqaaaakiaabkfacaqGPbGaae4yamaaDaaaleaa caWGbbGaamOqaaqaaiaadEcaaaGccaaI9aGaaG4waiaabAeadaahaa WcbeqaaiabgkHiTiaaigdaaaGccaaIDbWaa0baaSqaaiaadofaaeaa caWGbbaaaOGaaG4waiaabAeadaahaaWcbeqaaiabgkHiTiaaigdaaa GccaaIDbWaa0baaSqaaiaadcfaaeaacaWGcbaaaOGaae4ramaaCaaa leqabaGaam4uaiaadcfaaaGccaqGgbWaa0baaSqaaiaadgeaaeaaca WGmbaaaOGaaeOramaaDaaaleaacaWGcbaabaGaamytaaaakiaabkfa caqGPbGaae4yamaaBaaaleaacaWGmbaabeaakmaaBaaaleaacaWGnb aabeaakiaaiYhadaWgaaWcbaGaeq4SdCMaaGikaiabgwSixlaaiMca aeqaaOGaaGypaiaabEeadaahaaWcbeqaaiaadYeacaWGnbaaaOGaae OuaiaabMgacaqGJbWaaSbaaSqaaiaadYeaaeqaaOWaaSbaaSqaaiaa d2eaaeqaaOGaaGiFamaaBaaaleaacqaHZoWzcaaIOaGaeyyXICTaaG ykaaqabaGccaaI9aGaae4uaiaabogacaqGHbGaaeiBaiablIHiVjab eo7aNjaaiYcaaaa@7CE1@

как и ожидалось.

(c) В этом случае имеются равенство H= H B A c A d X B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaiaai2dacaqGibWaa0baaSqaaiaadkeaaeaa caWGbbaaaOGaaC4yamaaBaaaleaacaWGbbaabeaakiabgEPielaads gacaWGybWaaWbaaSqabeaacaWGcbaaaaaa@42CF@ и соответствующее разложение H ' = H B A F C B c A d Y B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaCaaaleqabaGaam4jaaaakiaai2dacaqG ibWaa0baaSqaaiaadkeaaeaacaWGbbaaaOGaaeOramaaDaaaleaaca WGdbaabaGaamOqaaaakiaahogadaWgaaWcbaGaamyqaaqabaGccqGH xkcXcaWGKbGaamywamaaCaaaleqabaGaamOqaaaaaaa@4642@ . В координатном репере ( Y A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabgkGi2oaaBaaaleaacaWGzbWaaWbaaeqa baGaamyqaaaaaeqaaOGaaGykamaaDaaaleaacaWGbbGaaGypaiaaig daaeaacaWGUbaaaaaa@4032@ коэффициенты (15) связности Вайценбока на многообразии M R ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaDaaaleaacaWGsbaabaGaam4jaaaaaaa@3A85@ представлены выражением Γ ' J I K =[ H ' 1 ] C I Y J H ' K C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aaWbaaSqabeaacaWGNaaaaOWaa0baaSqa aiaadQeaaeaacaWGjbaaaOWaaSbaaSqaaiaadUeaaeqaaOGaaGypai aaiUfacaqGibWaaWbaaSqabeaacaWGNaaaaOWaaWbaaSqabeaacqGH sislcaaIXaaaaOGaaGyxamaaDaaaleaacaWGdbaabaGaamysaaaaki abgkGi2oaaBaaaleaacaWGzbWaaWbaaeqabaGaamOsaaaaaeqaaOGa aeisamaaCaaaleqabaGaam4jaaaakmaaDaaaleaacaWGlbaabaGaam 4qaaaaaaa@4BF3@ . Здесь

Y J H ' K C = F K L F J M X M H L C + H L C Y J F L K , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOaIy7aaSbaaSqaaiaadMfadaahaaqabeaacaWG kbaaaaqabaGccaqGibWaaWbaaSqabeaacaWGNaaaaOWaa0baaSqaai aadUeaaeaacaWGdbaaaOGaaGypaiaabAeadaqhaaWcbaGaam4saaqa aiaadYeaaaGccaqGgbWaa0baaSqaaiaadQeaaeaacaWGnbaaaOGaey OaIy7aaSbaaSqaaiaadIfadaahaaqabeaacaWGnbaaaaqabaGccaqG ibWaa0baaSqaaiaadYeaaeaacaWGdbaaaOGaey4kaSIaaeisamaaDa aaleaacaWGmbaabaGaam4qaaaakiabgkGi2oaaBaaaleaacaWGzbWa aWbaaeqabaGaamOsaaaaaeqaaOGaaeOramaaDaaaleaacaWGmbaaba Gaam4saaaakiaaiYcaaaa@553B@

поскольку Y J H L C = F J M X M H L C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOaIy7aaSbaaSqaaiaadMfadaahaaqabeaacaWG kbaaaaqabaGccaqGibWaa0baaSqaaiaadYeaaeaacaWGdbaaaOGaaG ypaiaabAeadaqhaaWcbaGaamOsaaqaaiaad2eaaaGccqGHciITdaWg aaWcbaGaamiwamaaCaaabeqaaiaad2eaaaaabeaakiaabIeadaqhaa WcbaGaamitaaqaaiaadoeaaaaaaa@476F@ . Тогда приходим к формуле

Γ ' J I K =[ F 1 ] R I F K L F J M Γ M R L + [ F 1 ] R I Y J F K R , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aaWbaaSqabeaacaWGNaaaaOWaa0baaSqa aiaadQeaaeaacaWGjbaaaOWaaSbaaSqaaiaadUeaaeqaaOGaaGypai aaiUfacaqGgbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaGyxamaa DaaaleaacaWGsbaabaGaamysaaaakiaabAeadaqhaaWcbaGaam4saa qaaiaadYeaaaGccaqGgbWaa0baaSqaaiaadQeaaeaacaWGnbaaaOGa eu4KdC0aa0baaSqaaiaad2eaaeaacaWGsbaaaOWaaSbaaSqaaiaadY eaaeqaaOGaey4kaSIaaG4waiaabAeadaahaaWcbeqaaiabgkHiTiaa igdaaaGccaaIDbWaa0baaSqaaiaadkfaaeaacaWGjbaaaOGaeyOaIy 7aaSbaaSqaaiaadMfadaahaaqabeaacaWGkbaaaaqabaGccaqGgbWa a0baaSqaaiaadUeaaeaacaWGsbaaaOGaaGilaaaa@5BCA@

в которой Γ M R L MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaad2eaaeaacaWGsbaaaOWa aSbaaSqaaiaadYeaaeqaaaaa@3C48@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ коэффициенты связности Вайценбока на многообразии M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ . Соотношение (16) для компонент тензора кручения, таким образом, принимает вид

null

Сравнивая его с выражением (21), в котором R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOuaaaa@38D8@ следует заменить на T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ubaa@44A0@ , получаем T ' = γ T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8unaaCaaaleqabaGaam4jaaaakiaai2dacqaHZoWzda ahaaWcbeqaaiabgEHiQaaakiab=nb8ubaa@4AF3@ .

Пусть n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ . Компоненты тензора плотности дислокаций α ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySde2aaWbaaSqabeaacaWGNaaaaaaa@3A7B@ определены равенством (17): α ' IJ = 1 2 ε ' IMN T ' M J N MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySde2aaWbaaSqabeaacaWGNaaaaOWaaWbaaSqa beaacaWGjbGaamOsaaaakiaai2dacqGHsisldaWcaaqaaiaaigdaae aacaaIYaaaaiabew7aLnaaCaaaleqabaGaam4jaaaakmaaCaaaleqa baGaamysaiaad2eacaWGobaaamrr1ngBPrMrYf2A0vNCaeHbfv3ySL gzGyKCHTgD1jhaiuaakiab=nb8unaaCaaaleqabaGaam4jaaaakmaa DaaaleaacaWGnbaabaGaamOsaaaakmaaBaaaleaacaWGobaabeaaaa a@551F@ , где ε ' IMN = e IMN det G ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTdu2aaWbaaSqabeaacaWGNaaaaOWaaWbaaSqa beaacaWGjbGaamytaiaad6eaaaGccaaI9aWaaSaaaeaacaWGLbWaaW baaSqabeaacaWGjbGaamytaiaad6eaaaaakeaadaGcaaqaaiGacsga caGGLbGaaiiDaiaabEeadaahaaWcbeqaaiaadEcaaaaabeaaaaaaaa@4620@ . Определитель det G ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaciizaiaacwgacaGG0bGaae4ramaaCaaaleqabaGa am4jaaaaaaa@3C71@ связан с определителем detG MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaciizaiaacwgacaGG0bGaae4raaaa@3B98@ по формуле det G ' = Δ 2 detG MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaciizaiaacwgacaGG0bGaae4ramaaCaaaleqabaGa am4jaaaakiaai2dacqqHuoardaahaaWcbeqaaiaaikdaaaGcciGGKb GaaiyzaiaacshacaqGhbaaaa@4330@ , в которой[10] Δ=detF>0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuiLdqKaaGypaiGacsgacaGGLbGaaiiDaiaabAea caaI+aGaaGimaaaa@3F46@ . Подставляя соотношения для T ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8unaaCaaaleqabaGaam4jaaaaaaa@4579@ и det G ' MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaciizaiaacwgacaGG0bGaae4ramaaCaaaleqabaGa am4jaaaaaaa@3C71@ в α ' IJ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySde2aaWbaaSqabeaacaWGNaaaaOWaaWbaaSqa beaacaWGjbGaamOsaaaaaaa@3C4F@ , получаем

α ' IJ = 1 2 e IMN detG [ F 1 ] R J F M A F N B T A R B Δ . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySde2aaWbaaSqabeaacaWGNaaaaOWaaWbaaSqa beaacaWGjbGaamOsaaaakiaai2dacqGHsisldaWcaaqaaiaaigdaae aacaaIYaaaamaalaaabaGaamyzamaaCaaaleqabaGaamysaiaad2ea caWGobaaaaGcbaWaaOaaaeaaciGGKbGaaiyzaiaacshacaqGhbaale qaaaaakmaalaaabaGaaG4waiaabAeadaahaaWcbeqaaiabgkHiTiaa igdaaaGccaaIDbWaa0baaSqaaiaadkfaaeaacaWGkbaaaOGaaeOram aaDaaaleaacaWGnbaabaGaamyqaaaakiaabAeadaqhaaWcbaGaamOt aaqaaiaadkeaaaWefv3ySLgzgjxyRrxDYbqeguuDJXwAKbIrYf2A0v NCaGqbaOGae83eWt1aa0baaSqaaiaadgeaaeaacaWGsbaaaOWaaSba aSqaaiaadkeaaeqaaaGcbaGaeuiLdqeaaiaai6caaaa@6411@ (25)

Для того чтобы преобразовать полученное выражение к иному виду, вначале заметим, что T A R B = 1 2 T S R L ( δ A S δ B L δ A L δ B S ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8unaaDaaaleaacaWGbbaabaGaamOuaaaakmaaBaaale aacaWGcbaabeaakiaai2dadaWcaaqaaiaaigdaaeaacaaIYaaaaiab =nb8unaaDaaaleaacaWGtbaabaGaamOuaaaakmaaBaaaleaacaWGmb aabeaakiaaiIcacqaH0oazdaqhaaWcbaGaamyqaaqaaiaadofaaaGc cqaH0oazdaqhaaWcbaGaamOqaaqaaiaadYeaaaGccqGHsislcqaH0o azdaqhaaWcbaGaamyqaaqaaiaadYeaaaGccqaH0oazdaqhaaWcbaGa amOqaaqaaiaadofaaaGccaaIPaaaaa@5EB6@ . Вместе с тем[11] δ A S δ B L δ A L δ B S = e TAB e TSL MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiTdq2aa0baaSqaaiaadgeaaeaacaWGtbaaaOGa eqiTdq2aa0baaSqaaiaadkeaaeaacaWGmbaaaOGaeyOeI0IaeqiTdq 2aa0baaSqaaiaadgeaaeaacaWGmbaaaOGaeqiTdq2aa0baaSqaaiaa dkeaaeaacaWGtbaaaOGaaGypaiaadwgadaWgaaWcbaGaamivaiaadg eacaWGcbaabeaakiaadwgadaahaaWcbeqaaiaadsfacaWGtbGaamit aaaaaaa@4EB2@ , и тогда получаем соотношение T A R B = 1 2 e TAB e TSL T S R L MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8unaaDaaaleaacaWGbbaabaGaamOuaaaakmaaBaaale aacaWGcbaabeaakiaai2dadaWcaaqaaiaaigdaaeaacaaIYaaaaiaa dwgadaWgaaWcbaGaamivaiaadgeacaWGcbaabeaakiaadwgadaahaa WcbeqaaiaadsfacaWGtbGaamitaaaakiab=nb8unaaDaaaleaacaWG tbaabaGaamOuaaaakmaaBaaaleaacaWGmbaabeaaaaa@55A7@ . Учитывая его в (25) и принимая во внимание, что [ F 1 ] T I = 1 2Δ e IMN e tab F M A F N B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4waiaabAeadaahaaWcbeqaaiabgkHiTiaaigda aaGccaaIDbWaa0baaSqaaiaadsfaaeaacaWGjbaaaOGaaGypamaala aabaGaaGymaaqaaiaaikdacqqHuoaraaGaamyzamaaCaaaleqabaGa amysaiaad2eacaWGobaaaOGaamyzamaaBaaaleaacaWG0bGaamyyai aadkgaaeqaaOGaaeOramaaDaaaleaacaWGnbaabaGaamyqaaaakiaa bAeadaqhaaWcbaGaamOtaaqaaiaadkeaaaaaaa@4EAB@ , приходим к равенству

α ' IJ =[ F 1 ] T I [ F 1 ] R J α TR | γ() , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySde2aaWbaaSqabeaacaWGNaaaaOWaaWbaaSqa beaacaWGjbGaamOsaaaakiaai2dacaaIBbGaaeOramaaCaaaleqaba GaeyOeI0IaaGymaaaakiaai2fadaqhaaWcbaGaamivaaqaaiaadMea aaGccaaIBbGaaeOramaaCaaaleqabaGaeyOeI0IaaGymaaaakiaai2 fadaqhaaWcbaGaamOuaaqaaiaadQeaaaGccqaHXoqydaahaaWcbeqa aiaadsfacaWGsbaaaOGaaGiFamaaBaaaleaacqaHZoWzcaaIOaGaey yXICTaaGykaaqabaGccaaISaaaaa@5491@

где α TR MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySde2aaWbaaSqabeaacaWGubGaamOuaaaaaaa@3B7F@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ компоненты тензора плотности дислокаций αSec T M R T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySdeMaeyicI4Saae4uaiaabwgacaqGJbWaaeWa aeaacaWGubGaamytamaaBaaaleaacaWGsbaabeaakiabgEPielaads facaWGnbWaaSbaaSqaaiaadkfaaeqaaaGccaGLOaGaayzkaaaaaa@46CC@ . Сравнивая полученное выражение с формулой (19), в которой P MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiuaaaa@38D6@ следует заменить на α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySdegaaa@39A2@ , окончательно получаем, что α ' = γ α MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqySde2aaWbaaSqabeaacaWGNaaaaOGaaGypaiab eo7aNnaaCaaaleqabaGaey4fIOcaaOGaeqySdegaaa@3FB8@ . Этим теорема полностью доказана.

2.5.Конторсия связности

Как показано в разд. 2.3, на многообразии M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ можно построить различные материальные связности. Вместе с тем различие материальных связностей между собой приводит к различию соответствующих уравнений баланса массы и импульса, сформулированных относительно неевклидовой отсчетной формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ [23; 61]. В этой связи для сравнения уравнений баланса друг с другом целесообразно иметь меру отличия одной связности от другой. Эту меру можно задать следующим образом [26]. Предположим, что на многообразии M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ выбраны две связности MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ и ˜ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqGHhis0aiaawoWaaaaa@3A4B@ . Тогда разность

K= ˜ , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8ljaai2dacqGHhis0cqGHsisldaaiaaqaaiabgEGird Gaay5adaGaaGilaaaa@4AC6@ (26)

по определению, характеризует отличие связности ˜ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqGHhis0aiaawoWaaaaa@3A4B@ от связности MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ . Рассмотрим поле K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lbaa@448E@ более детально.

