Введение
Определяет понятие параллельного переноса на расслоении, соединения на векторных расслоениях.
connections on vector bundles
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi Civita connection on the tangent bundle of a pseudo Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear. Linear connections are also called Koszul connections after Jean Louis Koszul, who gave an algebraic framework for describing them
This article defines the connection on a vector bundle using a common mathematical notation which de emphasizes coordinates. However, other notations are also regularly used: in general relativity, vector bundle computations are usually written using indexed tensors; in gauge theory, the endomorphisms of the vector space fibers are emphasized. The different notations are equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles).
В математике, и особенно в дифференциальной геометрии и теории калибровочных полей, соединение на волокнистом расслоении — это механизм, определяющий понятие параллельного переноса на расслоении, то есть способ "соединить" или отождествить волокна над близкими точками. Наиболее распространенным случаем является линейное соединение на векторном расслоении, для которого понятие параллельного переноса должно быть линейным. Линейное соединение эквивалентно задается ковариантной производной, оператором, который дифференцирует сечения расслоения вдоль касательных направлений в базовом многообразии таким образом, что параллельные сечения имеют нулевую производную. Линейные соединения обобщают, для произвольных векторных расслоений, соединение Леви-Чивиты на тангенциальном расслоении псевдориманова многообразия, которое дает стандартный способ дифференцирования векторных полей. Нелинейные соединения обобщают эту концепцию на расслоения, волокна которых не обязательно линейны. Линейные соединения также называются соединениями Козуля в честь Жана Луи Козуля, который предложил алгебраическую основу для их описания.
connections on vector bundles
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi Civita connection on the tangent bundle of a pseudo Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear. Linear connections are also called Koszul connections after Jean Louis Koszul, who gave an algebraic framework for describing them
This article defines the connection on a vector bundle using a common mathematical notation which de emphasizes coordinates. However, other notations are also regularly used: in general relativity, vector bundle computations are usually written using indexed tensors; in gauge theory, the endomorphisms of the vector space fibers are emphasized. The different notations are equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles).
В данной статье соединение на векторном расслоении определяется с использованием общепринятой математической нотации, которая минимизирует использование координат. Однако регулярно используются и другие обозначения: в общей теории относительности вычисления с векторными расслоениями обычно записываются с использованием индексированных тензоров; в теории калибровочных полей подчеркиваются эндоморфизмы волокон векторного пространства. Различные обозначения эквивалентны, как обсуждается в статье о метрических соединениях (замечания, сделанные там, применимы ко всем векторным расслоениям).
connections on vector bundles
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi Civita connection on the tangent bundle of a pseudo Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear. Linear connections are also called Koszul connections after Jean Louis Koszul, who gave an algebraic framework for describing them
This article defines the connection on a vector bundle using a common mathematical notation which de emphasizes coordinates. However, other notations are also regularly used: in general relativity, vector bundle computations are usually written using indexed tensors; in gauge theory, the endomorphisms of the vector space fibers are emphasized. The different notations are equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles).
Индуцированные соединения
При заданном векторном расслоении существует множество связанных с ним расслоений, которые можно построить, например, двойное векторное расслоение, тензорные степени, симметричные и антисимметричные тензорные степени, и прямые суммы. Соединение на расслоении E индуцирует соединение на любом из этих связанных расслоений. Простота перехода между соединениями на связанных расслоениях более изящно описывается теорией главных расслоений, но здесь мы представляем некоторые из основных индуцированных соединений.
Любой связанный пакет
При заданном векторном расслоении ранга *r* и любом представлении ρ в линейную группу *G*, существует индуцированное соединение на ассоциированном векторном расслоении. Эта теория наиболее кратко формулируется, переходя к соединению главного расслоения на расслоении кадров расслоения *E* и используя теорию главных расслоений. Каждый из вышеприведенных примеров можно рассматривать как частный случай этой конструкции: двойственное расслоение соответствует обратному транспонированному (или обратному сопряженному) представлению, тензорное произведение – представлению тензорного произведения, прямая сумма – представлению прямой суммы и так далее.
Сродственные свойства множества соединений
Каждый векторный расслоение над многообразием допускает связь, что может быть доказано с помощью разбиений единицы. Однако связи не единственны. Если ∇₁ и ∇₂ – две связи на E, то их разность является линейным оператором. То есть,
для всех гладких функций f на M и всех гладких сечений s из E. Отсюда следует, что разность ∇₁ - ∇₂ может быть однозначно отождествлена с одной формой на M со значениями в расслоение эндоморфизмов End(E):
Обратно, если ∇ – связь на E, а ω – одна форма на M со значениями в End(E), то ∇ + ω – связь на E.
Иными словами, пространство связей на E является аффинным пространством для End(E). Это аффинное пространство обычно обозначается ∇.
In other words, the space of connections on is an affine space for This affine space is commonly denoted .
