Connection (vector bundle) (original) (raw)
미분기하학에서 코쥘 접속(Koszul接續, 영어: Koszul connection)은 벡터 다발의 각 올들을 이어붙여, 벡터장의 미분을 정의할 수 있게 하는 구조이다.
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| dbo:abstract | 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). (en) En géométrie différentielle, une connexion (de Koszul) est un opérateur sur les sections d'un fibré vectoriel. Cette notion a été introduite par Jean-Louis Koszul en 1950[réf. nécessaire] et formalise le transport parallèle de vecteurs le long d'une courbe en termes d'équation différentielle ordinaire. Les connexions sont des objets localement définis auxquels sont associées les notions de courbure et de torsion. L'un des exemples les plus simples de connexions de Koszul sans torsion est la connexion de Levi-Civita naturellement définie sur le fibré tangent de toute variété riemannienne. L'ensemble des connexions de Koszul forme un espace affine réel dont l'espace directeur est l'espace des 1-formes différentielles de la base B du fibré E à valeurs dans End(E), le fibré vectoriel des endomorphismes de E. Une connexion sur E induit des connexions sur les fibrés construits à partir de E par des opérations algébriques élémentaires (produit extérieur, produit tensoriel, ...). L'utilisation des connexions permet en particulier d'effectuer un calcul différentiel extérieur raisonnable sur les sections de E. Elles sont fortement utilisées en analyse. (fr) 미분기하학에서 코쥘 접속(Koszul接續, 영어: Koszul connection)은 벡터 다발의 각 올들을 이어붙여, 벡터장의 미분을 정의할 수 있게 하는 구조이다. (ko) Зв'язність на векторних розшаруваннях в диференціальній геометрії дозволяє ввести на довільних векторних розшаруваннях такі поняття як паралельне перенесення, тензори кривини і кручення і інші. Таким чином значна частина теорії і ідей може бути перенесена з гладких многовидів і їх дотичних розшарувань на векторні розшарування. Для зв'язності на векторних розшарування часто також використовується термін зв'язність Кошуля на честь французького математика Жана-Луї Кошуля. (uk) 在数学中，纤维丛上一个联络是一个定义丛上平行移动的装置；即将邻近点连接或等价的一种方法。如果纤维丛是向量丛，则平行移动的概念要求线性。这样的联络等价于一个共变导数，共变导数是一个能对截面关于底流形的切方向求微分的算子。联络在这个意义下，对任意向量丛，推广了光滑流形切丛的概念，经常叫做线性联络。 向量丛上的联络也经常称为科斯居尔联络，以让-路易·科斯居尔命名，他给出了描述这个联络的一个代数框架（）。 (zh) |
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| rdfs:comment | 미분기하학에서 코쥘 접속(Koszul接續, 영어: Koszul connection)은 벡터 다발의 각 올들을 이어붙여, 벡터장의 미분을 정의할 수 있게 하는 구조이다. (ko) Зв'язність на векторних розшаруваннях в диференціальній геометрії дозволяє ввести на довільних векторних розшаруваннях такі поняття як паралельне перенесення, тензори кривини і кручення і інші. Таким чином значна частина теорії і ідей може бути перенесена з гладких многовидів і їх дотичних розшарувань на векторні розшарування. Для зв'язності на векторних розшарування часто також використовується термін зв'язність Кошуля на честь французького математика Жана-Луї Кошуля. (uk) 在数学中，纤维丛上一个联络是一个定义丛上平行移动的装置；即将邻近点连接或等价的一种方法。如果纤维丛是向量丛，则平行移动的概念要求线性。这样的联络等价于一个共变导数，共变导数是一个能对截面关于底流形的切方向求微分的算子。联络在这个意义下，对任意向量丛，推广了光滑流形切丛的概念，经常叫做线性联络。 向量丛上的联络也经常称为科斯居尔联络，以让-路易·科斯居尔命名，他给出了描述这个联络的一个代数框架（）。 (zh) 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 fi (en) En géométrie différentielle, une connexion (de Koszul) est un opérateur sur les sections d'un fibré vectoriel. Cette notion a été introduite par Jean-Louis Koszul en 1950[réf. nécessaire] et formalise le transport parallèle de vecteurs le long d'une courbe en termes d'équation différentielle ordinaire. Les connexions sont des objets localement définis auxquels sont associées les notions de courbure et de torsion. L'un des exemples les plus simples de connexions de Koszul sans torsion est la connexion de Levi-Civita naturellement définie sur le fibré tangent de toute variété riemannienne. (fr) |
| rdfs:label | Connection (vector bundle) (en) Connexion de Koszul (fr) 코쥘 접속 (ko) Зв'язність на векторних розшаруваннях (uk) 联络 (向量丛) (zh) |
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