Если ( A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabgkGi2oaaBaaaleaacaWGbbaabeaakiaa iMcadaqhaaWcbaGaamyqaiaai2dacaaIXaaabaGaamOBaaaaaaa@3F32@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ координатный репер на M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ , то ковариантные производные, определяемые связностями ˜ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqGHhis0aiaawoWaaaaa@3A4B@ и MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ , представлены выражениями

u v= u A A v B + v C Γ A B C B è ˜ u v= u A A v B + v C Γ ˜ A B C B , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaSbaaSqaaiaabwhaaeqaaOGaaeODaiaa i2dacaWG1bWaaWbaaSqabeaacaWGbbaaaOWaaeWaaeaacqGHciITda WgaaWcbaGaamyqaaqabaGccaWG2bWaaWbaaSqabeaacaWGcbaaaOGa ey4kaSIaamODamaaCaaaleqabaGaam4qaaaakiabfo5ahnaaDaaale aacaWGbbaabaGaamOqaaaakmaaBaaaleaacaWGdbaabeaaaOGaayjk aiaawMcaaiabgkGi2oaaBaaaleaacaWGcbaabeaakiaaywW7caqGOd GaaGzbVpaaGaaabaGaey4bIenacaGLdmaadaWgaaWcbaGaaeyDaaqa baGccaqG2bGaaGypaiaadwhadaahaaWcbeqaaiaadgeaaaGcdaqada qaaiabgkGi2oaaBaaaleaacaWGbbaabeaakiaadAhadaahaaWcbeqa aiaadkeaaaGccqGHRaWkcaWG2bWaaWbaaSqabeaacaWGdbaaaOWaaa caaeaacqqHtoWraiaawoWaaiaaywW7daqhaaWcbaGaamyqaaqaaiaa dkeaaaGcdaWgaaWcbaGaam4qaaqabaaakiaawIcacaGLPaaacqGHci ITdaWgaaWcbaGaamOqaaqabaGccaaISaaaaa@6BC2@

в которых u,vVec( M R ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyDaiaaiYcacaaMc8UaaeODaiabgIGiolaabAfa caqGLbGaae4yaiaaiIcacaWGnbWaaSbaaSqaaiaadkfaaeqaaOGaaG ykaaaa@43A4@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ произвольные векторные поля, а Γ A B C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaadgeaaeaacaWGcbaaaOWa aSbaaSqaaiaadoeaaeqaaaaa@3C23@ и Γ ˜ A B C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqqHtoWraiaawoWaaiaaywW7daqhaaWc baGaamyqaaqaaiaadkeaaaGcdaWgaaWcbaGaam4qaaqabaaaaa@3E73@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ коэффициенты связности. Поэтому для значения поля (26) на тех же векторах u MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyDaaaa@38FB@ , v MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeODaaaa@38FC@ справедливо равенство

K u v= u v ˜ u v= u A v C ( Γ A B C Γ ˜ A B C ) B , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lnaaBaaaleaacaqG1baabeaakiaabAhacaaI9aGaey 4bIe9aaSbaaSqaaiaabwhaaeqaaOGaaeODaiabgkHiTmaaGaaabaGa ey4bIenacaGLdmaadaWgaaWcbaGaaeyDaaqabaGccaqG2bGaaGypai aadwhadaahaaWcbeqaaiaadgeaaaGccaWG2bWaaWbaaSqabeaacaWG dbaaaOGaaGikaiabfo5ahnaaDaaaleaacaWGbbaabaGaamOqaaaakm aaBaaaleaacaWGdbaabeaakiabgkHiTmaaGaaabaGaeu4KdCeacaGL dmaacaaMf8+aa0baaSqaaiaadgeaaeaacaWGcbaaaOWaaSbaaSqaai aadoeaaeqaaOGaaGykaiabgkGi2oaaBaaaleaacaWGcbaabeaakiaa iYcaaaa@654C@

и поле K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lbaa@448E@ , таким образом, имеет следующее полиадное представление:

K= K B A C A d X B d X C ,ãäå K B A C = Γ B A C Γ ˜ B A C . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8ljaai2dacqWFka=sdaqhaaWcbaGaamOqaaqaaiaadg eaaaGcdaWgaaWcbaGaam4qaaqabaGccqGHciITdaWgaaWcbaGaamyq aaqabaGccqGHxkcXcaWGKbGaamiwamaaCaaaleqabaGaamOqaaaaki abgEPielaadsgacaWGybWaaWbaaSqabeaacaWGdbaaaOGaaGilaiaa ywW7caqGJdGaaei5aiaabwoacqWFka=sdaqhaaWcbaGaamOqaaqaai aadgeaaaGcdaWgaaWcbaGaam4qaaqabaGccaaI9aGaeu4KdC0aa0ba aSqaaiaadkeaaeaacaWGbbaaaOWaaSbaaSqaaiaadoeaaeqaaOGaey OeI0YaaacaaeaacqqHtoWraiaawoWaaiaaywW7daqhaaWcbaGaamOq aaqaaiaadgeaaaGcdaWgaaWcbaGaam4qaaqabaGccaaIUaaaaa@6DF3@

Заметим, что несмотря на то что аффинная связность не образует тензорного поля третьего ранга, разность между любыми двумя связностями, напротив, является тензорным полем [55]. По этой причине K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lbaa@448E@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ тензорное поле третьего ранга: KSec T M R T M R T M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8ljabgIGiolaabofacaqGLbGaae4yamaabmaabaGaam ivamaaCaaaleqabaGaey4fIOcaaOGaamytamaaBaaaleaacaWGsbaa beaakiabgEPielaadsfacaWGnbWaaSbaaSqaaiaadkfaaeqaaOGaey 4LIqSaamivaiaad2eadaWgaaWcbaGaamOuaaqabaaakiaawIcacaGL Paaaaaa@579F@ .

Используя поле K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lbaa@448E@ , можно связать кручения, кривизны и неметричности различных материальных связностей следующим образом. Тензор кручения T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ubaa@44A0@ (10) связности MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ может быть выражен через тензор кручения T ˜ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgi gjxyRrxDYbacfaGae83eWtfacaGLdmaaaaa@4562@ второй связности ˜ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqGHhis0aiaawoWaaaaa@3A4B@ в соответствии с формулой

T(u,v)= T ˜ (u,v)+ K u v K v u, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ujaaiIcacaqG1bGaaGilaiaaykW7caqG2bGaaGykai aai2dadaaiaaqaaiab=nb8ubGaay5adaGaaGzaVlaaiIcacaqG1bGa aGilaiaaykW7caqG2bGaaGykaiabgUcaRiab=Pa8lnaaBaaaleaaca qG1baabeaakiaabAhacqGHsislcqWFka=sdaWgaaWcbaGaaeODaaqa baGccaqG1bGaaGilaaaa@5F24@

при выводе которой использовалось равенство u v= ˜ u v+ K u v MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaSbaaSqaaiaabwhaaeqaaOGaaeODaiaa i2dadaaiaaqaaiabgEGirdGaay5adaWaaSbaaSqaaiaabwhaaeqaaO GaaeODaiabgUcaRmrr1ngBPrMrYf2A0vNCaeHbfv3ySLgzGyKCHTgD 1jhaiuaacqWFka=sdaWgaaWcbaGaaeyDaaqabaGccaqG2baaaa@507A@ . В координатном репере

T B A C = T ˜ B A C + K B A C K C A B . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8unaaDaaaleaacaWGcbaabaGaamyqaaaakmaaBaaale aacaWGdbaabeaakiaai2dadaaiaaqaaiab=nb8ubGaay5adaGaaGzb VpaaDaaaleaacaWGcbaabaGaamyqaaaakmaaBaaaleaacaWGdbaabe aakiabgUcaRiab=Pa8lnaaDaaaleaacaWGcbaabaGaamyqaaaakmaa BaaaleaacaWGdbaabeaakiabgkHiTiab=Pa8lnaaDaaaleaacaWGdb aabaGaamyqaaaakmaaBaaaleaacaWGcbaabeaakiaai6caaaa@5AB6@

Аналогично тензор кривизны Римана MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Xrisbaa@43D8@ (11) связности MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ может быть выражен через тензор кривизны ˜ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgi gjxyRrxDYbacfaGae8hhHifacaGLdmaaaaa@449A@ связности ˜ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqGHhis0aiaawoWaaaaa@3A4B@ согласно соотношению

(uv)= ˜ (u,v)+ ˜ u , K v + K u , ˜ v K [u,v] +[ K u , K v ]. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=XrisjaaiIcacaqG1bGaaGPaVlaabAhacaaIPaGaaGypam aaGaaabaGae8hhHifacaGLdmaacaaIOaGaaeyDaiaaiYcacaaMc8Ua aeODaiaaiMcacqGHRaWkdaWadaqaamaaGaaabaGaey4bIenacaGLdm aadaWgaaWcbaGaaeyDaaqabaGccaaISaGaaGPaVlab=Pa8lnaaBaaa leaacaqG2baabeaaaOGaay5waiaaw2faaiabgUcaRmaadmaabaGae8 NcWV0aaSbaaSqaaiaabwhaaeqaaOGaaGilaiaaykW7daaiaaqaaiab gEGirdGaay5adaWaaSbaaSqaaiaabAhaaeqaaaGccaGLBbGaayzxaa GaeyOeI0Iae8NcWV0aaSbaaSqaaiaaiUfacaqG1bGaaGilaiaaykW7 caqG2bGaaGyxaaqabaGccqGHRaWkcaaIBbGae8NcWV0aaSbaaSqaai aabwhaaeqaaOGaaGilaiaaykW7cqWFka=sdaWgaaWcbaGaaeODaaqa baGccaaIDbGaaGOlaaaa@7C78@

В координатном репере

A B C D = ˜ A B C D + A K B D C B K A D C + Γ ˜ B L C K A D L + K B L C Γ ˜ A D L + K B L C K A D L Γ ˜ A L C K B D L MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=XrisnaaBaaaleaacaWGbbaabeaakmaaBaaaleaacaWGcb aabeaakmaaDaaaleaacaWGdbaabaGaamiraaaakiaai2dadaaiaaqa aiab=XrisbGaay5adaGaaGzbVpaaBaaaleaacaWGbbaabeaakmaaBa aaleaacaWGcbaabeaakmaaDaaaleaacaWGdbaabaGaamiraaaakiab gUcaRiabgkGi2oaaBaaaleaacaWGbbaabeaakiab=Pa8lnaaDaaale aacaWGcbaabaGaamiraaaakmaaBaaaleaacaWGdbaabeaakiabgkHi TiabgkGi2oaaBaaaleaacaWGcbaabeaakiab=Pa8lnaaDaaaleaaca WGbbaabaGaamiraaaakmaaBaaaleaacaWGdbaabeaakiabgUcaRmaa GaaabaGaeu4KdCeacaGLdmaacaaMf8+aa0baaSqaaiaadkeaaeaaca WGmbaaaOWaaSbaaSqaaiaadoeaaeqaaOGae8NcWV0aa0baaSqaaiaa dgeaaeaacaWGebaaaOWaaSbaaSqaaiaadYeaaeqaaOGaey4kaSIae8 NcWV0aa0baaSqaaiaadkeaaeaacaWGmbaaaOWaaSbaaSqaaiaadoea aeqaaOWaaacaaeaacqqHtoWraiaawoWaaiaaywW7daqhaaWcbaGaam yqaaqaaiaadseaaaGcdaWgaaWcbaGaamitaaqabaGccqGHRaWkcqWF ka=sdaqhaaWcbaGaamOqaaqaaiaadYeaaaGcdaWgaaWcbaGaam4qaa qabaGccqWFka=sdaqhaaWcbaGaamyqaaqaaiaadseaaaGcdaWgaaWc baGaamitaaqabaGccqGHsisldaaiaaqaaiabfo5ahbGaay5adaGaaG zbVpaaDaaaleaacaWGbbaabaGaamitaaaakmaaBaaaleaacaWGdbaa beaakiab=Pa8lnaaDaaaleaacaWGcbaabaGaamiraaaakmaaBaaale aacaWGmbaabeaakiabgkHiTaaa@8E38@

K A L C Γ ˜ B D L K A L C K B D L . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOeI0Yefv3ySLgzgjxyRrxDYbqeguuDJXwAKbIr Yf2A0vNCaGqbaiab=Pa8lnaaDaaaleaacaWGbbaabaGaamitaaaakm aaBaaaleaacaWGdbaabeaakmaaGaaabaGaeu4KdCeacaGLdmaacaaM f8+aa0baaSqaaiaadkeaaeaacaWGebaaaOWaaSbaaSqaaiaadYeaae qaaOGaeyOeI0Iae8NcWV0aa0baaSqaaiaadgeaaeaacaWGmbaaaOWa aSbaaSqaaiaadoeaaeqaaOGae8NcWV0aa0baaSqaaiaadkeaaeaaca WGebaaaOWaaSbaaSqaaiaadYeaaeqaaOGaaGOlaaaa@59A0@

Наконец, тензор неметричности Q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rbaa@449A@ (12) связности MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ может быть выражен через тензор неметричности Q ˜ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgi gjxyRrxDYbacfaGae8heWhfacaGLdmaaaaa@455C@ связности ˜ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqGHhis0aiaawoWaaaaa@3A4B@ по формуле:

Q(u,v,w)= Q ˜ (u,v,w)+G( K u v,w)+G(v, K u w). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rjaaiIcacaqG1bGaaGilaiaaykW7caqG2bGaaGilai aaykW7caqG3bGaaGykaiaai2dadaaiaaqaaiab=bb8rbGaay5adaGa aGikaiaabwhacaaISaGaaGPaVlaabAhacaaISaGaaGPaVlaabEhaca aIPaGaey4kaSIaae4raiaaiIcacqWFka=sdaWgaaWcbaGaaeyDaaqa baGccaqG2bGaaGilaiaaykW7caqG3bGaaGykaiabgUcaRiaabEeaca aIOaGaaeODaiaaiYcacaaMc8Uae8NcWV0aaSbaaSqaaiaabwhaaeqa aOGaae4DaiaaiMcacaaIUaaaaa@6ECF@

При ее выводе предполагалось, что материальная метрика G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ на геометрических пространствах ( M R ,G,) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaad2eadaWgaaWcbaGaamOuaaqabaGccaaI SaGaaGPaVlaabEeacaaISaGaaGPaVlabgEGirlaaiMcaaaa@4219@ и ( M R ,G, ˜ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaad2eadaWgaaWcbaGaamOuaaqabaGccaaI SaGaaGPaVlaabEeacaaISaGaaGPaVpaaGaaabaGaey4bIenacaGLdm aaaaa@4228@ ) одна и та же. В координатном репере

Q ABC = Q ˜ ABC + G CD K A D B + G BD K A D C . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rnaaBaaaleaacaWGbbGaamOqaiaadoeaaeqaaOGaaG ypamaaGaaabaGae8heWhfacaGLdmaacaaMf8+aaSbaaSqaaiaadgea caWGcbGaam4qaaqabaGccqGHRaWkcaqGhbWaaSbaaSqaaiaadoeaca WGebaabeaakiab=Pa8lnaaDaaaleaacaWGbbaabaGaamiraaaakmaa BaaaleaacaWGcbaabeaakiabgUcaRiaabEeadaWgaaWcbaGaamOqai aadseaaeqaaOGae8NcWV0aa0baaSqaaiaadgeaaeaacaWGebaaaOWa aSbaaSqaaiaadoeaaeqaaOGaaGOlaaaa@5F55@

Рассмотрим частный случай. Предположим, что на многообразии M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ фиксирована материальная метрика G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ (6), которая определяет связность Леви-Чивита ˜ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqGHhis0aiaawoWaaaaa@3A4B@ . Коэффициенты этой связности в координатном репере ( A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabgkGi2oaaBaaaleaacaWGbbaabeaakiaa iMcadaqhaaWcbaGaamyqaiaai2dacaaIXaaabaGaamOBaaaaaaa@3F32@ представлены выражениями (13) Γ ˜ B A C = G AD 2 B G CD + C G BD D G BC MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqqHtoWraiaawoWaaiaaywW7daqhaaWc baGaamOqaaqaaiaadgeaaaGcdaWgaaWcbaGaam4qaaqabaGccaaI9a WaaSaaaeaacaqGhbWaaWbaaSqabeaacaWGbbGaamiraaaaaOqaaiaa ikdaaaWaaeWaaeaacqGHciITdaWgaaWcbaGaamOqaaqabaGccaqGhb WaaSbaaSqaaiaadoeacaWGebaabeaakiabgUcaRiabgkGi2oaaBaaa leaacaWGdbaabeaakiaabEeadaWgaaWcbaGaamOqaiaadseaaeqaaO GaeyOeI0IaeyOaIy7aaSbaaSqaaiaadseaaeqaaOGaae4ramaaBaaa leaacaWGcbGaam4qaaqabaaakiaawIcacaGLPaaaaaa@54D4@ . Другую связность MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ будем считать произвольной. Если T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ubaa@44A0@ и Q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rbaa@449A@ , соответственно, кручение (10) и неметричность (12) связности MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ , то тогда справедлива следующая общая формула в координатном репере ( A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabgkGi2oaaBaaaleaacaWGbbaabeaakiaa iMcadaqhaaWcbaGaamyqaiaai2dacaaIXaaabaGaamOBaaaaaaa@3F32@ [23]:

Γ B A C = Γ ˜ B A C + G AD 2 ( T D B C + T C D B + T B D C )+ G AD 2 ( Q B C D + Q C D B Q D B C ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaadkeaaeaacaWGbbaaaOWa aSbaaSqaaiaadoeaaeqaaOGaaGypamaaGaaabaGaeu4KdCeacaGLdm aacaaMf8+aa0baaSqaaiaadkeaaeaacaWGbbaaaOWaaSbaaSqaaiaa doeaaeqaaOGaey4kaSYaaSaaaeaacaqGhbWaaWbaaSqabeaacaWGbb GaamiraaaaaOqaaiaaikdaaaGaaGikamrr1ngBPrMrYf2A0vNCaeHb fv3ySLgzGyKCHTgD1jhaiuaacqWFtapvdaWgaaWcbaGaamiraaqaba GcdaWgaaWcbaGaamOqaaqabaGcdaWgaaWcbaGaam4qaaqabaGccqGH RaWkcqWFtapvdaWgaaWcbaGaam4qaaqabaGcdaWgaaWcbaGaamiraa qabaGcdaWgaaWcbaGaamOqaaqabaGccqGHRaWkcqWFtapvdaWgaaWc baGaamOqaaqabaGcdaWgaaWcbaGaamiraaqabaGcdaWgaaWcbaGaam 4qaaqabaGccaaIPaGaey4kaSYaaSaaaeaacaqGhbWaaWbaaSqabeaa caWGbbGaamiraaaaaOqaaiaaikdaaaGaaGikaiab=bb8rnaaBaaale aacaWGcbaabeaakmaaBaaaleaacaWGdbaabeaakmaaBaaaleaacaWG ebaabeaakiabgUcaRiab=bb8rnaaBaaaleaacaWGdbaabeaakmaaBa aaleaacaWGebaabeaakmaaBaaaleaacaWGcbaabeaakiabgkHiTiab =bb8rnaaBaaaleaacaWGebaabeaakmaaBaaaleaacaWGcbaabeaakm aaBaaaleaacaWGdbaabeaakiaaiMcacaaIUaaaaa@7AB2@ (27)

Отсюда вытекает соотношение для компонент тензора K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lbaa@448E@ :

K B A C = G AD 2 ( T D B C + T C D B + T B D C )+ G AD 2 ( Q B C D + Q C D B Q D B C ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lnaaDaaaleaacaWGcbaabaGaamyqaaaakmaaBaaale aacaWGdbaabeaakiaai2dadaWcaaqaaiaabEeadaahaaWcbeqaaiaa dgeacaWGebaaaaGcbaGaaGOmaaaacaaIOaGae83eWt1aaSbaaSqaai aadseaaeqaaOWaaSbaaSqaaiaadkeaaeqaaOWaaSbaaSqaaiaadoea aeqaaOGaey4kaSIae83eWt1aaSbaaSqaaiaadoeaaeqaaOWaaSbaaS qaaiaadseaaeqaaOWaaSbaaSqaaiaadkeaaeqaaOGaey4kaSIae83e Wt1aaSbaaSqaaiaadkeaaeqaaOWaaSbaaSqaaiaadseaaeqaaOWaaS baaSqaaiaadoeaaeqaaOGaaGykaiabgUcaRmaalaaabaGaae4ramaa CaaaleqabaGaamyqaiaadseaaaaakeaacaaIYaaaaiaaiIcacqWFqa FudaWgaaWcbaGaamOqaaqabaGcdaWgaaWcbaGaam4qaaqabaGcdaWg aaWcbaGaamiraaqabaGccqGHRaWkcqWFqaFudaWgaaWcbaGaam4qaa qabaGcdaWgaaWcbaGaamiraaqabaGcdaWgaaWcbaGaamOqaaqabaGc cqGHsislcqWFqaFudaWgaaWcbaGaamiraaqabaGcdaWgaaWcbaGaam OqaaqabaGcdaWgaaWcbaGaam4qaaqabaGccaaIPaGaaGOlaaaa@73B8@ (28)

Если связность MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ является связностью Вайценбока, порожденной полем локальных деформаций H= H B A c A d X B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaiaai2dacaqGibWaa0baaSqaaiaadkeaaeaa caWGbbaaaOGaaC4yamaaBaaaleaacaWGbbaabeaakiabgEPielaads gacaWGybWaaWbaaSqabeaacaWGcbaaaaaa@42CF@ , то Q=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rjaai2dacaqGWaaaaa@4614@ и формула (28) принимает вид

K B A C = G AD 2 ( T D B C + T C D B + T B D C ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lnaaDaaaleaacaWGcbaabaGaamyqaaaakmaaBaaale aacaWGdbaabeaakiaai2dadaWcaaqaaiaabEeadaahaaWcbeqaaiaa dgeacaWGebaaaaGcbaGaaGOmaaaacaaIOaGae83eWt1aaSbaaSqaai aadseaaeqaaOWaaSbaaSqaaiaadkeaaeqaaOWaaSbaaSqaaiaadoea aeqaaOGaey4kaSIae83eWt1aaSbaaSqaaiaadoeaaeqaaOWaaSbaaS qaaiaadseaaeqaaOWaaSbaaSqaaiaadkeaaeqaaOGaey4kaSIae83e Wt1aaSbaaSqaaiaadkeaaeqaaOWaaSbaaSqaaiaadseaaeqaaOWaaS baaSqaaiaadoeaaeqaaOGaaGykaiaai6caaaa@5DD6@ (29)

Таким образом, K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lbaa@448E@ полностью определяется тензором кручения T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ubaa@44A0@ связности Вайценбока MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ . По этой причине будем называть K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lbaa@448E@ тензором конторсии [13]. С геометрической точки зрения отклонение связности Вайценбока от связности Леви-Чивита заключается в несогласованности малых элементов, на которые разбивается многообразие M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ в соответствии с той или иной его геометрией. Именно в случае связности Леви-Чивита эти элементы, пусть даже будучи малыми, обладают кривизной (пример MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ часть поверхности сферы), а в случае связности Вайценбока малые элементы представляют часть плоскости. Конторсия связности (29) определяет дополнительный изгиб плоского элемента с тем, чтобы он совпал с малым римановым элементом многообразия M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ .

Замечание 5 .То, что соотношение для коэффициентов связности Вайценбока (получаемое из равенства (27) путем отбрасывания слагаемых с неметричностью) содержит в себе коэффициенты связности Леви-Чивита, может быть интерпретировано в рамках континуальной теории дефектов. Действительно, риманово пространство соответствует телу с дисклинациями, в то время как пространство Вайценбока соответствует телу с дислокациями. Вместе с тем, если в кристалле присутствуют дискретно распределенные дислокации, то они индуцируют дисклинации, которые расположены на концах линий дислокации [62]. В случае непрерывного распределения дефектов поля дислокаций и дисклинаций непрерывны и могут быть представлены совокупностью геометрических понятий: кривизна, кручение и неметричность. Математически это и означает, что связность Вайценбока содержит в некотором смысле связность Леви-Чивита.

Наконец, если связность MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ является связностью Вейля, порожденной материальной метрикой G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ и полем 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaaaa@38BE@ -форм ν MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyVd4gaaa@39BB@ , то, согласно (18), тензорное поле K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lbaa@448E@ представлено компонентами

K B A C = 1 2 ( ν B δ C A + ν C δ B A ν D G BC G AD ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lnaaDaaaleaacaWGcbaabaGaamyqaaaakmaaBaaale aacaWGdbaabeaakiaai2dacqGHsisldaWcaaqaaiaaigdaaeaacaaI YaaaaiaaiIcacqaH9oGBdaWgaaWcbaGaamOqaaqabaGccqaH0oazda qhaaWcbaGaam4qaaqaaiaadgeaaaGccqGHRaWkcqaH9oGBdaWgaaWc baGaam4qaaqabaGccqaH0oazdaqhaaWcbaGaamOqaaqaaiaadgeaaa GccqGHsislcqaH9oGBdaWgaaWcbaGaamiraaqabaGccaqGhbWaaSba aSqaaiaadkeacaWGdbaabeaakiaabEeadaahaaWcbeqaaiaadgeaca WGebaaaOGaaGykaiaai6caaaa@628B@

Из полученной формулы следует, что поле K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lbaa@448E@ полностью определяется полем ковекторов ν MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyVd4gaaa@39BB@ . Поскольку кривизна связности Вейля может быть отлична от нуля, нет причин связывать отклонение связности Вейля от связности Леви-Чивита с геометрией малых элементов. Вместе с тем отличие двух связностей друг от друга может быть интерпретировано в терминах формы объема. Действительно, в случае риманова пространства форма объема dV= detG d X 1 d X n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamizaiaadAfacaaI9aWaaOaaaeaaciGGKbGaaiyz aiaacshacaqGhbaaleqaaOGaaGjcVlaadsgacaWGybWaaWbaaSqabe aacaaIXaaaaOGaey4jIKTaeS47IWKaey4jIKTaamizaiaadIfadaah aaWcbeqaaiaad6gaaaaaaa@4AC1@ ковариантно постоянна, т. е.

L dV=0. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaWbaaSqabeaacaWGmbaaaOGaamizaiaa dAfacaaI9aGaaeimaiaai6caaaa@3E87@

В противоположность этому в случае связности Вейля риманова форма объема эволюционирует при параллельном перенесении, т. е.

Wl dV0. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIe9aaWbaaSqabeaacaWGxbGaamiBaaaakiaa dsgacaWGwbGaeyiyIKRaaeimaiaai6caaaa@4083@

С физической точки зрения изменение римановой формы объема при переносе от точки к точке означает, что представительные объемы, составляющие тело, содержат точечные дефекты типа пор либо независимо от других элементарных объемов изменяются в силу химических реакций или тепловых процессов [21; 60]. В этой связи возникают <<метрические аномалии>> (по терминологии [63]), которые могут быть формализованы в виде отличной от нуля 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaaaa@38BE@ -формы Вейля ν MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyVd4gaaa@39BB@ .

Замечание 6. Существование различных способов задания материальной связности на теле приводит к проблеме формулировки критерия выбора одной из возможных связностей. По-видимому, в рамках теории простого материала, когда отклик тела зависит лишь от градиента первого порядка, такого критерия не существует [25; 64; 65]. Несмотря на отсутствие строго обоснования этого заключения, можно предложить следующий ход рассуждений. Несовместные деформации определяются полем H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ локальных деформаций, которое имеет n 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBamaaCaaaleqabaGaaGOmaaaaaaa@39DF@ компонент. Вместе с тем аффинная связность для своего полного определения требует задания n 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBamaaCaaaleqabaGaaG4maaaaaaa@39E0@ коэффициентов связности. Таким образом, задание одного лишь поля локальных деформаций не дает достаточно данных для единственного определения материальной связности.

2.6. Физический смысл конторсии связности Вайценбока

Остановимся более подробно на связности Вайценбока. При этом будем полагать, что размерность тела равна размерности физического пространства, т. е. n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ . Геометрия на многообразии M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ естественным образом индуцирует геометрию на теле B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ , поскольку многообразия M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ и B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ топологически эквивалентны. В этой связи для полей связности и метрики на теле B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ , определяющих геометрию, будем использовать те же обозначения, что использовались для полей на M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ . В частности, на B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ определено поле конторсии (29).

Имея дело с телом, вместо поля локальных деформаций H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ определим единообразную отсчетную [13] P MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiuaaaa@38D6@ , которая является полем Bp P p Hom( T p B;V) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cjabg2GiNiab=Lc8WjablAAiHjaabcfadaWgaaWcba Gae8xkWdhabeaakiabgIGiolaabIeacaqGVbGaaeyBaiaaiIcacaWG ubWaaSbaaSqaaiab=Lc8WbqabaGccqWFbaVqcaaI7aGaaGPaVprr1n gBPrwtHrhAXaqehuuDJXwAKbstHrhAG8KBLbacgaGae4xfXBLaaGyk aaaa@6463@ локальных конфигураций P p MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiuamaaBaaaleaatuuDJXwAKzKCHTgD1jharyqr 1ngBPrgigjxyRrxDYbacfaGae8xkWdhabeaaaaa@45D1@ . В свою очередь, каждая локальная конфигурация определяется равенством P p = T p ϰ (p) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiuamaaBaaaleaatuuDJXwAKzKCHTgD1jharyqr 1ngBPrgigjxyRrxDYbacfaGae8xkWdhabeaakiaai2dacaWGubWaaS baaSqaaiab=Lc8WbqabaWefv3ySLgznfgDOfdarCqr1ngBPrginfgD ObYtUvgaiyaakiab+b=a5paaCaaaleqabaGaaGikaiab=Lc8WjaaiM caaaaaaa@595B@ , где ϰ (p) :BE MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+daahaaWcbeqaaiaaiIcatuuDJXwAKzKCHTgD1jharC qr1ngBPrgigjxyRrxDYbacgaGae4xkWdNaaGykaaaakiaaiQdacaaM c8Uae4xaWlKaeyOKH4Qae8hmHueaaa@5978@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ конфигурация, при которой инфинитезимальная окрестность точки p MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Lc8Wbaa@44D2@ переходит в натуральное состояние. В явном виде, используя элементы семейства { γ (X) } X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4Eaiabeo7aNnaaCaaaleqabaGaaGikamrr1ngB PrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae83fXJLaaGykaa aakiaai2hadaWgaaWcbaGae83fXJLaeyicI4Sae8NeXp1aaSbaaeaa caWGsbaabeaaaeqaaaaa@4F29@ и обозначая через ϰ R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+daWgaaWcbaGaamOuaaqabaaaaa@4501@ конфигурацию, образом которой является форма S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ , получим

ϰ (p) = γ ( ϰ R (p)) ϰ R . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+daahaaWcbeqaaiaaiIcatuuDJXwAKzKCHTgD1jharC qr1ngBPrgigjxyRrxDYbacgaGae4xkWdNaaGykaaaakiaai2dacqaH ZoWzdaahaaWcbeqaaiaaiIcacqWFWpq+daWgaaqaaiaadkfaaeqaai aaiIcacqGFPapCcaaIPaGaaGykaaaakiablIHiVjab=b=a5paaBaaa leaacaWGsbaabeaakiaai6caaaa@62BF@

Тензору конторсии K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Pa8lbaa@448E@ отвечает прямой образ K:BEnd(V) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4saiaaiQdacaaMc8+efv3ySLgzgjxyRrxDYbqe guuDJXwAKbIrYf2A0vNCaGqbaiab=fa8cjabgkziUkaabweacaqGUb GaaeizaiaaiIcatuuDJXwAK1uy0HwmaeXbfv3ySLgzG0uy0Hgip5wz aGGbaiab+vr8wjaaiMcaaaa@58FD@ в физическом пространстве E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ согласно следующей формуле:

Ku=P K P 1 u P 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4saiaahwhacaaI9aGaaeiuamrr1ngBPrMrYf2A 0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaacqWFka=sdaWgaaWcbaGaae iuamaaCaaabeqaaiabgkHiTiaaigdaaaGaaCyDaaqabaGccaqGqbWa aWbaaSqabeaacqGHsislcaaIXaaaaaaa@4E73@ (30)

для всех uV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCyDaiabgIGioprr1ngBPrwtHrhAXaqeguuDJXwA KbstHrhAG8KBLbacfaGae8xfXBfaaa@45F0@ . Будем называть поле K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4saaaa@38D7@ конторсией в смысле Нолла, поскольку оно соответствует полю D MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCiraaaa@38D0@ , определенному в [13].

Для получения физической интерпретации поля конторсии рассмотрим частную единообразную отсчетную P MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiuaaaa@38D6@ , для которой существует конфигурация ϰ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+aaa@43FE@ , удовлетворяющая следующему условию:

Q=(Tϰ) P 1 :Bp Q p O(3)End(V). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCyuaiaai2dacaaIOaGaamivamrr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae8h8dKVaaGykaiaabcfada ahaaWcbeqaaiabgkHiTiaaigdaaaGccaaI6aGaaGPaVprr1ngBPrMr Yf2A0vNCaeXbfv3ySLgzGyKCHTgD1jhaiyaacqGFbaVqcqGHniYjcq GFPapCcqWIMgsycaWHrbWaaSbaaSqaaiab+Lc8WbqabaGccqGHiiIZ caqGpbGaaGikaiaaiodacaaIPaGaeyOGIWSaaeyraiaab6gacaqGKb GaaGikaiab=vr8wjaaiMcacaaIUaaaaa@6E2F@

Иными словами, композиция полей P 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiuamaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@3AAB@ и Tϰ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8h8dKpaaa@44D7@ есть не более, чем поле вращений в физическом пространстве E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ . В этом случае действие конторсии по Ноллу K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4saaaa@38D7@ (30) дает выражение для производной[12] Q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCyuaaaa@38DD@ [13]:

Ku= Q T ( D P Q)u, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4saiaahwhacaaI9aGaeyOeI0IaaCyuamaaCaaa leqabaGaaeivaaaakiaaiIcacaWGebWaaSbaaSqaaiaabcfaaeqaaO GaaCyuaiaaiMcacaWH1bGaaGilaaaa@4336@

где uV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCyDaiabgIGioprr1ngBPrwtHrhAXaqeguuDJXwA KbstHrhAG8KBLbacfaGae8xfXBfaaa@45F0@ . С физической точки зрения последнее равенство может быть интерпретировано следующим образом: несовместность локальных деформаций возникает в силу несогласованности поворотов представительных объемов в физическом пространстве. Эта несогласованность ведет к невозможности собрать повернутые объемы в сплошное тело. Обратные вращения, которые делают подобную сборку возможной, определяются полем Q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCyuaaaa@38DD@ . В свою очередь, темп изменения значений Q MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCyuaaaa@38DD@ относительно пространственных сдвигов определяется конторсией по Ноллу K MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaC4saaaa@38D7@ .