Связь с основными и Эресманнскими соединениями
Пусть – векторное расслоение ранга , а – расслоение кадров. Тогда (главная) связность на порождает связность на . Заметим, что сечения находятся во взаимно однозначном соответствии с право-эквивариантными отображениями (это можно увидеть, рассмотрев обратное отображение над , которое изоморфно тривиальному расслоению). Пусть – сечение , тогда соответствующее эквивариантное отображение равно . Ковариантная производная на тогда задается выражением
где – горизонтальный подъем из в (вспомним, что горизонтальный подъем определяется связностью на ). Обратно, связность на определяет связность на , и эти две конструкции взаимно обратимы. Связность на также эквивалентно определяется линейной связностью Эресмана на . Это предоставляет один из способов построения соответствующей главной связности. Индуцированные связности, обсуждаемые в #Индуцированные связности, могут быть построены как связности на других ассоциированных расслоениях над расслоением кадров , используя представления, отличные от стандартного представления, используемого выше. Например, если обозначает стандартное представление на , то ассоциированное расслоение к представлению на является прямым суммарным расслоением , и индуцированная связность точно та, что была описана выше.
Параллельный транспорт и голономия
Соединение на векторном расслоении определяет понятие параллельного переноса вдоль кривой в Пусть γ будет гладким путем в ℝ. Сечение E вдоль γ называется параллельным, если для всех t выполняется
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
d/dt s(t) = ∇γ’ s(t).
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
Эквивалентно, можно рассматривать обратное расслоение γ*E расслоения E посредством γ. Это векторное расслоение над ℝ с волокном E в каждой точке. Соединение ∇ на E переносится на соединение на γ*E. Сечение s из γ*E параллельно тогда и только если
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
∇γ’ s = 0.
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
Предположим, что γ — путь из a в b в ℝ. Вышеуказанное уравнение, определяющее параллельные сечения, является обыкновенным дифференциальным уравнением первого порядка (см. локальное выражение выше) и поэтому имеет единственное решение для каждого возможного начального условия. То есть для каждого вектора v в E_a существует единственный параллельный срез s вдоль γ, такой что s(a) = v. Определим карту параллельного переноса
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
Φγ : E_a → E_b
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
посредством Φγ(v) = s(b). Можно показать, что Φγ является линейным изоморфизмом, с обратным, заданным следованием той же процедуры с обратным путем γ⁻¹ от b к a.
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
Параллельный перенос может быть использован для определения голономической группы соединения ∇, базирующейся в точке x в ℝ. Это подгруппа GL(E_x), состоящая из всех карт параллельного переноса, исходящих из петель, базирующихся в x:
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
Hol_x = {Φγ | γ — петля, базирующаяся в x}.
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
Голономическая группа соединения тесно связана с кривизной соединения.
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
Соединение можно восстановить из его операторов параллельного переноса следующим образом. Пусть X — векторное поле и s — сечение E. В точке x выберите интегральную кривую γ для X, начинающуюся в x. Для каждого t мы будем писать Φγ(t) для карты параллельного переноса, проходящей вдоль γ от x до γ(t). В частности, для каждого t, Φγ(t) : E_x → E_γ(t). Тогда Φγ(t) определяет кривую в векторном пространстве Hom(E_x, E_γ(t)), которая может быть дифференцирована. Ковариантная производная восстанавливается как
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
∇_X s = d/dt (Φγ(t) s(x)).
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
Это показывает, что эквивалентное определение соединения дано путем указания всех параллельных транспортных изоморфизмов Φγ между волокнами E и принятия вышеуказанного выражения в качестве определения ∇.
for all Equivalently, one can consider the pullback bundle of by This is a vector bundle over with fiber over The connection on pulls back to a connection on A section of is parallel if and only if
Suppose is a path from to in The above equation defining parallel sections is a first order ordinary differential equation (cf. local expression above) and so has a unique solution for each possible initial condition. That is, for each vector in there exists a unique parallel section of with Define a parallel transport map
by It can be shown that is a linear isomorphism, with inverse given by following the same procedure with the reversed path from to
Parallel transport can be used to define the holonomy group of the connection based at a point in This is the subgroup of consisting of all parallel transport maps coming from loops based at :
The holonomy group of a connection is intimately related to the curvature of the connection
The connection can be recovered from its parallel transport operators as follows. If is a vector field and a section, at a point pick an integral curve for at For each we will write for the parallel transport map traveling along from to In particular for every , we have Then defines a curve in the vector space , which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphisms between fibres of and taking the above expression as the definition of .
Личность Бьянки
Версия второго (дифференциального) тождества Бьянки из римановой геометрии справедлива для связности на любом векторном расслоении. Напомним, что связность на векторном расслоении индуцирует эндоморфную связность на этом расслоении. Эта эндоморфная связность сама имеет внешнюю ковариантную производную, которую мы условно называем . Поскольку кривизна является глобально определенной формой ранга 2 со значениями в , мы можем применить к ней внешнюю ковариантную производную. Тождество Бьянки утверждает, что это так. Это лаконично отражает сложные тензорные формулы тождества Бьянки в случае римановых многообразий, и из этого уравнения можно перейти к стандартным тождествам Бьянки, раскрыв связность и кривизну в локальных координатах. В общем случае аналога первого (алгебраического) тождества Бьянки для произвольной связности нет, поскольку оно использует специальные симметрии связности Леви-Чивиты. А именно, используется тот факт, что индексы векторного расслоения в тензоре кривизны можно переставить с индексами кокасательного расслоения, получаемыми после использования метрики для понижения или повышения индексов. Например, это позволяет определить условие отсутствия кручения для связности Леви-Чивиты, но для общего векторного расслоения индекс относится к локальному координатному базису , а индексы – к локальному координатному каркасу и , возникающему из расщепления. Однако в особых случаях, например, когда ранг равен размерности и выбрана форма привязки, можно использовать привязку для перестановки индексов и определить понятие кручения для аффинных связностей, которые не являются связностью Леви-Чивиты.
