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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Connection (principal bundle) ： ウィキペディア英語版 Connection (principal bundle) In mathematics, a connection 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. A principal ''G''-connection on a principal G-bundle ''P'' over a smooth manifold ''M'' is a particular type of connection which is compatible with the action of the group ''G''. A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any fiber bundle associated to ''P'' via the associated bundle construction. In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the frame bundle of a smooth manifold. ==Formal definition== Let ''π'':''P''→''M'' be a smooth principal ''G''-bundle over a smooth manifold ''M''. Then a principal ''G''-connection on ''P'' is a differential 1-form on ''P'' with values in the Lie algebra $\backslash mathfrak\; g$ of ''G'' which is ''G-equivariant'' and ''reproduces'' the ''Lie algebra generators'' of the ''fundamental vector fields'' on ''P''. In other words, it is an element ''ω'' of $\backslash Omega^1(P,\backslash mathfrak\; g)\backslash cong\; C^\backslash infty(P,\; T^$ *P\otimes\mathfrak g) such that # $\backslash hbox\_g(R\_g^$ *\omega)=\omega where ''R''''g'' denotes right multiplication by ''g'', and $\backslash operatorname\_g$ is the adjoint representation on $\backslash mathfrak\; g$ (explicitly, $\backslash operatorname\_gX\; =\; \backslash fracg\backslash exp(tX)g^\backslash bigl|\_$); # if $\backslash xi\backslash in\; \backslash mathfrak\; g$ and ''X''''ξ'' is the vector field on ''P'' associated to ''ξ'' by differentiating the ''G'' action on ''P'', then ''ω''(''X''''ξ'') = ''ξ'' (identically on ''P''). Sometimes the term ''principal G-connection'' refers to the pair (''P'',''ω'') and ''ω'' itself is called the connection form or connection 1-form of the principal connection. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Connection (principal bundle)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.