2.7. Восстановление поля локальных деформаций по известным метрике и кручению

До сих пор поле локальных деформаций H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ предполагалось известным. По нему синтезировались метрика G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ и связность MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ , определяющие геометрию отсчетной формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ . Отклонение этой геометрии от евклидовой (а с физической точки зрения степень несовместности) определялось тензорными полями кручения, кривизны и неметричности. Рассмотрим теперь обратную задачу. Предположим, что известны: 1) материальная метрика G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ и 2) меры несовместности локальных деформаций. Поставим целью восстановить по этим данным поле H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ . В дальнейших рассуждениях ограничимся случаем связности Вайценбока.

Пусть известны метрика G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ и тензорное поле третьего ранга T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ubaa@44A0@ , которое ассоциируется с кручением связности Вайценбока (в частности, в координатном репере, T B A C = T C A B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8unaaDaaaleaacaWGcbaabaGaamyqaaaakmaaBaaale aacaWGdbaabeaakiaai2dacqGHsislcqWFtapvdaqhaaWcbaGaam4q aaqaaiaadgeaaaGcdaWgaaWcbaGaamOqaaqabaaaaa@4DAA@ ). Используя поля G MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raaaa@38CD@ и T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=nb8ubaa@44A0@ , из формулы (27) (в которой надо положить Q=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=bb8rjaai2dacaqGWaaaaa@4614@ ) получаем значения полей Γ B A C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaadkeaaeaacaWGbbaaaOWa aSbaaSqaaiaadoeaaeqaaaaa@3C23@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ коэффициентов связности. В дальнейшем будем полагать эти поля известными.

Формула (15) влечет соотношения

B H C D = Γ B A C H A D ,B,C,D=1,,n, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOaIy7aaSbaaSqaaiaadkeaaeqaaOGaaeisamaa DaaaleaacaWGdbaabaGaamiraaaakiaai2dacqqHtoWrdaqhaaWcba GaamOqaaqaaiaadgeaaaGcdaWgaaWcbaGaam4qaaqabaGccaqGibWa a0baaSqaaiaadgeaaeaacaWGebaaaOGaaGilaiaaywW7caWGcbGaaG ilaiaaykW7caWGdbGaaGilaiaaykW7caWGebGaaGypaiaaigdacaaI SaGaaGPaVlablAciljaaiYcacaaMc8UaamOBaiaaiYcaaaa@5668@ (31)

которые образуют систему n 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBamaaCaaaleqabaGaaG4maaaaaaa@39E0@ линейных однородных уравнений в частных производных первого порядка относительно n 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBamaaCaaaleqabaGaaGOmaaaaaaa@39DF@ неизвестных H B A MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaDaaaleaacaWGcbaabaGaamyqaaaaaaa@3A88@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ компонент поля локальных деформаций. Поскольку число уравнений больше числа неизвестных, то для однозначного решения системы (31) должны выполняться некоторые условия интегрируемости. Эти условия вытекают из того, что функции H B A MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaDaaaleaacaWGcbaabaGaamyqaaaaaaa@3A88@ удовлетворяют равенствам B L H C D = L B H C D MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOaIy7aaSbaaSqaaiaadkeaaeqaaOGaeyOaIy7a aSbaaSqaaiaadYeaaeqaaOGaaeisamaaDaaaleaacaWGdbaabaGaam iraaaakiaai2dacqGHciITdaWgaaWcbaGaamitaaqabaGccqGHciIT daWgaaWcbaGaamOqaaqabaGccaqGibWaa0baaSqaaiaadoeaaeaaca WGebaaaaaa@4786@ :

L ( Γ B A C H A D )= B ( Γ L A C H A D ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOaIy7aaSbaaSqaaiaadYeaaeqaaOGaaGikaiab fo5ahnaaDaaaleaacaWGcbaabaGaamyqaaaakmaaBaaaleaacaWGdb aabeaakiaabIeadaqhaaWcbaGaamyqaaqaaiaadseaaaGccaaIPaGa aGypaiabgkGi2oaaBaaaleaacaWGcbaabeaakiaaiIcacqqHtoWrda qhaaWcbaGaamitaaqaaiaadgeaaaGcdaWgaaWcbaGaam4qaaqabaGc caqGibWaa0baaSqaaiaadgeaaeaacaWGebaaaOGaaGykaiaai6caaa a@4E9C@ (32)

Используя правило дифференцирования произведения и равенства (31), из (32) получаем соотношения

H A D L A B C =0, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaDaaaleaacaWGbbaabaGaamiraaaatuuD JXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbacfaGccqWFCeIuda qhaaWcbaGaamitaaqaaiaadgeaaaGcdaWgaaWcbaGaamOqaaqabaGc daWgaaWcbaGaam4qaaqabaGccaaI9aGaaGimaiaaiYcaaaa@4C69@

где L A B C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=XrisnaaDaaaleaacaWGmbaabaGaamyqaaaakmaaBaaale aacaWGcbaabeaakmaaBaaaleaacaWGdbaabeaaaaa@4797@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ компоненты тензора кривизны MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Xrisbaa@43D8@ . Таким образом, условия интегрируемости равносильны равенству

=0, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=Xrisjaai2dacaqGWaGaaGilaaaa@4608@ (33)

что геометрически очевидно: если Γ B A C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeu4KdC0aa0baaSqaaiaadkeaaeaacaWGbbaaaOWa aSbaaSqaaiaadoeaaeqaaaaa@3C23@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ коэффициенты связности Вайценбока, то определяемая ими кривизна должна быть равна нулю [66].

Предположим, что условие (33) выполнено. Для интегрирования системы (31) зафиксируем некоторую точку X 0 M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwamaaBaaaleaacaaIWaaabeaakiabgIGiolaa d2eadaWgaaWcbaGaamOuaaqabaaaaa@3D29@ и выберем произвольную гладкую кривую[13]

χ:]ε,ε[ M R , X A = X A (s),A=1,,n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4XdmMaaGOoaiaaykW7caaIDbGaeyOeI0IaeqyT duMaaGilaiaaykW7cqaH1oqzcaaIBbGaeyOKH4QaamytamaaBaaale aacaWGsbaabeaakiaaiYcacaaMf8UaamiwamaaCaaaleqabaGaamyq aaaakiaai2dacaWGybWaaWbaaSqabeaacaWGbbaaaOGaaGikaiaado hacaaIPaGaaGilaiaaykW7caWGbbGaaGypaiaaigdacaaISaGaaGPa VlablAciljaaiYcacaaMc8UaamOBaaaa@5C59@

такую, что χ(0)= X 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4XdmMaaGikaiaaicdacaaIPaGaaGypaiaadIfa daWgaaWcbaGaaGimaaqabaaaaa@3E63@ . Домножая обе части (31) на компоненты X B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwamaaCaaaleqabaGaamOqaaaaaaa@39D4@ вектора скорости χ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdmgaaa@39BA@ , получаем

d H C D ds = Γ B A C X B H A D ,C,D=1,,n. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaSaaaeaacaWGKbGaaeisamaaDaaaleaacaWGdbaa baGaamiraaaaaOqaaiaadsgacaWGZbaaaiaai2dacqqHtoWrdaqhaa WcbaGaamOqaaqaaiaadgeaaaGcdaWgaaWcbaGaam4qaaqabaGccaWG ybWaaWbaaSqabeaacaWGcbaaaOGaaeisamaaDaaaleaacaWGbbaaba GaamiraaaakiaaiYcacaaMf8Uaam4qaiaaiYcacaaMc8Uaamiraiaa i2dacaaIXaGaaGilaiaaykW7cqWIMaYscaaISaGaaGPaVlaad6gaca aIUaaaaa@55B4@

В матричном виде

dH ds =H.P, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaSaaaeaacaWGKbGaamisaaqaaiaadsgacaWGZbaa aiaai2dacaWGibGaaGOlaiaadcfacaaISaaaaa@3F81@ (34)

где H=[ H B A ] MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamisaiaai2dacaaIBbGaaeisamaaDaaaleaacaWG cbaabaGaamyqaaaakiaai2faaaa@3DF2@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ неизвестная матрица, а P=[ Γ B A C X B ] MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuaiaai2dacaaIBbGaeu4KdC0aa0baaSqaaiaa dkeaaeaacaWGbbaaaOWaaSbaaSqaaiaadoeaaeqaaOGaamiwamaaCa aaleqabaGaamOqaaaakiaai2faaaa@4170@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ известная матрица коэффициентов. Таким образом, система (31) уравнений в частных производных преобразована к одному матричному обыкновенному дифференциальному уравнению (34) вдоль кривой χ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdmgaaa@39BA@ . Решение этого уравнения может быть представлено в виде [67]

H= H 0 .Ω(s), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamisaiaai2dacaWGibWaaSbaaSqaaiaaicdaaeqa aOGaaGOlaiabfM6axjaaiIcacaWGZbGaaGykaiaaiYcaaaa@40AD@ (35)

где H 0 =H(0) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamisamaaBaaaleaacaaIWaaabeaakiaai2dacaWG ibGaaGikaiaaicdacaaIPaaaaa@3D73@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ заданное значение матрицы H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamisaaaa@38D0@ в точке X 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwamaaBaaaleaacaaIWaaabeaaaaa@39C6@ , а Ω(s) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuyQdCLaaGikaiaadohacaaIPaaaaa@3BEE@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ матрицант, представленный рядом

Ω(s)=E+ n=1 0< s 1 < s n <s k=1 n P( s k ) k=1 n d s k , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuyQdCLaaGikaiaadohacaaIPaGaaGypaiaadwea cqGHRaWkdaaeWbqabSqaaiaad6gacaaI9aGaaGymaaqaaiabg6HiLc qdcqGHris5aOGaaGjcVpaapefabeWcbaGaaGimaiaaiYdacaWGZbWa aSbaaeaacaaIXaaabeaacqWIVlctcaaI8aGaam4CamaaBaaabaGaam OBaaqabaGaaGipaiaadohaaeqaniabgUIiYdGcdaqeWbqabSqaaiaa dUgacaaI9aGaaGymaaqaaiaad6gaa0Gaey4dIunakiaadcfacaaIOa Gaam4CamaaBaaaleaacaWGRbaabeaakiaaiMcadaqeWbqabSqaaiaa dUgacaaI9aGaaGymaaqaaiaad6gaa0Gaey4dIunakiaadsgacaWGZb WaaSbaaSqaaiaadUgaaeqaaOGaaGilaaaa@65A3@ (36)

в котором E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyraaaa@38CD@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ единичная матрица.

Таким образом, при условии (33) решение системы (31) не зависит от выбора кривой, соединяющей точки X 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwamaaBaaaleaacaaIWaaabeaaaaa@39C6@ и X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ , и однозначно определяется соотношениями (35) и (36) с точностью до жесткого вращения.

3. Обобщенная деформация и ее градиент

3.1. Отображение уплощения

Поскольку неевклидова единообразная форма S R =( M R ,G,) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaakiaai2dacaaI OaGaamytamaaBaaaleaacaWGsbaabeaakiaaiYcacaaMc8Uaae4rai aaiYcacaaMc8Uaey4bIeTaaGykaaaa@44C5@ синтезируется относительно некоторой наблюдаемой евклидовой формы S R =( M R ,g | S , | S ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaGccaaI9aGaaGikaiaad2 eadaWgaaWcbaGaamOuaaqabaGccaaISaGaaGPaVlaahEgacaaI8bWa aSbaaSqaaiaadofaaeqaaOGaaGilaiaaykW7cqGHhis0caaI8bWaaS baaSqaaiaadofaaeqaaOGaaGykaaaa@53A0@ , можно ожидать, что формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ и S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ как специфические геометрические структуры над общим многообразием M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ связаны между собой посредством некоторого отображения. Действительно, вернемся к примеру криволинейной мембраны (раздел 1.5), формы которой изображены на рис. 2. Неевклидова отсчетная форма S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ мембраны является полусферой, в то время как евклидовы формы представлены регионами евклидовой плоскости. Одним из этих регионов является форма S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ . В рамках рис. 2 деформацию неевклидовой отсчетной формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ в некоторую евклидову форму S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ можно представить в виде двух последовательных трансформаций. В ходе первой трансформации полусфера <<уплощается>> на физическую плоскость таким образом, чтобы полученный плоский регион совпадал с S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ . Вторая трансформация преобразует евклидову форму S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ в евклидову форму S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ и является, таким образом, обычной евклидовой деформацией.

Обобщая пример с криволинейной мембраной на случай произвольного деформируемого твердого тела, определим отображение λ R : S R S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaOGaaGOoaiaa ykW7caWGtbWaaSbaaSqaaiaadkfaaeqaaOGaeyOKH46efv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFse=udaWgaaWcbaGa amOuaaqabaaaaa@4D4D@ , которое заменяет неевклидову геометрию над многообразием M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ на геометрию над тем же многообразием, индуцированную из физического пространства E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ . В пределах гладких структур отображение λ R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaaaa@3ABA@ является не чем иным, как тождественным отображением Id M R : M R M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeysaiaabsgadaWgaaWcbaGaamytamaaBaaabaGa amOuaaqabaaabeaakiaaiQdacaaMc8UaamytamaaBaaaleaacaWGsb aabeaakiabgkziUkaad2eadaWgaaWcbaGaamOuaaqabaaaaa@43A6@ на общем для рассматриваемых геометрических пространств многообразии M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ . Вместе с тем относительно геометрических структур над этим многообразием λ R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaaaa@3ABA@ не является тождественным отображением: оно преобразует каждую точку X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ неевклидовой формы с ее окрестностью и геометрией, индуцированной на эту окрестность, в ту же самую точку X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwaaa@4372@ с той же окрестностью, но геометрия которой индуцирована из объемлющего пространства E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ . Будем называть отображение λ R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaaaa@3ABA@ отображением уплощения [26].

Для формальной интерпретации отображения уплощения λ R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaaaa@3ABA@ представляется уместным использовать подход теории категорий. Поскольку этот подход не является общепринятым в механике континуума, изложим его более детально. Произвольная категория Cat MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4qaiaabggacaqG0baaaa@3AA4@ состоит из следующих двух совокупностей [68]:

MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFIaca aa@3C81@ Класса Ob(Cat) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4taiaabkgacaaMi8UaaGikaiaaboeacaqGHbGa aeiDaiaaiMcaaaa@3F51@ , элементы которого называются объектами категории Cat MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4qaiaabggacaqG0baaaa@3AA4@ .

MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFIaca aa@3C81@ Класса Hom(Cat) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaiaab+gacaqGTbGaaGjcVlaaiIcacaqGdbGa aeyyaiaabshacaaIPaaaaa@4047@ , элементы которого называются морфизмами категории Cat MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4qaiaabggacaqG0baaaa@3AA4@ .

Каждому морфизму fHom(Cat) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzaiabgIGiolaabIeacaqGVbGaaeyBaiaayIW7 caaIOaGaae4qaiaabggacaqG0bGaaGykaaaa@42B6@ соответствует пара (X,Y) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadIfacaaISaGaaGPaVlaadMfacaaIPaaa aa@3D64@ , состоящая из объектов X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ и Y MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamywaaaa@38E1@ . Первый из них называется объектом отправления, а второй MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ объектом прибытия морфизма[14] f MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzaaaa@38EE@ . Кроме того, для любых объектов X,Y,ZOb(Cat) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaiaaiYcacaaMc8UaamywaiaaiYcacaaMc8Ua amOwaiabgIGiolaab+eacaqGIbGaaGjcVlaaiIcacaqGdbGaaeyyai aabshacaaIPaaaaa@47F1@ существует бинарная операция : Hom Cat (X;Y)× Hom Cat (Y;Z) Hom Cat (X;Z) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeSigI8MaaGOoaiaaykW7caqGibGaae4Baiaab2ga daWgaaWcbaGaae4qaiaabggacaqG0baabeaakiaayIW7caaIOaGaam iwaiaaiUdacaaMc8UaamywaiaaiMcacqGHxdaTcaqGibGaae4Baiaa b2gadaWgaaWcbaGaae4qaiaabggacaqG0baabeaakiaayIW7caaIOa GaamywaiaaiUdacaaMc8UaamOwaiaaiMcacqGHsgIRcaqGibGaae4B aiaab2gadaWgaaWcbaGaae4qaiaabggacaqG0baabeaakiaayIW7ca aIOaGaamiwaiaaiUdacaaMc8UaamOwaiaaiMcaaaa@6522@ , называемая композицией. Здесь символ Hom Cat (X;Y) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisaiaab+gacaqGTbWaaSbaaSqaaiaaboeacaqG HbGaaeiDaaqabaGccaaMi8UaaGikaiaadIfacaaI7aGaaGPaVlaadM facaaIPaaaaa@4488@ обозначает класс всех морфизмов из Cat MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4qaiaabggacaqG0baaaa@3AA4@ с объектом отправления X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ и объектом прибытия Y MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamywaaaa@38E1@ . Предполагается, что композиция удовлетворяет следующим двум аксиомам:

1. Ассоциативность: (fg)h=f(gh) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadAgacqWIyiYBcaWGNbGaaGykaiablIHi VjaadIgacaaI9aGaamOzaiablIHiVjaaiIcacaWGNbGaeSigI8Maam iAaiaaiMcaaaa@4604@ .