Трансформации калибра
Учитывая две связи на векторном расслоении, естественно спросить, когда они могут считаться эквивалентными. Существует четко определенное понятие автоморфизма векторного расслоения. Сечение является автоморфизмом, если обратимо в каждой точке. Такой автоморфизм называется калибровочным преобразованием, а группа всех автоморфизмов называется калибровочной группой, часто обозначаемой как или . Калибровочную группу можно аккуратно охарактеризовать как пространство сечений сопряженного расслоения расслоения кадров векторного расслоения. Это не следует путать с сопряженным расслоением (малой буквы), которое естественным образом отождествляется с самим собой. Расслоение является ассоциированным расслоением над главным расслоением кадров посредством сопряженного представления на себя, , и имеет в качестве волокна ту же общую линейную группу , где. Обратите внимание, что, несмотря на то, что оно имеет то же волокно, что и расслоение кадров, и ассоциировано с ним, оно не равно расслоению кадров и даже не является главным расслоением. Калибровочную группу можно также охарактеризовать как калибровочное преобразование, действующее на сечения, и, следовательно, действующее на связи посредством сопряжения. Явно, если – связь на , то определяется как
A gauge transformation of acts on sections , and therefore acts on connections by conjugation. Explicitly, if is a connection on , then one defines by
for To check that is a connection, one verifies the product rule
It may be checked that this defines a left group action of on the affine space of all connections
Since is an affine space modelled on , there should exist some endomorphism valued one form such that Using the definition of the endomorphism connection induced by , it can be seen that
which is to say that
Two connections are said to be gauge equivalent if they differ by the action of the gauge group, and the quotient space is the moduli space of all connections on In general this topological space is neither a smooth manifold or even a Hausdorff space, but contains inside it the moduli space of Yang–Mills connections on , which is of significant interest in gauge theory and physics.
для Чтобы проверить, что является связностью, проверяется правило произведения:
A gauge transformation of acts on sections , and therefore acts on connections by conjugation. Explicitly, if is a connection on , then one defines by
for To check that is a connection, one verifies the product rule
It may be checked that this defines a left group action of on the affine space of all connections
Since is an affine space modelled on , there should exist some endomorphism valued one form such that Using the definition of the endomorphism connection induced by , it can be seen that
which is to say that
Two connections are said to be gauge equivalent if they differ by the action of the gauge group, and the quotient space is the moduli space of all connections on In general this topological space is neither a smooth manifold or even a Hausdorff space, but contains inside it the moduli space of Yang–Mills connections on , which is of significant interest in gauge theory and physics.
Можно проверить, что это определяет левое групповое действие на аффинном пространстве всех связей. Поскольку является аффинным пространством, моделируемым на , должно существовать некоторое эндоморфно-значное одномерное дифференциальное уравнение такое, что. Используя определение эндоморфной связности, индуцированной , можно увидеть, что
A gauge transformation of acts on sections , and therefore acts on connections by conjugation. Explicitly, if is a connection on , then one defines by
for To check that is a connection, one verifies the product rule
It may be checked that this defines a left group action of on the affine space of all connections
Since is an affine space modelled on , there should exist some endomorphism valued one form such that Using the definition of the endomorphism connection induced by , it can be seen that
which is to say that
Two connections are said to be gauge equivalent if they differ by the action of the gauge group, and the quotient space is the moduli space of all connections on In general this topological space is neither a smooth manifold or even a Hausdorff space, but contains inside it the moduli space of Yang–Mills connections on , which is of significant interest in gauge theory and physics.
что означает, что две связи называются калибровочно эквивалентными, если они отличаются действием калибровочной группы, а фактор-пространство является пространством модулей всех связей на . В общем случае это топологическое пространство не является ни гладким многообразием, ни даже пространством Хаусдорфа, но содержит в себе пространство модулей связей Янга–Миллса на , которое представляет значительный интерес в калибровочной теории и физике.
A gauge transformation of acts on sections , and therefore acts on connections by conjugation. Explicitly, if is a connection on , then one defines by
for To check that is a connection, one verifies the product rule
It may be checked that this defines a left group action of on the affine space of all connections
Since is an affine space modelled on , there should exist some endomorphism valued one form such that Using the definition of the endomorphism connection induced by , it can be seen that
which is to say that
Two connections are said to be gauge equivalent if they differ by the action of the gauge group, and the quotient space is the moduli space of all connections on In general this topological space is neither a smooth manifold or even a Hausdorff space, but contains inside it the moduli space of Yang–Mills connections on , which is of significant interest in gauge theory and physics.