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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 Lie group action ： ウィキペディア英語版 Lie group action In differential geometry, a Lie group action on a manifold ''M'' is a group action by a Lie group ''G'' on ''M'' that is a differentiable map; in particular, it is a continuous group action. Together with a Lie group action by ''G'', ''M'' is called a ''G''-manifold. The orbit types of ''G'' form a stratification of ''M'' and this can be used to understand the geometry of ''M''. Let $\backslash sigma:\; G\; \backslash times\; M\; \backslash to\; M,\; (g,\; x)\; \backslash to\; g\; \backslash cdot\; x$ be a group action. It is a Lie group action if it is differentiable. Thus, in particular, the orbit map $\backslash sigma\_x\; :\; G\; \backslash to\; M,\; g\; \backslash cdot\; x$ is differentiable and one can compute its differential at the identity element of ''G'': :$\backslash mathfrak\; \backslash to\; T\_x\; M$. If ''X'' is in $\backslash mathfrak$, then its image under the above is a tangent vector at ''x'' and, varying ''x'', one obtains a vector field on ''M''; the minus of this vector field is called the fundamental vector field associated with ''X'' and is denoted by $X^\backslash \#$. (The "minus" ensures that $\backslash mathfrak\; \backslash to\; \backslash Gamma(TM)$ is a Lie algebra homomorphism.) The kernel of the map can be easily shown (cf. Lie correspondence) to be the Lie algebra $\backslash mathfrak\_x$ of the stabilizer $G\_x$ (which is closed and thus a Lie subgroup of ''G''.) Let $P\; \backslash to\; M$ be a principal ''G''-bundle. Since ''G'' has trivial stabilizers in ''P'', for ''u'' in ''P'', $a\; \backslash mapsto\; a^\backslash \#\_u:\; \backslash mathfrak\; \backslash to\; T\_u\; P$ is an isomorphism onto a subspace; this subspace is called the vertical subspace. A fundamental vector field on ''P'' is thus vertical. In general, the orbit space $M/G$ does not admit a manifold structure since, for example, it may not be Hausdorff. However, if ''G'' is compact, then $M/G$ is Hausdorff and if, moreover, the action is free, then $M/G$ is a manifold (in fact, $M\; \backslash to\; M/G$ is a principal ''G''-bundle.)〔.〕 This is a consequence of the slice theorem. If the "free action" is relaxed to "finite stabilizer", one instead obtains an orbifold (or quotient stack.) A substitute for the construction of the quotient is the Borel construction from algebraic topology: assume ''G'' is compact and let $EG$ denote the universal bundle, which we can assume to be a manifold since ''G'' is compact, and let ''G'' act on $EG\; \backslash times\; M$ diagonally; the action is free since it is so on the first factor. Thus, one can form the ''quotient manifold'' $M\_G\; =\; (EG\; \backslash times\; M)/G$. The constriction in particular allows one to define the equivariant cohomology of ''M''; namely, one sets :$H^$ *_G(M) = H^ *_{\text{dr}}(M_G), where the right-hand side denotes the de Rham cohomology, which makes sense since $M\_G$ has a structure of manifold (thus there is the notion of differential forms.) If ''G'' is compact, then any ''G''-manifold admits an invariant metric; i.e., a Riemannian metric with respect to which ''G'' acts on ''M'' as isometries. == See also == *Hamiltonian group action *Equivariant differential form 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「Lie group action」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.