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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 Fundamental vector field ： ウィキペディア英語版 Fundamental vector field In the study of mathematics and especially differential geometry, fundamental vector fields are an instrument that describes the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry, and the study of Hamiltonian group actions. ==Motivation== Important to applications in mathematics and physics is the notion of a flow on a manifold. In particular, if $M$ is a smooth manifold and $X$ is a smooth vector field, one is interested in finding integral curves to $X$. More precisely, given $p\; \backslash in\; M$ one is interested in curves $\backslash gamma\_p:\; \backslash mathbb\; R\; \backslash to\; M$ such that :$\backslash gamma\_p\text{'}(t)\; =\; X\_,\; \backslash qquad\; \backslash gamma\_p(0)\; =\; p,$ for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations. If $X$ is furthermore a complete vector field, then the flow of $X$, defined as the collection of all integral curves for $X$, is a diffeomorphism of $M$. The flow $\backslash phi\_X:\; \backslash mathbb\; R\; \backslash times\; M\; \backslash to\; M$ given by $\backslash phi\_X(t,p)\; =\; \backslash gamma\_p(t)$ is in fact an action of the additive Lie group $(\backslash mathbb\; R,+)$ on $M$. Conversely, every smooth action $A:\backslash mathbb\; R\; \backslash times\; M\; \backslash to\; M$ defines a complete vector field $X$ via the equation : $X\_p\; =\; \backslash left.\backslash frac\backslash right|\_\; A(t,p).$ It is then a simple result that there is a bijective correspondence between $\backslash mathbb\; R$ actions on $M$ and complete vector fields on $M$. In the language of flow theory, the vector field $X$ is called the ''infinitesimal generator''. Intuitively, the behaviour of the flow at each point corresponds to the "direction" indicated by the vector field. It is a natural question to ask whether one may establish a similar correspondence between vector fields and more arbitrary Lie group actions on $M$. 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「Fundamental vector field」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.