2. Существование нейтрального элемента: для любого объекта XOb(Cat) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaiabgIGiolaab+eacaqGIbGaaGjcVlaaiIca caqGdbGaaeyyaiaabshacaaIPaaaaa@41B2@ существует морфизм Id X Hom Cat (X;X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeysaiaabsgadaWgaaWcbaGaamiwaaqabaGccqGH iiIZcaqGibGaae4Baiaab2gadaWgaaWcbaGaae4qaiaabggacaqG0b aabeaakiaayIW7caaIOaGaamiwaiaaiUdacaaMc8UaamiwaiaaiMca aaa@48D1@ , называемый тождественным морфизмом, такой, что Id Y f=f Id X =f MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeysaiaabsgadaWgaaWcbaGaamywaaqabaGccqWI yiYBcaWGMbGaaGypaiaadAgacqWIyiYBcaqGjbGaaeizamaaBaaale aacaWGybaabeaakiaai2dacaWGMbaaaa@4453@ для любого f Hom Cat (X;Y) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzaiabgIGiolaabIeacaqGVbGaaeyBamaaBaaa leaacaqGdbGaaeyyaiaabshaaeqaaOGaaGjcVlaaiIcacaWGybGaaG 4oaiaaykW7caWGzbGaaGykaaaa@46F7@ .

Замечание 7. В определении категории используется терминология, принятая в аксиоматической теории множеств [69]. В рамках нее различаются множества и классы, где последние представляют структуру более общую, нежели множества, что позволяет избежать проблемы с известными противоречиями (<<множество всех множеств>> и др.).

Замечание 8. В общем случае морфизм не является отображением в привычном понимании этого термина. Например, можно рассмотреть категорию, представленную частично упорядоченным множеством (X,) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadIfacaaISaGaaGPaVprr1ngBPrwtHrhA YaqeguuDJXwAKbstHrhAGq1DVbacfaGae8xFQqOaaGykaaaa@47D9@ . Объектами этой категории являются элементы множества X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ , а морфизмами MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbcKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C75@ упорядоченные пары (x,y) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadIhacaaISaGaaGPaVlaadMhacaaIPaaa aa@3DA4@ :

xyåñëèxy. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiEaiabgkziUkaadMhacaaMf8Uaaey5aiaabgpa caqGRdGaaei6aiaaywW7caWG4bWefv3ySLgznfgDOjdaryqr1ngBPr ginfgDObcv39gaiuaacqWF9PcHcaWG5bGaaGOlaaaa@52C2@

Композиция двух морфизмов осуществляется удалением среднего элемента по аксиоме транзитивности отношения MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb a8aacqWF9PcHaaa@4356@ , а тождественным морфизмом является пара (x,x) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadIhacaaISaGaaGPaVlaadIhacaaIPaaa aa@3DA3@ .

Таким образом, морфизм является обобщением понятия отображения. В работе используется более узкая интерпретация морфизма как отображения с некоторыми свойствами, одинаковыми для всех морфизмов рассматриваемой категории. Такое представление позволяет с единых позиций говорить, в частности, о стирании геометрических свойств, что будет рассмотрено далее.

Пусть C 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4qamaaBaaaleaacaaIXaaabeaaaaa@39B0@ и C 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4qamaaBaaaleaacaaIYaaabeaaaaa@39B1@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ произвольные категории. Функтором (ковариантным функтором) из C 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4qamaaBaaaleaacaaIXaaabeaaaaa@39B0@ в C 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4qamaaBaaaleaacaaIYaaabeaaaaa@39B1@ называется совокупность отображений (обычно обозначаемых одним и тем же символом F MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFXeIraaa@42AA@ ) таких, что имеется отображение F:Ob( C 1 )Ob( C 2 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFXeIrcaaI6aGaaGPaVlaab+eacaqGIbGaaGjcVlaaiIcaca qGdbWaaSbaaSqaaiaaigdaaeqaaOGaaGykaiabgkziUkaab+eacaqG IbGaaGjcVlaaiIcacaqGdbWaaSbaaSqaaiaaikdaaeqaaOGaaGykaa aa@53AF@ и для всех X,YOb( C 1 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaiaaiYcacaaMc8UaamywaiabgIGiolaab+ea caqGIbGaaGjcVlaaiIcacaqGdbWaaSbaaSqaaiaaigdaaeqaaOGaaG ykaaaa@43E7@ имеется другое отображение F: Hom C 1 (X;Y) Hom C 2 (F(X);F(Y)) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFXeIrcaaI6aGaaGPaVlaabIeacaqGVbGaaeyBamaaBaaale aacaqGdbWaaSbaaeaacaaIXaaabeaaaeqaaOGaaGjcVlaaiIcacaWG ybGaaG4oaiaaykW7caWGzbGaaGykaiabgkziUkaabIeacaqGVbGaae yBamaaBaaaleaacaqGdbWaaSbaaeaacaaIYaaabeaaaeqaaOGaaGjc VlaaiIcacqWFXeIrcaaIOaGaamiwaiaaiMcacaaI7aGaaGPaVlab=f tigjaaiIcacaWGzbGaaGykaiaaiMcaaaa@62F7@ такие, что

F(fg)=F(f)F(g),F( Id X )= Id F(X) , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFXeIrcaaIOaGaamOzaiablIHiVjaadEgacaaIPaGaaGypai ab=ftigjaaiIcacaWGMbGaaGykaiablIHiVjab=ftigjaaiIcacaWG NbGaaGykaiaaiYcacaaMf8Uae8xmHyKaaGikaiaabMeacaqGKbWaaS baaSqaaiaadIfaaeqaaOGaaGykaiaai2dacaqGjbGaaeizamaaBaaa leaacqWFXeIrcaaIOaGaamiwaiaaiMcaaeqaaOGaaGilaaaa@5E4D@

для всех f Hom C 1 (X;Y) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzaiabgIGiolaabIeacaqGVbGaaeyBamaaBaaa leaacaqGdbWaaSbaaeaacaaIXaaabeaaaeqaaOGaaGjcVlaaiIcaca WGybGaaG4oaiaaykW7caWGzbGaaGykaaaa@45F8@ и g Hom C 1 (Y;Z) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4zaiabgIGiolaabIeacaqGVbGaaeyBamaaBaaa leaacaqGdbWaaSbaaeaacaaIXaaabeaaaeqaaOGaaGjcVlaaiIcaca WGzbGaaG4oaiaaykW7caWGAbGaaGykaaaa@45FB@ . Весь функтор обозначается символом F: C 1 C 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFXeIrcaaI6aGaaGPaVlaaboeadaWgaaWcbaGaaGymaaqaba GccqGHsgIRcaqGdbWaaSbaaSqaaiaaikdaaeqaaaaa@4A4B@ .

В рамках формализма теории категорий отображение уплощения можно охарактеризовать следующим образом. Пусть Geom MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raiaabwgacaqGVbGaaeyBaaaa@3B97@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ категория, объектами которой являются геометрические пространства[15] (M,g,) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaad2eacaaISaGaaGPaVlaabEgacaaISaGa aGPaVlabgEGirlaaiMcaaaa@412C@ , а морфизмами которой являются гладкие отображения. Далее, пусть Diff MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiraiaabMgacaqGMbGaaeOzaaaa@3B88@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ другая категория. Ее объектами являются гладкие многообразия, а морфизмами MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ гладкие отображения. Определим стирающий функтор F:GeomDiff MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFXeIrcaaI6aGaaGPaVlaabEeacaqGLbGaae4Baiaab2gacq GHsgIRcaqGebGaaeyAaiaabAgacaqGMbaaaa@4DFF@ , который <<стирает>> информацию о геометрии подлежащего многообразия, т. е. если f:( M 1 , g 1 , 1 )( M 2 , g 2 , 2 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzaiaaiQdacaaMc8UaaGikaiaad2eadaWgaaWc baGaaGymaaqabaGccaaISaGaaGPaVlaabEgadaWgaaWcbaGaaGymaa qabaGccaaISaGaaGPaVlabgEGirpaaBaaaleaacaaIXaaabeaakiaa iMcacqGHsgIRcaaIOaGaamytamaaBaaaleaacaaIYaaabeaakiaaiY cacaaMc8Uaae4zamaaBaaaleaacaaIYaaabeaakiaaiYcacaaMc8Ua ey4bIe9aaSbaaSqaaiaaikdaaeqaaOGaaGykaaaa@5525@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ морфизм категории Geom MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raiaabwgacaqGVbGaaeyBaaaa@3B97@ , то F(f): M 1 M 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFXeIrcaaIOaGaamOzaiaaiMcacaaI6aGaaGPaVlaad2eada WgaaWcbaGaaGymaaqabaGccqGHsgIRcaWGnbWaaSbaaSqaaiaaikda aeqaaaaa@4CB3@ является морфизмом категории Diff MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiraiaabMgacaqGMbGaaeOzaaaa@3B88@ . Здесь морфизм f MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzaaaa@38EE@ преобразует точки и подмножества одного геометрического пространства в точки и подмножества другого геометрического пространства, в то время как морфизм F(f) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFXeIrcaaIOaGaamOzaiaaiMcaaaa@44FA@ действует лишь на уровне подлежащих гладких многообразий. Тогда, используя стирающий функтор, определим отображение уплощения λ R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaaaa@3ABA@ как морфизм λ R :( M R ,G,)( M R ,g | S R , | S R ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaOGaaGOoaiaa ykW7caaIOaGaamytamaaBaaaleaacaWGsbaabeaakiaaiYcacaaMc8 Uaae4raiaaiYcacaaMc8Uaey4bIeTaaGykaiabgkziUkaaiIcacaWG nbWaaSbaaSqaaiaadkfaaeqaaOGaaGilaiaaykW7caWHNbGaaGiFam aaBaaaleaacaWGtbWaaSbaaeaacaWGsbaabeaaaeqaaOGaaGilaiaa ykW7cqGHhis0caaI8bWaaSbaaSqaaiaadofadaWgaaqaaiaadkfaae qaaaqabaGccaaIPaaaaa@596A@ , применение стирающего функтора к которому дает тождественный морфизм, то есть F( λ R )= Id M R : M R M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFXeIrcaaIOaGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaOGaaG ykaiaai2dacaqGjbGaaeizamaaBaaaleaacaWGnbWaaSbaaeaacaWG sbaabeaaaeqaaOGaaGOoaiaaykW7caWGnbWaaSbaaSqaaiaadkfaae qaaOGaeyOKH4QaamytamaaBaaaleaacaWGsbaabeaaaaa@533A@ .

3.2. Обобщенная деформация

Использование неевклидовой формы позволяет определить кинематику самонапряженного тела подобно классическому подходу. Действительно, пусть S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ неевклидова форма, рассматриваемая как образ вложения ϰ R :BR MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+daWgaaWcbaGaamOuaaqabaGccaaI6aGaaGPaVprr1n gBPrMrYf2A0vNCaeXbfv3ySLgzGyKCHTgD1jhaiyaacqGFbaVqcqGH sgIRcaWGsbaaaa@5698@ тела B MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbac faWdaiab=fa8cbaa@447C@ в соответствующий неевклидов аналог R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOuaaaa@38DA@ физического пространства[16]. Если ϰ:BE MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+caaI6aGaaGPaVprr1ngBPrMrYf2A0vNCaeXbfv3ySL gzGyKCHTgD1jhaiyaacqGFbaVqcqGHsgIRcqWFWesraaa@55CF@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ некоторая <<евклидова>> конфигурация, образ которой равен S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ , то под обобщенной деформацией неевклидовой отсчетной формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ в евклидову актуальную форму S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ будем понимать композицию

λ:= ϰ ^ ϰ ^ R 1 : S R S. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWMaaGOoaiaai2dadaqiaaqaamrr1ngBPrwt HrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae8h8dKpacaGLcmaacq WIyiYBdaqiaaqaaiab=b=a5dGaayPadaWaa0baaSqaaiaadkfaaeaa cqGHsislcaaIXaaaaOGaaGOoaiaaykW7caWGtbWaaSbaaSqaaiaadk faaeqaaOGaeyOKH4Qae8NeXpLaaGOlaaaa@57D6@ (1)

С другой стороны, используя отображение уплощения, можно представить такое же отображение λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWgaaa@39B7@ как композицию

λ=γ λ R : S R S. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWMaaGypaiabeo7aNjablIHiVjabeU7aSnaa BaaaleaacaWGsbaabeaakiaaiQdacaaMc8Uaam4uamaaBaaaleaaca WGsbaabeaakiabgkziUorr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8NeXpLaaGOlaaaa@525E@ (2)

Здесь λ R : S R S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaOGaaGOoaiaa ykW7caWGtbWaaSbaaSqaaiaadkfaaeqaaOGaeyOKH46efv3ySLgznf gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFse=udaWgaaWcbaGa amOuaaqabaaaaa@4D4D@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ отображение уплощения, а γ: S R S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdCMaaGOoaiaaykW7tuuDJXwAK1uy0HwmaeHb fv3ySLgzG0uy0Hgip5wzaGqbaiab=jr8tnaaBaaaleaacaWGsbaabe aakiabgkziUkab=jr8tbaa@4C33@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ <<евклидова>> деформация. Таким образом, деформация неевклидовой отсчетной формы в евклидову актуальную форму состоит из уплощения неевклидовой формы, т. е. вложения в евклидово пространство E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ , и последующей евклидовой деформации, которая преобразует формы в евклидовом пространстве.

Соотношение (2) могло бы показаться тривиальной переформулировкой равенства (1), в которой λ R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaaaa@3ABA@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ тождественное отображение. В действительности, с геометрической точки зрения это не так. Для уяснения обстоятельств, позволивших перейти к равенству (2), следует все отображения, составляющие композицию (1), рассматривать как морфизмы категории Geom MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raiaabwgacaqGVbGaaeyBaaaa@3B97@ . Далее, вспомним, что евклидова форма S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ является образом евклидовой конфигурации ϰ R :BE MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWpq+daWgaaWcbaGaamOuaaqabaGccaaI6aGaaGPaVprr1n gBPrMrYf2A0vNCaeXbfv3ySLgzGyKCHTgD1jhaiyaacqGFbaVqcqGH sgIRcqWFWesraaa@56DC@ (обратите внимание, здесь нижний индекс R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOuaaaa@38DA@ , а не R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOuaaaa@38DA@ ). Тогда, вставляя тождественный морфизм Id B = ϰ ^ R 1 ϰ ^ R :BB MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeysaiaabsgadaWgaaWcbaWefv3ySLgzgjxyRrxD YbqeguuDJXwAKbIrYf2A0vNCaGqbaiab=fa8cbqabaGccaaI9aWaae caaeaatuuDJXwAK1uy0HwmaeXbfv3ySLgzG0uy0Hgip5wzaGGbaiab +b=a5dGaayPadaWaa0baaSqaaiaadkfaaeaacqGHsislcaaIXaaaaO GaeSigI82aaecaaeaacqGFWpq+aiaawkWaamaaBaaaleaacaWGsbaa beaakiaaiQdacaaMc8Uae8xaWlKaeyOKH4Qae8xaWleaaa@63C5@ между отображениями ϰ ^ R 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaecaaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy 0Hgip5wzaGqbaiab=b=a5dGaayPadaWaa0baaSqaaiaadkfaaeaacq GHsislcaaIXaaaaaaa@476C@ и ϰ ^ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaecaaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy 0Hgip5wzaGqbaiab=b=a5dGaayPadaaaaa@44C0@ , получаем из (1) формулу (2):

λ= ϰ ^ ϰ ^ R 1 ϰ ^ R ϰ ^ R 1 =γ λ R , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWMaaGypamaaHaaabaWefv3ySLgznfgDOfda ryqr1ngBPrginfgDObYtUvgaiuaacqWFWpq+aiaawkWaaiablIHiVn aaHaaabaGae8h8dKpacaGLcmaadaqhaaWcbaGaamOuaaqaaiabgkHi TiaaigdaaaGccqWIyiYBdaqiaaqaaiab=b=a5dGaayPadaWaaSbaaS qaaiaadkfaaeqaaOGaeSigI82aaecaaeaacqWFWpq+aiaawkWaamaa DaaaleaacaWGsbaabaGaeyOeI0IaaGymaaaakiaai2dacqaHZoWzcq WIyiYBcqaH7oaBdaWgaaWcbaGaamOuaaqabaGccaaISaaaaa@621A@

где γ= ϰ ^ ϰ ^ R 1 : S R S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdCMaaGypamaaHaaabaWefv3ySLgznfgDOfda ryqr1ngBPrginfgDObYtUvgaiuaacqWFWpq+aiaawkWaaiablIHiVn aaHaaabaGae8h8dKpacaGLcmaadaqhaaWcbaGaamOuaaqaaiabgkHi TiaaigdaaaGccaaI6aGaaGPaVlab=jr8tnaaBaaaleaacaWGsbaabe aakiabgkziUkab=jr8tbaa@5750@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ деформация евклидовых форм, а λ R := ϰ ^ R ϰ ^ R 1 : S R S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaOGaaGOoaiaa i2dadaqiaaqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLb acfaGae8h8dKpacaGLcmaadaWgaaWcbaGaamOuaaqabaGccqWIyiYB daqiaaqaaiab=b=a5dGaayPadaWaa0baaSqaaiaadkfaaeaacqGHsi slcaaIXaaaaOGaaGOoaiaaykW7caWGtbWaaSbaaSqaaiaadkfaaeqa aOGaeyOKH4Qae8NeXp1aaSbaaSqaaiaadkfaaeqaaaaa@5A3B@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ искомое отображение уплощения.

