About: Connection (vector bundle)

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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

Attributes	Values
rdf:type	Thing yago:WikicatVectorBundles yago:Abstraction100002137 yago:Collection107951464 yago:Group100031264 yago:Package108008017 device
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)
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:seeAlso	Curvature form Gauge group (mathematics) Connection (principal bundle) Exterior covariant derivative
foaf:depiction
dcterms:subject	Vector bundles Connection (mathematics)
Wikipage page ID	391816 (xsd:integer)
Wikipage revision ID	1124793518 (xsd:integer)
Link from a Wikipage to another Wikipage	Product rule Pushforward (differential) Module (mathematics) Metric connection Holonomy group D-module Integral curve Commutator Connection (mathematics) Mathematics Gauge theory Gauge theory (mathematics) Tangent bundle Yang–Mills theory Pullback bundle Christoffel symbols Ehresmann connection General relativity Connection (principal bundle) Contorsion tensor Levi-Civita connection Lie algebra Linear isomorphism Loop (topology) Smooth function Smooth functions Path (topology) Tangent space Automorphism Adjoint bundle Vector bundles Hausdorff space Linear map Affine connection Affine space Euclidean space Exterior covariant derivative Exterior derivative Fiber bundle Parallel transport Partial differential equation Foundations of Differential Geometry Frame bundle Equivariant Riemann curvature tensor Pseudo-Riemannian manifold Riemannian connection Riemannian manifold Riemannian metric Ring (mathematics) Group action Jean-Louis Koszul Teleparallelism Covariant derivative Tensor product Connection (mathematics) Torsion tensor Moduli space Differentiable manifold Differential geometry Associated bundle Bundle metric Ordinary differential equation Yang–Mills equations Topological manifold Section (fiber bundle) Smooth manifold Vertical and horizontal bundles Tensor algebra Vector bundle Linear Flat vector bundle Partitions of unity

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