翻訳と辞書 Words near each other ・ Cartajima ・ Cartal River ・ Cartaletis gracilis ・ Cartamundi ・ Cartan ・ Cartan (crater) ・ Cartan connection ・ Cartan decomposition ・ Cartan formalism (physics) ・ Cartan matrix ・ Cartan model ・ Cartan pair ・ Cartan subalgebra ・ Cartan subgroup ・ Cartan's criterion ・ Cartan's equivalence method ・ Cartan's lemma ・ Cartan's lemma (potential theory) ・ Cartan's theorem ・ Cartan's theorems A and B ・ Cartanal ・ Cartanul River ・ Cartan–Brauer–Hua theorem ・ Cartan–Dieudonné theorem ・ Cartan–Eilenberg resolution ・ Cartan–Hadamard theorem ・ Cartan–Karlhede algorithm ・ Cartan–Kuranishi prolongation theorem ・ Cartan–Kähler theorem ・ Cartap Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cartan's equivalence method ： ウィキペディア英語版 Cartan's equivalence method In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism. For example, if ''M'' and ''N'' are two Riemannian manifolds with metrics ''g'' and ''h'', respectively, when is there a diffeomorphism :$\backslash phi:M\backslash rightarrow\; N$ such that :$\backslash phi^$ *h=g? Although the answer to this particular question was known in dimension 2 to Gauss and in higher dimensions to Christoffel and perhaps Riemann as well, Élie Cartan and his intellectual heirs developed a technique for answering similar questions for radically different geometric structures. (For example see the Cartan–Karlhede algorithm.) Cartan successfully applied his equivalence method to many such structures, including projective structures, CR structures, and complex structures, as well as ostensibly non-geometrical structures such as the equivalence of Lagrangians and ordinary differential equations. (His techniques were later developed more fully by many others, such as D. C. Spencer and Shiing-Shen Chern.) The equivalence method is an essentially algorithmic procedure for determining when two geometric structures are identical. For Cartan, the primary geometrical information was expressed in a coframe or collection of coframes on a differentiable manifold. See method of moving frames. == Overview of Cartan's method == Specifically, suppose that ''M'' and ''N'' are a pair of manifolds each carrying a G-structure for a structure group ''G''. This amounts to giving a special class of coframes on ''M'' and ''N''. Cartan's method addresses the question of whether there exists a local diffeomorphism φ:''M''→''N'' under which the ''G''-structure on ''N'' pulls back to the given ''G''-structure on ''M''. An equivalence problem has been ''"solved"'' if one can give a complete set of structural invariants for the ''G''-structure: meaning that such a diffeomorphism exists if and only if all of the structural invariants agree in a suitably defined sense. Explicitly, local systems of one-forms θ''i'' and γ''i'' are given on ''M'' and ''N'', respectively, which span the respective cotangent bundles (i.e., are coframes). The question is whether there is a local diffeomorphism φ:''M''→''N'' such that the pullback of the coframe on ''N'' satisfies :$\backslash phi^$ *\gamma^i(y)=g^i_j(x)\theta^j(x),\ (g^i_j)\in G (1) where the coefficient ''g'' is a function on ''M'' taking values in the Lie group ''G''. For example, if ''M'' and ''N'' are Riemannian manifolds, then ''G''=''O''(''n'') is the orthogonal group and θ''i'' and γ''i'' are orthonormal coframes of ''M'' and ''N'' respectively. The question of whether two Riemannian manifolds are isometric is then a question of whether there exists a diffeomorphism φ satisfying (1). The first step in the Cartan method is to express the pullback relation (1) in as invariant a way as possible through the use of a "''prolongation''". The most economical way to do this is to use a ''G''-subbundle ''PM'' of the principal bundle of linear coframes ''LM'', although this approach can lead to unnecessary complications when performing actual calculations. In particular, later on this article uses a different approach. But for the purposes of an overview, it is convenient to stick with the principal bundle viewpoint. The second step is to use the diffeomorphism invariance of the exterior derivative to try to isolate any other higher-order invariants of the ''G''-structure. Basically one obtains a connection in the principal bundle ''PM'', with some torsion. The components of the connection and of the torsion are regarded as invariants of the problem. The third step is that if the remaining torsion coefficients are not constant in the fibres of the principal bundle ''PM'', it is often possible (although sometimes difficult), to normalize them by setting them equal to a convenient constant value and solving these normalization equations, thereby reducing the effective dimension of the Lie group ''G''. If this occurs, one goes back to step one, now having a Lie group of one lower dimension to work with. 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「Cartan's equivalence method」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.