Таким образом, в рамках категории Geom MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4raiaabwgacaqGVbGaaeyBaaaa@3B97@ отображение λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWgaaa@39B7@ переводит неевклидово отсчетное состояние в актуальное евклидово напряженное состояние. Каждое из этих состояний формализуется в чисто геометрических терминах, а поля, характеризующие отсчетное и актуальное состояния, трансформируются друг в друга посредством операций прямого и обратного образов. Вместе с тем стирающий функтор F MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFXeIraaa@42AA@ , примененный к λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWgaaa@39B7@ , дает морфизм категории Diff MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeiraiaabMgacaqGMbGaaeOzaaaa@3B88@ , F(λ): M R M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFXeIrcaaIOaGaeq4UdWMaaGykaiaaiQdacaaMc8Uaamytam aaBaaaleaacaWGsbaabeaakiabgkziUkaad2eaaaa@4CB0@ , между подлежащими многообразиями форм, в рамках которого операции прямого и обратного образов пересчитывают лишь дифференциальные свойства полей на подлежащих многообразиях.

3.3. Координатное представление обобщенной деформации

Рассмотрим представление обобщенной деформации λ: S R S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWMaaGOoaiaaykW7caWGtbWaaSbaaSqaaiaa dkfaaeqaaOGaeyOKH46efv3ySLgznfgDOfdaryqr1ngBPrginfgDOb YtUvgaiuaacqWFse=uaaa@4B3D@ в локальных координатах. Хотя λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWgaaa@39B7@ является гладким отображением многообразий и его координатное представление может быть рассмотрено с общих позиций теории гладких многообразий, имеется особенность, связанная с тем, что подструктуры M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ и M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytaaaa@38D5@ области определения S R =( M R ,G,) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaakiaai2dacaaI OaGaamytamaaBaaaleaacaWGsbaabeaakiaaiYcacaaMc8Uaae4rai aaiYcacaaMc8Uaey4bIeTaaGykaaaa@44C5@ и области прибытия S=(M,g | S , | S ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=ucaaI9aGaaGikaiaad2eacaaISaGaaGPaVlaahEgaca aI8bWaaSbaaSqaaiaadofaaeqaaOGaaGilaiaaykW7cqGHhis0caaI 8bWaaSbaaSqaaiaadofaaeqaaOGaaGykaaaa@5186@ являются n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ -мерными подмногообразиями E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ .

Для многообразия M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ существует семейство {( U X R , φ X R )} X M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4EaiaaiIcacaWGvbWaa0baaSqaamrr1ngBPrwt HrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae83fXJfabaGaamOuaa aakiaaiYcacaaMc8UaeqOXdO2aa0baaSqaaiab=Dr8ybqaaiaadkfa aaGccaaIPaGaaGyFamaaBaaaleaacqWFxepwcqGHiiIZcaWGnbWaaS baaeaacaWGsbaabeaaaeqaaaaa@551B@ карт из максимального атласа пространства E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ такое, что [39, теорема 5.8]

MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFIaca aa@3C81@ для любой точки X M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcqGHiiIZcaWGnbWaaSbaaSqaaiaadkfaaeqaaaaa@46CB@ справедливо включение X U X R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcqGHiiIZcaWGvbWaa0baaSqaaiab=Dr8ybqaaiaadk faaaaaaa@48B9@ , и, таким образом, семейство { U X R } X M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4EaiaadwfadaqhaaWcbaWefv3ySLgznfgDOfda ryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwaeaacaWGsbaaaOGaaG yFamaaBaaaleaacqWFxepwcqGHiiIZcaWGnbWaaSbaaeaacaWGsbaa beaaaeqaaaaa@4CC5@ покрывает M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ ,

MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFIaca aa@3C81@ пересечение M R U X R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaakiabgMIihlaa dwfadaqhaaWcbaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUv gaiuaacqWFxepwaeaacaWGsbaaaaaa@48CD@ является n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ -срезкой U X R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyvamaaDaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqaaiaadkfaaaaaaa@4550@ .

Последнее условие означает, что для всякой точки Y U X R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFyeFwcqGHiiIZcaWGvbWaa0baaSqaaiab=Dr8ybqaaiaadk faaaaaaa@48BB@ образ φ X (Y) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOXdO2aaSbaaSqaamrr1ngBPrwtHrhAXaqeguuD JXwAKbstHrhAG8KBLbacfaGae83fXJfabeaakiaaiIcacqWFyeFwca aIPaaaaa@48B1@ имеет нулевыми последние 3n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4maiabgkHiTiaad6gaaaa@3AA0@ координат[17]. Таким образом, семейство A M R ={( M R U X R ,π φ X R | M R U X R )} X M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb a8aacqWFacFqdaWgaaWcbaGaamytamaaBaaabaGaamOuaaqabaaabe aakiaai2dacaaI7bGaaGikaiaad2eadaWgaaWcbaGaamOuaaqabaGc cqGHPiYXcaWGvbWaa0baaSqaamrr1ngBPrwtHrhAXaqehuuDJXwAKb stHrhAG8KBLbacgaGae43fXJfabaGaamOuaaaakiaaiYcacaaMc8Ua eqiWdaNaeSigI8MaeqOXdO2aa0baaSqaaiab+Dr8ybqaaiaadkfaaa GccaaI8bWaaSbaaSqaaiaad2eadaWgaaqaaiaadkfaaeqaaiabgMIi hlaadwfadaqhaaqaaiab+Dr8ybqaaiaadkfaaaaabeaakiaaiMcaca aI9bWaaSbaaSqaaiab+Dr8yjabgIGiolaad2eadaWgaaqaaiaadkfa aeqaaaqabaaaaa@7289@ является гладким атласом на M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ . Здесь π: 3 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiWdaNaaGOoaiaaykW7tuuDJXwAK1uy0HMmaeHb fv3ySLgzG0uy0HgiuD3BaGqbaiab=1risnaaCaaaleqabaGaaG4maa aakiabgkziUkab=1risnaaCaaaleqabaGaamOBaaaaaaa@4BDD@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ проекция на первые n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ координат. Отбрасывая повторяющиеся карты ( M R U X R ,π φ X R | M R U X R ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaad2eadaWgaaWcbaGaamOuaaqabaGccqGH PiYXcaWGvbWaa0baaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHr hAG8KBLbacfaGae83fXJfabaGaamOuaaaakiaaiYcacaaMc8UaeqiW daNaeSigI8MaeqOXdO2aa0baaSqaaiab=Dr8ybqaaiaadkfaaaGcca aI8bWaaSbaaSqaaiaad2eadaWgaaqaaiaadkfaaeqaaiabgMIihlaa dwfadaqhaaqaaiab=Dr8ybqaaiaadkfaaaaabeaakiaaiMcaaaa@5C80@ и обозначая через U α = M R U X R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyvamaaBaaaleaacqaHXoqyaeqaaOGaaGypaiaa d2eadaWgaaWcbaGaamOuaaqabaGccqGHPiYXcaWGvbWaa0baaSqaam rr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae83fXJfa baGaamOuaaaaaaa@4C43@ , φ α =π φ X R | M R U X R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOXdO2aaSbaaSqaaiabeg7aHbqabaGccaaI9aGa eqiWdaNaeSigI8MaeqOXdO2aa0baaSqaamrr1ngBPrwtHrhAXaqegu uDJXwAKbstHrhAG8KBLbacfaGae83fXJfabaGaamOuaaaakiaaiYha daWgaaWcbaGaamytamaaBaaabaGaamOuaaqabaGaeyykICSaamyvam aaDaaabaGae83fXJfabaGaamOuaaaaaeqaaaaa@55DF@ , элементы оставшихся карт, окончательно приходим к следующему представлению для атласа A M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb a8aacqWFacFqdaWgaaWcbaGaamytamaaBaaabaGaamOuaaqabaaabe aaaaa@45D2@ : A M R ={( U α , φ α )} αI MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb a8aacqWFacFqdaWgaaWcbaGaamytamaaBaaabaGaamOuaaqabaaabe aakiaai2dacaaI7bGaaGikaiaadwfadaWgaaWcbaGaeqySdegabeaa kiaaiYcacaaMc8UaeqOXdO2aaSbaaSqaaiabeg7aHbqabaGccaaIPa GaaGyFamaaBaaaleaacqaHXoqycqGHiiIZcaWGjbaabeaaaaa@56B3@ . Хотя упоминание о гладкой структуре объемлющего пространства было стерто из этого атласа, следует иметь в виду, что на самом деле карты этого атласа были получены из карт физического пространства. Аналогичные рассуждения, примененные к многообразию M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytaaaa@38D5@ , дают атлас A M ={( V β , ψ β )} βJ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyqamaaBaaaleaacaWGnbaabeaakiaai2dacaaI 7bGaaGikaiaadAfadaWgaaWcbaGaeqOSdigabeaakiaaiYcacaaMc8 UaeqiYdK3aaSbaaSqaaiabek7aIbqabaGccaaIPaGaaGyFamaaBaaa leaacqaHYoGycqGHiiIZcaWGkbaabeaaaaa@4AC1@ .

Пусть теперь X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaiabgIGiolaadofadaWgaaWcbaGaamOuaaqa baaaaa@3C3F@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ некоторая точка неевклидовой формы. В силу непрерывности λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWgaaa@39B7@ , существует пара карт (U,φ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadwfacaaISaGaaGPaVlabeA8aQjaaiMca aaa@3E40@ и (V,ψ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadAfacaaISaGaaGPaVlabeI8a5jaaiMca aaa@3E52@ из гладких структур S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ и S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ такая, что XU MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaiabgIGiolaadwfaaaa@3B3E@ и λ(U)V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWMaaGikaiaadwfacaaIPaGaeyOGIWSaamOv aaaa@3ECD@ . Тогда композиция

λ~ψλφ1 : φ(U)ψ(V),λ~ : (Q1,,Qn)(q1,,qn) (3)

соответствует координатному представлению λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWgaaa@39B7@ в окрестности X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ . По определению, она является отображением между открытыми подмножествами n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb a8aacqWFDeIudaahaaWcbeqaaiaad6gaaaaaaa@43DB@ .

3.4. Некоторые замечания о представлении реперов

Поскольку неевклидова форма S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ рассматривается как отдельное многообразие, приходится определять абстрактные координатный репер ( Q A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabgkGi2oaaBaaaleaacaWGrbWaaWbaaeqa baGaamyqaaaaaeqaaOGaaGykamaaDaaaleaacaWGbbGaaGypaiaaig daaeaacaWGUbaaaaaa@402A@ и корепер (d Q A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadsgacaWGrbWaaWbaaSqabeaacaWGbbaa aOGaaGykamaaDaaaleaacaWGbbGaaGypaiaaigdaaeaacaWGUbaaaa aa@3F8C@ , порожденные локальными координатами ( Q A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadgfadaahaaWcbeqaaiaadgeaaaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EA3@ . Иначе обстоит дело в случае евклидовой формы S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ , поскольку она рассматривается как подпространство физического пространства E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ .

Локальные координаты (qA)A=1n на S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ порождают абстрактные поля (qA)A=1n и (dqA)An координатного репера и корепера. С другой стороны, каноническая инъекция ι S :SE MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyUdK2aaSbaaSqaamrr1ngBPrwtHrhAXaqeguuD JXwAKbstHrhAG8KBLbacfaGae8NeXpfabeaakiaaiQdacaaMc8Uae8 NeXpLae83OHaRae8hmHueaaa@4C8B@ индуцирует касательное отображение T ι S :TSTE MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivaiabeM7aPnaaBaaaleaatuuDJXwAK1uy0Hwm aeHbfv3ySLgzG0uy0Hgip5wzaGqbaiab=jr8tbqabaGccaaI6aGaaG PaVlaadsfacqWFse=ucqGHsgIRcaWGubGae8hmHueaaa@4F0A@ , которое является инъективным линейным отображением на каждом касательном слое: TxιSHom(TxS; V). По этой причине касательное пространство TxS* может быть отождествлено с n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaaaa@38F6@ -мерным векторным подпространством V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFveVvaaa@436E@ . В силу этого, полагаем eA|x :=TxιS[qA], что приводит к семейству ( e A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahwgadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EBA@ векторных полей e A :SV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCyzamaaBaaaleaacaWGbbaabeaakiaaiQdacaaM c8+efv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFse =ucqGHsgIRcqWFveVvaaa@4B6F@ , касательных к S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ . Это семейство есть не более чем образ абстрактного локального репера (qA)A=1n при вложении многообразия M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytaaaa@38D5@ , подлежащего для S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ , в физическое пространство.

3.5. Градиент обобщенной деформации

Если λ: S R S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWMaaGOoaiaaykW7caWGtbWaaSbaaSqaaiaa dkfaaeqaaOGaeyOKH46efv3ySLgznfgDOfdaryqr1ngBPrginfgDOb YtUvgaiuaacqWFse=uaaa@4B3D@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ обобщенная деформация и если X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaiabgIGiolaadofadaWgaaWcbaGaamOuaaqa baaaaa@3C3F@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ некоторая точка отсчетной формы, то обобщение градиента евклидовой деформации представлено касательным отображением T X λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaacaWGybaabeaakiabeU7aSjab gIGiodaa@3D27@ Hom( T X S R ; T λ(X) S) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyicI4Saaeisaiaab+gacaqGTbGaaGikaiaadsfa daWgaaWcbaGaamiwaaqabaGccaWGtbWaaSbaaSqaaiaadkfaaeqaaO GaaG4oaiaaykW7caWGubWaaSbaaSqaaiabeU7aSjaaiIcacaWGybGa aGykaaqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiu aakiab=jr8tjaaiMcaaaa@5424@ . Это MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ линейное отображение, которое в координатах ( Q A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadgfadaahaaWcbeqaaiaadgeaaaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EA3@ и (qA)A=1n на формах S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ и S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ , определенных в окрестностях X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ и λ(X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWMaaGikaiaadIfacaaIPaaaaa@3BF9@ соответственно, имеет следующее диадное представление:

TXλqAQBXeAdQB,

где qAQB  MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ частные производные координатного представления (3). Таким образом, приходим к семейству { T X λ} X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4EaiaadsfadaWgaaWcbaGaamiwaaqabaGccqaH 7oaBcaaI9bWaaSbaaSqaaiaadIfacqGHiiIZcaWGtbWaaSbaaeaaca WGsbaabeaaaeqaaaaa@420C@ касательных отображений. По нему синтезируется глобальное касательное отображение Tλ:T S R TS MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivaiabeU7aSjaaiQdacaaMc8Uaamivaiaadofa daWgaaWcbaGaamOuaaqabaGccqGHsgIRcaWGubWefv3ySLgznfgDOf daryqr1ngBPrginfgDObYtUvgaiuaacqWFse=uaaa@4DC8@ , которое действует между касательными расслоениями T S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivaiaadofadaWgaaWcbaGaamOuaaqabaaaaa@3AB7@ и TS MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8NeXpfaaa@4441@ .

Можно получить простейшее координатное представление для градиента Tλ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivaiabeU7aSbaa@3A90@ обобщенной деформации, которое аналогично соответствующему представлению градиента евклидовой деформации, используемому в ряде монографий по теории упругости [50]. Это представление может быть получено путем переноса локальных координат из актуальной формы S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ на отсчетную S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ следующим способом.

