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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 Exponential map (Lie theory) ： ウィキペディア英語版 Exponential map (Lie theory) In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group to the group which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary justifications for the study of Lie groups at the level of Lie algebras. The ordinary exponential function of mathematical analysis is a special case of the exponential map when ''G'' is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects. ==Definitions== Let $G$ be a Lie group and $\backslash mathfrak\; g$ be its Lie algebra (thought of as the tangent space to the identity element of $G$). The exponential map is a map :$\backslash exp\backslash colon\; \backslash mathfrak\; g\; \backslash to\; G$ which can be defined in several different ways as follows: *It is given by $\backslash exp(X)\; =\; \backslash gamma(1)$ where ::$\backslash gamma\backslash colon\; \backslash mathbb\; R\; \backslash to\; G$ :is the unique one-parameter subgroup of $G$ whose tangent vector at the identity is equal to $X$. It follows easily from the chain rule that $\backslash exp(tX)\; =\; \backslash gamma(t)$. The map $\backslash gamma$ may be constructed as the integral curve of either the right- or left-invariant vector field associated with $X$. That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero. *It is the exponential map of a canonical left-invariant affine connection on ''G'', such that parallel transport is given by left translation. That is, $\backslash exp(X)\; =\; \backslash gamma(1)$ where $\backslash gamma$ is the unique geodesic with the initial point at the identity element and the initial velocity ''X'' (thought of as a tangent vector). *It is the exponential map of a canonical right-invariant affine connection on ''G''. This is usually different from the canonical left-invariant connection, but both connections have the same geodesics (orbits of 1-parameter subgroups acting by left or right multiplication) so give the same exponential map. * If $G$ is a matrix Lie group, then the exponential map coincides with the matrix exponential and is given by the ordinary series expansion: ::$\backslash exp\; (X)\; =\; \backslash sum\_^\backslash infty\backslash frac\; =\; I\; +\; X\; +\; \backslash fracX^2\; +\; \backslash fracX^3\; +\; \backslash cdots$ :(here $I$ is the identity matrix). *If ''G'' is compact, it has a Riemannian metric invariant under left and right translations, and the exponential map is the exponential map of this Riemannian metric. *The Lie group–Lie algebra correspondence also gives the definition: for ''X'' in $\backslash mathfrak\; g$, $t\; \backslash mapsto\; \backslash exp(tX)$ is the unique Lie group homomorphism correspondening to the Lie algebra homomorphism $t\; \backslash mapsto\; tX.$ (note: $\backslash operatorname(\backslash mathbb)\; =\; \backslash mathbb$.) 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「Exponential map (Lie theory)」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.