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In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of a coordinate frame, and so is not a tensorial object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it i

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rdfs:label	Connection form (en) Forme de connexion (fr) 接続形式 (ja) Форма связности (ru) 联络形式 (zh)
rdfs:comment	En géométrie différentielle, une 1-forme de connexion est une forme différentielle sur un -fibré principal qui vérifie certains axiomes.La donnée d'une forme de connexion permet de parler, entre autres, de courbure, de torsion, de dérivée covariante, de relevé horizontal, de transport parallèle, d'holonomie et de théorie de jauge.La notion de forme de connexion est intimement reliée à la notion de connexion d'Ehresmann. (fr) 接続形式（せつぞくけいしき、connection form）は、数学、特に微分幾何学における概念の1つで、微分形式や（moving frame）のことばを使うことにより、接続のデータを構成する方法である。 (ja) 在数学，特别是微分几何中，一个联络形式（connection form）是用活动标架与微分形式的语言处理联络数据的一种方式。历史上联络形式由埃利·嘉当在二十世纪上半叶引入，作为他活动标架方法的一部分，也是其主要促进因素之一。联络形式一般取决于标架的选取，从而不是一个张量性对象。在嘉当最初的工作之后，涌现出联络形式的各种推广与重新解释。特别地，在一个主丛上，一个主联络是将联络形式自然重新解释为一个张量性对象。另一方面，联络形式作为定义在微分流形上的微分形式与在一个抽象的主丛上相比，有其优越性。从而，尽管它们不满足张量性，联络形式依然被使用，因为利用它们计算相对简单。在物理学中，联络形式在规范理论中通过也广泛应用。与向量丛的每个基相伴的联络形式是微分 1-形式矩阵。联络形式没有张量性因为在基变化下，联络形式的变换涉及到转移函数的外微分，与列维-奇维塔联络的克里斯托费尔符号非常类似。一个联络形式的主要张量性不变量是其曲率形式。如果有将向量丛与切丛等价的一个，则有另一个不变量：挠率形式。在许多情形，考虑有附加结构的向量丛上的联络形式：即带有结构群的纤维丛。 (zh) In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of a coordinate frame, and so is not a tensorial object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it i (en)
rdfs:seeAlso	Connection (vector bundle)
dcterms:subject	Differential geometry Connection (mathematics) Fiber bundles Maps of manifolds Smooth functions
Wikipage page ID	706412 (xsd:integer)
Wikipage revision ID	1122494053 (xsd:integer)
Link from a Wikipage to another Wikipage	Canonical one-form Pushforward (differential) Endomorphism ring Metric connection Vector-valued differential form Torsion (differential geometry) Bianchi identities De Rham cohomology Almost complex manifold Holonomic basis Curvature form Lie group Metric (vector bundle) Differential geometry Connection (mathematics) Covariance and contravariance of vectors Mathematics Matrix (mathematics) Gauge covariant derivative Gauge theory Tangent bundle Pullback (differential geometry) Christoffel symbols Classical mechanics Ehresmann connection General relativity Connection (principal bundle) Connection (vector bundle) Contorsion tensor Levi-Civita connection Lie algebra Complex manifold Élie Cartan Derivation (abstract algebra) Orthonormal basis Physics Spin group Dual basis Adjoint representation Affine connection Equivalence relation Exterior algebra Exterior derivative Fiber bundle Fibre bundle Parallel transport Differential operator Frame bundle Riemann curvature tensor Moving frame Riemannian connection Riemannian metric Cotangent bundle Tensor Atlas (topology) Connection (mathematics) Fiber bundles Maps of manifolds Smooth functions Change of basis Charles Ehresmann Tetrad formalism Torsion tensor Differentiable manifold Differential form Differential geometry Associated bundle Bundle metric CR manifold Spinor Kronecker delta Linear form Orthogonal group

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