Пусть A S ={( V β , ψ β )} βJ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb a8aacqWFacFqdaWgaaWcbaWefv3ySLgznfgDOfdarCqr1ngBPrginf gDObYtUvgaiyaacqGFse=uaeqaaOGaaGypaiaaiUhacaaIOaGaamOv amaaBaaaleaacqaHYoGyaeqaaOGaaGilaiaaykW7cqaHipqEdaWgaa WcbaGaeqOSdigabeaakiaaiMcacaaI9bWaaSbaaSqaaiabek7aIjab gIGiolaadQeaaeqaaaaa@6068@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ гладкий атлас на S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ , индуцированный криволинейными координатами из E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ . Здесь множество V β S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOvamaaBaaaleaacqaHYoGyaeqaaOGaeyOGIW8e fv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFse=uaa a@4816@ открыто в S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ , а вся совокупность { V β } βJ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4EaiaadAfadaWgaaWcbaGaeqOSdigabeaakiaa i2hadaWgaaWcbaGaeqOSdiMaeyicI4SaamOsaaqabaaaaa@40E1@ образует открытое покрытие S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ . Более того, для любого βJ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOSdiMaeyicI4SaamOsaaaa@3BF7@ отображение

ψβ : VβOβ,ψβ(x)=(q1,, qn)

является гомеоморфизмом между V β MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOvamaaBaaaleaacqaHYoGyaeqaaaaa@3AAB@ и открытым подмножеством O β MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4tamaaBaaaleaacqaHYoGyaeqaaaaa@3AA4@ пространства n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb a8aacqWFDeIudaahaaWcbeqaaiaad6gaaaaaaa@43DB@ .

Для каждого βJ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOSdiMaeyicI4SaamOsaaaa@3BF7@ положим

U β = λ 1 ( V β ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyvamaaBaaaleaacqaHYoGyaeqaaOGaaGypaiab eU7aSnaaCaaaleqabaGaeyOeI0IaaGymaaaakiaaiIcacaWGwbWaaS baaSqaaiabek7aIbqabaGccaaIPaaaaa@4325@

и

φ β = ψ β λ | U β : U β O β . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOXdO2aaSbaaSqaaiabek7aIbqabaGccaaI9aGa eqiYdK3aaSbaaSqaaiabek7aIbqabaGccqWIyiYBcqaH7oaBcaaI8b WaaSbaaSqaaiaadwfadaWgaaqaaiabek7aIbqabaaabeaakiaaiQda caaMc8UaamyvamaaBaaaleaacqaHYoGyaeqaaOGaeyOKH4Qaam4tam aaBaaaleaacqaHYoGyaeqaaOGaaGOlaaaa@5119@

Тогда U β S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyvamaaBaaaleaacqaHYoGyaeqaaOGaeyOGIWSa am4uamaaBaaaleaacaWGsbaabeaaaaa@3E8B@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ открытое множество, семейство { V β } βJ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4EaiaadAfadaWgaaWcbaGaeqOSdigabeaakiaa i2hadaWgaaWcbaGaeqOSdiMaeyicI4SaamOsaaqabaaaaa@40E1@ образует покрытие S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ , а φ β MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOXdO2aaSbaaSqaaiabek7aIbqabaaaaa@3B8D@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ гомеоморфизм. Следовательно, совокупность A S R ={( U β , φ β )} βJ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb a8aacqWFacFqdaWgaaWcbaGaam4uamaaBaaabaGaamOuaaqabaaabe aakiaai2dacaaI7bGaaGikaiaadwfadaWgaaWcbaGaeqOSdigabeaa kiaaiYcacaaMc8UaeqOXdO2aaSbaaSqaaiabek7aIbqabaGccaaIPa GaaGyFamaaBaaaleaacqaHYoGycqGHiiIZcaWGkbaabeaaaaa@56C0@ является гладким атласом на S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ . Это означает, что точки из S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ и S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ характеризуются одинаковыми координатами (qA)A=1n. Рисунок 4 иллюстрирует идею такой арифметизации S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ .

 

Рис. 3.1. Специальная арифметизация неевклидовой формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@

Fig. 3.1. Special arithmetization of non-Euclidean shape S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@

 

Рассмотрим обобщенную деформацию λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWgaaa@39B7@ в паре карт ( U β , φ β ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadwfadaWgaaWcbaGaeqOSdigabeaakiaa iYcacaaMc8UaeqOXdO2aaSbaaSqaaiabek7aIbqabaGccaaIPaaaaa@41EE@ и ( V β , ψ β ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadAfadaWgaaWcbaGaeqOSdigabeaakiaa iYcacaaMc8UaeqiYdK3aaSbaaSqaaiabek7aIbqabaGccaaIPaaaaa@4200@ :

λ ˜ = ψ β λ φ β 1 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqaH7oaBaiaawoWaaiaai2dacqaHipqE daWgaaWcbaGaeqOSdigabeaakiablIHiVjabeU7aSjablIHiVjabeA 8aQnaaDaaaleaacqaHYoGyaeaacqGHsislcaaIXaaaaOGaaGOlaaaa @4902@

Раскрывая определение φ β MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOXdO2aaSbaaSqaaiabek7aIbqabaaaaa@3B8D@ и принимая во внимание, что

λ λ ^ 1 | V β = ι V β : V β S, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWMaeSigI82aaecaaeaacqaH7oaBaiaawkWa amaaCaaaleqabaGaeyOeI0IaaGymaaaakiaaiYhadaWgaaWcbaGaam OvamaaBaaabaGaeqOSdigabeaaaeqaaOGaaGypaiabeM7aPnaaBaaa leaacaWGwbWaaSbaaeaacqaHYoGyaeqaaaqabaGccaaI6aGaaGPaVl aadAfadaWgaaWcbaGaeqOSdigabeaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaOGae83OHaRae8NeXpLaaGilaaaa@5B7D@

где ι V β MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyUdK2aaSbaaSqaaiaadAfadaWgaaqaaiabek7a Ibqabaaabeaaaaa@3C7B@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ каноническая инъекция, получаем

λ ˜ = ψ β ι V β ψ β 1 = Id O β . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacqaH7oaBaiaawoWaaiaai2dacqaHipqE daWgaaWcbaGaeqOSdigabeaakiablIHiVjabeM7aPnaaBaaaleaaca WGwbWaaSbaaeaacqaHYoGyaeqaaaqabaGccqWIyiYBcqaHipqEdaqh aaWcbaGaeqOSdigabaGaeyOeI0IaaGymaaaakiaai2dacaqGjbGaae izamaaBaaaleaacaWGpbWaaSbaaeaacqaHYoGyaeqaaaqabaGccaaI Uaaaaa@5127@

Следовательно, координатное представление λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdWgaaa@39B7@ является тождественным отображением.

Пусть ( e A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahwgadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EBA@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ поле локальных базисов, которое соответствует локальным координатам (qA)A=1n на S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ , а (qA)A=1n  MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ локальный репер на S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ , порожденный теми же локальными координатами (qA)A=1n. Этим реперам соответствуют кореперы ( E A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadweadaahaaWcbeqaaiaadgeaaaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3E97@ и (dqA)A=1n. Поскольку относительно карт ( U β , φ β ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadwfadaWgaaWcbaGaeqOSdigabeaakiaa iYcacaaMc8UaeqOXdO2aaSbaaSqaaiabek7aIbqabaGccaaIPaaaaa@41EE@ и ( V β , ψ β ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadAfadaWgaaWcbaGaeqOSdigabeaakiaa iYcacaaMc8UaeqiYdK3aaSbaaSqaaiabek7aIbqabaGccaaIPaaaaa@4200@ справедливо соотношение λAqB=δBA, то приходим к равенствам

Tλ=δBAeAdqBeAdqA.

Таким образом, матрица градиента в каждой точке неевклидовой формы совпадает с единичной матрицей.

3.6. Имплант

Главная линейная часть отображения λ R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaaaa@3ABA@ в точке X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaiabgIGiolaadofadaWgaaWcbaGaamOuaaqa baaaaa@3C3F@ определяется касательным отображением, которое в настоящей работе обозначается через K X = T X λ R : T X S R T X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4samaaBaaaleaacaWGybaabeaakiaai2dacaWG ubWaaSbaaSqaaiaadIfaaeqaaOGaeq4UdW2aaSbaaSqaaiaadkfaae qaaOGaaGOoaiaaykW7caWGubWaaSbaaSqaaiaadIfaaeqaaOGaam4u amaaBaaaleaacaWGsbaabeaakiabgkziUkaadsfadaWgaaWcbaWefv 3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwaeqa aOGae8NeXp1aaSbaaSqaaiaadkfaaeqaaaaa@56C1@ , где X= λ R (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcaaI9aGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaOGaaG ikaiaadIfacaaIPaaaaa@493C@ . Следуя терминологии, используемой в монографии [18], будем называть отображение K X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4samaaBaaaleaacaWGybaabeaaaaa@39DA@ имплантом. В локальных координатах имплант имеет следующее диадное представление:

K X = Q A Q B E A d Q B . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4samaaBaaaleaacaWGybaabeaakiaai2dadaWc aaqaaiabgkGi2orr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLb acfaGae8heXh1aaWbaaSqabeaacaWGbbaaaaGcbaGaeyOaIyRaamyu amaaCaaaleqabaGaamOqaaaaaaGccaWHfbWaaSbaaSqaaiaadgeaae qaaOGaey4LIqSaamizaiaadgfadaahaaWcbeqaaiaadkeaaaGccaaI Uaaaaa@5301@

Здесь ( Q A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadgfadaahaaWcbeqaaiaadgeaaaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EA3@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ локальные координаты на неевклидовой форме S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ , а ( Q A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8heXh1aaWbaaSqabeaacaWGbbaaaOGaaGykamaaDa aaleaacaWGbbGaaGypaiaaigdaaeaacaWGUbaaaaaa@492E@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ локальные координаты на евклидовой форме S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ . Заметим, что хотя обе эти формы имеют одно и то же подлежащее многообразие M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ , локальные координаты на них могут быть выбраны различными. Семейство ( E A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahweadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3E9A@ является полем локальных базисов на S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ .

Поскольку по построению форма S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ глобально единообразна, имплант K X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4samaaBaaaleaacaWGybaabeaaaaa@39DA@ преобразует инфинитезимальную единообразную окрестность точки X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ в инфинитезимальную неединообразную (в общем случае) окрестность точки X= λ R (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcaaI9aGaeq4UdW2aaSbaaSqaaiaadkfaaeqaaOGaaG ikaiaadIfacaaIPaaaaa@493C@ . Значит, имплант действует подобно обратному отображению к локальной деформации H X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaaaaa@446F@ , которая преобразует инфинитезимальную неединообразную окрестность в единообразную. Вместе с тем H X 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaDaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqaaiabgkHiTiaaigdaaaaaaa@4612@ и K X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4samaaBaaaleaacaWGybaabeaaaaa@39DA@ являются отображениями между разными евклидовыми векторными пространствами:

H X 1 :(U, g U )( T X M R ,g | M R ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaDaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqaaiabgkHiTiaaigdaaaGcca aI6aGaaGPaVlaaiIcacqWFueFvcaaISaGaaGPaVlaahEgadaWgaaWc baGae8hfXxfabeaakiaaiMcacqGHsgIRcaaIOaGaamivamaaBaaale aacqWFxepwaeqaaOGaamytamaaBaaaleaacaWGsbaabeaakiaaiYca caaMc8UaaC4zaiaaiYhadaWgaaWcbaGaamytamaaBaaabaGaamOuaa qabaaabeaakiaaiMcaaaa@5F51@

для обратного отображения к локальной деформации и

K X :( T X M R , G X )( T X M R ,g | M R ), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4samaaBaaaleaacaWGybaabeaakiaaiQdacaaM c8UaaGikaiaadsfadaWgaaWcbaGaamiwaaqabaGccaWGnbWaaSbaaS qaaiaadkfaaeqaaOGaaGilaiaaykW7caqGhbWaaSbaaSqaaiaadIfa aeqaaOGaaGykaiabgkziUkaaiIcacaWGubWaaSbaaSqaamrr1ngBPr wtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae83fXJfabeaakiaa d2eadaWgaaWcbaGaamOuaaqabaGccaaISaGaaGPaVlaahEgacaaI8b WaaSbaaSqaaiaad2eadaWgaaqaaiaadkfaaeqaaaqabaGccaaIPaGa aGilaaaa@5E1D@

для импланта. В явном виде соотношение между имплантом и локальной деформацией может быть выражено как

H X K X = I ˜ X , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaBaaaleaacaWGybaabeaakiaabUeadaWg aaWcbaGaamiwaaqabaGccaaI9aWaaacaaeaacaqGjbaacaGLdmaada WgaaWcbaGaamiwaaqabaGccaaISaaaaa@3FE0@

где I ˜ X = e A d Q A MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacaqGjbaacaGLdmaadaWgaaWcbaGaamiw aaqabaGccaaI9aGaaCyzamaaBaaaleaacaWGbbaabeaakiabgEPiel aadsgacaWGrbWaaWbaaSqabeaacaWGbbaaaaaa@4210@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ изометрия. В этом диадном представлении семейство ( e A ) A=1 n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaahwgadaWgaaWcbaGaamyqaaqabaGccaaI PaWaa0baaSqaaiaadgeacaaI9aGaaGymaaqaaiaad6gaaaaaaa@3EBA@ отвечает базису U MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFueFvaaa@436C@ .

3.7. Сравнение с классическим подходом

Целесообразно сравнить подход, связанный с построением глобальной натуральной формы, со стандартными рассуждениями, в которых такая форма не используется. Классическая теория необратимых деформаций оперирует семейством { S (X) } X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaG4Eamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhA G8KBLbacfaGae8NeXp1aaWbaaSqabeaacaaIOaGae83fXJLaaGykaa aakiaai2hadaWgaaWcbaGae83fXJLaeyicI4Sae8NeXp1aaSbaaeaa caWGsbaabeaaaeqaaaaa@4F5D@ евклидовых форм MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ образов разгрузочных деформаций γ (X) : S R S (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdC2aaWbaaSqabeaacaaIOaWefv3ySLgznfgD Ofdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwcaaIPaaaaOGaaG OoaiaaykW7cqWFse=udaWgaaWcbaGaamOuaaqabaGccqGHsgIRcqWF se=udaahaaWcbeqaaiaaiIcacqWFxepwcaaIPaaaaaaa@532B@ . Если S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ актуальная форма, а γ: S R S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4SdCMaaGOoaiaaykW7tuuDJXwAK1uy0HwmaeHb fv3ySLgzG0uy0Hgip5wzaGqbaiab=jr8tnaaBaaaleaacaWGsbaabe aakiabgkziUkab=jr8tbaa@4C33@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ соответствующая деформация, то для каждой точки X S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFxepwcqGHiiIZcqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@47D4@ композиция

χ (X) =γ [ γ (X) ] 1 : S (X) S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdm2aaWbaaSqabeaacaaIOaWefv3ySLgznfgD Ofdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwcaaIPaaaaOGaaG ypaiabeo7aNjablIHiVjaaiUfacqaHZoWzdaahaaWcbeqaaiaaiIca cqWFxepwcaaIPaaaaOGaaGyxamaaCaaaleqabaGaeyOeI0IaaGymaa aakiaaiQdacaaMc8Uae8NeXp1aaWbaaSqabeaacaaIOaGae83fXJLa aGykaaaakiabgkziUkab=jr8tbaa@5EB3@ (4)

отвечает деформации, которая преобразует локально единообразную форму S (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaahaaWcbeqaaiaaiIcacqWFxepwcaaIPaaaaaaa@46DF@ в актуальную форму S MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=uaaa@4368@ .

Переходя к касательному отображению в точке Y S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFyeFwcqGHiiIZcqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@47D6@ , получаем равенство

T γ (X) (Y) χ (X) = F Y T γ (X) (Y) [ γ (X) ] 1 : T γ (X) (Y) S (X) T γ(Y) S, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaacqaHZoWzdaahaaqabeaacaaI OaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxe pwcaaIPaaaaiaaiIcacqWFyeFwcaaIPaaabeaakiabeE8aJnaaCaaa leqabaGaaGikaiab=Dr8yjaaiMcaaaGccaaI9aGaaCOramaaBaaale aacqWFyeFwaeqaaOGaeSigI8MaamivamaaBaaaleaacqaHZoWzdaah aaqabeaacaaIOaGae83fXJLaaGykaaaacaaIOaGae8hgXNLaaGykaa qabaGccaaIBbGaeq4SdC2aaWbaaSqabeaacaaIOaGae83fXJLaaGyk aaaakiaai2fadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaaI6aGaaG PaVlaadsfadaWgaaWcbaWaaSbaaeaacqaHZoWzdaahaaqabeaacaaI OaGae83fXJLaaGykaaaacaaIOaGae8hgXNLaaGykaaqabaaabeaaki ab=jr8tnaaCaaaleqabaGaaGikaiab=Dr8yjaaiMcaaaGccqGHsgIR caWGubWaaSbaaSqaaiabeo7aNjaaiIcacqWFyeFwcaaIPaaabeaaki ab=jr8tjaaiYcaaaa@831B@

где F Y = T Y γ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCOramaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Hr8zbqabaGccaaI9aGaamivamaaBa aaleaacqWFyeFwaeqaaOGaeq4SdCgaaa@49DD@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ градиент деформации. Полагая теперь Y=X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFyeFwcaaI9aGae83fXJfaaa@4620@ и используя определение (2) локальной деформации, приходим к соотношению

D X = F X H X 1 :U T γ(X) S. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCiramaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccaaI9aGaaCOramaaBa aaleaacqWFxepwaeqaaOGaeSigI8MaaCisamaaDaaaleaacqWFxepw aeaacqGHsislcaaIXaaaaOGaaGOoaiaaykW7cqWFueFvcqGHsgIRca WGubWaaSbaaSqaaiabeo7aNjaaiIcacqWFxepwcaaIPaaabeaakiab =jr8tjaai6caaaa@5CA3@ (5)

Тензор D X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCiramaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaaaaa@446B@ в левой части этой формулы определен как D X = T γ (X) (Y) χ (X) | Y=X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCiramaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccaaI9aGaamivamaaBa aaleaacqaHZoWzdaahaaqabeaacaaIOaGae83fXJLaaGykaaaacaaI OaGae8hgXNLaaGykaaqabaGccqaHhpWydaahaaWcbeqaaiaaiIcacq WFxepwcaaIPaaaaOGaaGiFamaaBaaaleaacqWFyeFwcaaI9aGae83f XJfabeaaaaa@59A7@ . Будем называть его полной дисторсией.

Рассмотрим гиперупругое тело. Его отклик характеризуется плотностью упругой энергии (1), которую в рамках настоящих рассуждений можно представить следующей зависимостью:

w=W(X, D X H X ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4Daiaai2dacaWGxbGaaGikamrr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae83fXJLaaGilaiaaykW7ca WHebWaaSbaaSqaaiab=Dr8ybqabaGccaWHibWaaSbaaSqaaiab=Dr8 ybqabaGccaaIPaGaaGOlaaaa@5043@

Поскольку как форма S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ , так и поле H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ фиксированы, можно положить

W ^ (X, D X ):=W(X, D X H X ), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaecaaeaacaWGxbaacaGLcmaacaaIOaWefv3ySLgz nfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwcaaISaGaaG PaVlaahseadaWgaaWcbaGae83fXJfabeaakiaaiMcacaaI6aGaaGyp aiaadEfacaaIOaGae83fXJLaaGilaiaaykW7caWHebWaaSbaaSqaai ab=Dr8ybqabaGccaWHibWaaSbaaSqaaiab=Dr8ybqabaGccaaIPaGa aGilaaaa@5A1A@

и отклик принимает вид

w= W ^ (X, D X ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4Daiaai2dadaqiaaqaaiaadEfaaiaawkWaaiaa iIcatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaiab=D r8yjaaiYcacaaMc8UaaCiramaaBaaaleaacqWFxepwaeqaaOGaaGyk aiaai6caaaa@4E19@

Таким образом, отклик был переопределен относительно семейства локально единообразных форм: аргументом отклика теперь является тензор полной дисторсии. Вместе с тем следует иметь в виду, что поле W ^ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaecaaeaacaWGxbaacaGLcmaaaaa@39A1@ рассматривается на евклидовой <<промежуточной>> форме S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ , которая в общем случае не единообразна.

Проведем то же самое рассуждение в рамках геометрического подхода. Соотношение (2) является аналогом для (4), а формула

T X λ= F X K X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaacaWGybaabeaakiabeU7aSjaa i2dacaWHgbWaaSbaaSqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHr hAG8KBLbacfaGae83fXJfabeaakiablIHiVjaabUeadaWgaaWcbaGa amiwaaqabaaaaa@4BEF@

служит аналогом для (5). Поскольку поля H X 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeisamaaDaaaleaacaWGybaabaGaeyOeI0IaaGym aaaaaaa@3B80@ и K X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4samaaBaaaleaacaWGybaabeaaaaa@39DA@ равны с точностью до изометрии I ˜ X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaacaaeaacaqGjbaacaGLdmaadaWgaaWcbaGaamiw aaqabaaaaa@3A9A@ , можно считать тензоры D X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCiramaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaaaaa@446B@ и T X λ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaacaWGybaabeaakiabeU7aSbaa @3BA3@ также равными (что допустимо, ибо отклик инвариантен по отношению к изометрии). В этом случае[18]

w= W ^ ^ (X, T X λ), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4Daiaai2dadaqiaaqaamaaHaaabaGaam4vaaGa ayPadaaacaGLcmaacaaIOaGaamiwaiaaiYcacaaMc8UaamivamaaBa aaleaacaWGybaabeaakiabeU7aSjaaiMcacaaISaaaaa@44FF@

и отклик был переопределен относительно неевклидовой единообразной формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4uamaaBaaaleaacaWGsbaabeaaaaa@39DE@ , т. е. полученная зависимость для отклика рассматривается не в точках самонапряженной евклидовой промежуточной формы S R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFse=udaWgaaWcbaGaamOuaaqabaaaaa@446B@ , а в точках неевклидовой отсчетной формы.

Заключение

Дифференциально-геометрический язык может быть плодотворно использован для моделирования несовместных деформаций в твердых телах. Он позволяет, в частности, отказаться от явного использования семейства локально единообразных евклидовых форм, заменяя последнее одной неевклидовой единообразной формой. Благодаря такой замене удается сохранить отсчетное описание состояния тела с несовместными деформациями, что дает, в свою очередь, возможность использовать привычную методологию нелинейной механики континуума.

Геометрия неевклидовой формы синтезируется на основе тензорного поля локальных деформаций (3), значение которого в каждой точке исходной формы является обратимым линейным оператором (2), переводящим представительный объем, окружающий эту точку, в натуральное состояние. Вместе с тем в классической литературе по теории дефектов (см., к примеру, [70]) отсутствует способ определения локальных деформаций, апеллирующий к некоторому эксперименту. Чтобы восполнить этот пробел, в работе развита идея локальной разгрузки.

Хотя риманова метрика на неевклидовой отсчетной форме однозначно восстанавливается по значениям поля локальных деформаций (формула (6)), для аффинной связности это не так. В работе рассмотрены различные способы синтезирования материальной связности. Используя их, можно получить связность Леви-Чивита (с нулевыми кручением и неметричностью), связность Вайценбока (с нулевыми кривизной и неметричностью) и связность Вейля (с нулевым кручением). В рамках теории простых тел (т. е. тел, отклик которых характеризуется лишь первым градиентом деформации) при известном поле локальных деформаций H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ все эти связности совершенно равноправны и, по-видимому, нет никакого способа предпочесть одну связность другой, опираясь лишь на поле H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCisaaaa@38D4@ . Если же дополнительно известно, что в теле присутствуют дефекты определенного типа, то тогда удается сделать однозначный выбор: связность Вайценбока соответствует непрерывному распределению дислокаций, связность Леви-Чивита характеризует тело с дисклинациями, а связность Вейля MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ тело с метрическими аномалиями.

Отличие одной связности от другой может быть охарактеризовано тензором конторсии (26), имеющим различный геометрический смысл для каждой пары материальных связностей. В настоящей работе в качестве одной из связностей выбиралась связность Леви-Чивита, которая сравнивалась с двумя другими связностями: Вайценбока и Вейля. Отличие связности Вайценбока от связности Леви-Чивита вызвано различными геометриями <<в малом>>. В случае первой связности малый элемент тела является плоским, а в случае связности Леви-Чивита MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ искривленным.

В свою очередь, отличие связности Вейля от связности Леви-Чивита вызвано различным поведением римановой формы объема при ее параллельном переносе вдоль кривой на теле. Если на отсчетной форме задана связность Леви-Чивита, то форма объема ковариантно постоянна, т. е. не меняется при параллельном переносе. В случае связности Вейля это не так: форма объема эволюционирует при параллельном переносе вдоль кривой, что физически интерпретируется как наличие метрических аномалий (точечных дефектов, неоднородного поля температур).

Различие между тремя рассмотренными связностями несущественно в рамках функционала отклика, поскольку последний использует лишь метрическую информацию, которая одна и та же для всех связностей. Вместе с тем при формулировке уравнений баланса импульса в отсчетном описании появляются дополнительные слагаемые, которые могут быть интерпретированы как фиктивные силы, возникающие в силу наличия несовместных деформаций [26]. Для получения явного выражения этих фиктивных сил необходимо учесть конторсию, которая присутствует в выражениях для связностей Вайценбока и Вейля. Таким образом, связности Вайценбока и Вейля дают более полное описание несовместных деформаций, чем связность Леви-Чивита: вместе с метрическими данными они содержат информацию о пространственном расположении представительных объемов, что определяет дополнительную часть уравнений баланса импульса в отсчетном описании.

Процедура синтезирования неевклидовой формы существенно опирается на выбор некоторой промежуточной формы, с которой стирается геометрия. Такой выбор неоднозначен и в общем случае может приводить к различным геометриям на теле. Вместе с тем при некоторых предположениях о связи полей локальных деформаций, соответствующих двум формам, геометрии на таких формах оказываются неразличимыми в том смысле, что соответствующие инварианты связностей отличаются на обратный образ.



[1] В котором V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOvaaaa@38DE@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzaeaeaaaaaaaaa8GacaWFuaca aa@3C13@ подлежащее множество, а + V :V×VV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4kaSYaaSbaaSqaaiaadAfaaeqaaOGaaGOoaiaa ykW7caWGwbGaey41aqRaamOvaiabgkziUkaadAfaaaa@42DA@ и V :×VV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyyXIC9aaSbaaSqaaiaadAfaaeqaaOGaaGOoaiaa ykW7tuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab=1 risjabgEna0kaadAfacqGHsgIRcaWGwbaaaa@4E1F@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzaeaeaaaaaaaaa8GacaWFuaca aa@3C13@ операции сложения и умножения на скаляр соответственно.

[2] Тем самым исключаются многообразия вида ленты Мебиуса.

[3] Можно представить себе, что у наблюдателя имеются различные <<очки>>, которые позволяют акцентировать внимание на тех или иных подструктурах геометрического пространства. Самые слабые очки позволяют различать лишь гладкие структуры, а самые сильные MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzaeaeaaaaaaaaa8GacaWFuaca aa@3C13@ всю структуру в целом.

[4] Если n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ , то в силу естественных отождествлений T X S R V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccqWFse=udaWgaaWcba GaamOuaaqabaGccqGHfjcqcqWFveVvaaa@4A7D@ и T γ(X) SV MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamivamaaBaaaleaacqaHZoWzcaaIOaWefv3ySLgz nfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFxepwcaaIPaaabe aakiab=jr8tjabgwKiajab=vr8wbaa@4C7C@ можно рассматривать F X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCOramaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaaaaa@446D@ как линейное отображение F X End(V) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCOramaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccqGHiiIZcaqGfbGaae OBaiaabsgacaaIOaGae8xfXBLaaGykaaaa@4BE1@ , соответствующее классическому градиенту деформации, то есть F X = γ ' (X) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaCOramaaBaaaleaatuuDJXwAK1uy0HwmaeHbfv3y SLgzG0uy0Hgip5wzaGqbaiab=Dr8ybqabaGccaaI9aGaeq4SdC2aaW baaSqabeaacaWGNaaaaOGaaGikaiab=Dr8yjaaiMcaaaa@4B12@ .

[5] То есть между этими пространствами существует биекция, не зависящая ни от какого выбора базиса, которая является изоморфизмом векторных пространств, сохраняющим скалярное произведение.

[6] При интерпретации векторных полей как дифференцирований алгебры C (M) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4qamaaCaaaleqabaGaeyOhIukaaOGaaGikaiaa d2eacaaIPaaaaa@3CAA@ гладких функций.

[7] Символ () # MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiabgwSixlaaiMcadaahaaWcbeqaaiaacoca aaaaaa@3C86@ обозначает музыкальный изоморфизм (операцию <<опускания>> индексов) [39, с.~342].

[8] Иными словами, e ABC =1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyzamaaCaaaleqabaGaamyqaiaadkeacaWGdbaa aOGaaGypaiaaigdaaaa@3CFB@ , если (A,B,C) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadgeacaaISaGaaGPaVlaadkeacaaISaGa aGPaVlaadoeacaaIPaaaaa@403F@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzaeaeaaaaaaaaa8GacaWFuaca aa@3C13@ четная перестановка, e ABC =1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyzamaaCaaaleqabaGaamyqaiaadkeacaWGdbaa aOGaaGypaiabgkHiTiaaigdaaaa@3DE8@ , если (A,B,C) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadgeacaaISaGaaGPaVlaadkeacaaISaGa aGPaVlaadoeacaaIPaaaaa@403F@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzaeaeaaaaaaaaa8GacaWFuaca aa@3C13@ нечетная перестановка, и e ABC =0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyzamaaCaaaleqabaGaamyqaiaadkeacaWGdbaa aOGaaGypaiaaicdaaaa@3CFA@ , если в тройке (A,B,C) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaadgeacaaISaGaaGPaVlaadkeacaaISaGa aGPaVlaadoeacaaIPaaaaa@403F@ хотя бы два элемента совпадают.

[9] Здесь и далее в доказательстве <<нештрихованные>> поля (за исключением F MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeOraaaa@38CC@ ) следует рассматривать с композицией | γ() MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGiFamaaBaaaleaacqaHZoWzcaaIOaGaeyyXICTa aGykaaqabaaaaa@3E8B@ . Для экономии места последний символ опускается в промежуточных выкладках и появляется лишь в финальных выражениях.

[10] Предполагается, что деформация сохраняет ориентацию.

[11] Здесь e TAB MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyzamaaBaaaleaacaWGubGaamyqaiaadkeaaeqa aaaa@3B7F@ и e TSL MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyzamaaCaaaleqabaGaamivaiaadofacaWGmbaa aaaa@3B9C@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzaeaeaaaaaaaaa8GacaWFuaca aa@3C13@ альтернаторы.

[12] Здесь D P ψ=(Tψ) P 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiramaaBaaaleaacaqGqbaabeaakiabeI8a5jaa i2dacaaIOaGaamivaiabeI8a5jaaiMcacqWIyiYBcaqGqbWaaWbaaS qabeaacqGHsislcaaIXaaaaaaa@4458@ для произвольного поля ψ:BW MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqiYdKNaaGOoaiaaykW7tuuDJXwAKzKCHTgD1jha ryqr1ngBPrgigjxyRrxDYbacfaGae8xaWlKaeyOKH46efv3ySLgznf gDOfdarCqr1ngBPrginfgDObYtUvgaiyaacqGFwe=vaaa@55F4@ , где W MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFwe=vaaa@4370@ обозначает E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFWesraaa@42A8@ , V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqb a8aacqWFveVvaaa@436E@ , или End(V) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaeyraiaab6gacaqGKbGaaGikamrr1ngBPrwtHrhA XaqeguuDJXwAKbstHrhAG8KBLbacfaGae8xfXBLaaGykaaaa@4773@ .

[13] Поскольку M R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytamaaBaaaleaacaWGsbaabeaaaaa@39D8@ как всякое многообразие линейно связно, хотя бы одна такая кривая всегда имеется.

[14] В этом случае принято использовать обозначение f:XY MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzaiaaiQdacaaMc8UaamiwaiabgkziUkaadMfa aaa@3EE5@ .

[15] В упорядоченном наборе (M,g,) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGikaiaad2eacaaISaGaaGPaVlaabEgacaaISaGa aGPaVlabgEGirlaaiMcaaaa@412C@ первый элемент соответствует подлежащему многообразию, а g MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaae4zaaaa@38ED@ и MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4bIenaaa@3989@ есть, соответственно, метрика и связность на M MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamytaaaa@38D5@ .

[16] На рис. 2 пространству R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOuaaaa@38DA@ соответствует сфера.

[17] Для трехмерного тела ( n=3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOBaiaai2dacaaIZaaaaa@3A7A@ ) локальные координаты точек евклидовых форм полностью определяются тройками чисел.

[18] Заметим, что выполненные преобразования корректны в силу свойства локальности отклика.

×

Об авторах

Сергей Александрович Лычев

Институт проблем механики им. А.Ю. Ишлинского РАН

Email: lychevsa@mail.ru
ORCID iD: 0000-0001-7590-1389

доктор физико-математических наук, доцент, ведущий научный сотрудник лаборатории механики технологических процессов

Россия, Москва

Константин Георгиевич Койфман

Московский государственный технический университет им. Н.Э. Баумана

Автор, ответственный за переписку.
Email: koifman.konstantin@gmail.com
ORCID iD: 0000-0002-7891-9995

тьютор по математике

Россия, Москва

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