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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 Gauss's lemma (Riemannian geometry) ： ウィキペディア英語版 Gauss's lemma (Riemannian geometry) In Riemannian geometry, Gauss's lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point. More formally, let ''M'' be a Riemannian manifold, equipped with its Levi-Civita connection, and ''p'' a point of ''M''. The exponential map is a mapping from the tangent space at ''p'' to ''M'': :$\backslash mathrm\; :\; T\_pM\; \backslash to\; M$ which is a diffeomorphism in a neighborhood of zero. Gauss' lemma asserts that the image of a sphere of sufficiently small radius in ''T''p''M'' under the exponential map is perpendicular to all geodesics originating at ''p''. The lemma allows the exponential map to be understood as a radial isometry, and is of fundamental importance in the study of geodesic convexity and normal coordinates. == Introduction == We define the exponential map at $p\backslash in\; M$ by :$$ \exp_p: T_pM\supset B_(0) \longrightarrow M,\quad v\longmapsto \gamma_(1), where $\backslash gamma\_\backslash$ is the unique geodesic with $\backslash gamma(0)=p$ and tangent $\backslash gamma\_\text{'}(0)=v\; \backslash in\; T\_pM$ and $\backslash epsilon\_0$ is chosen small enough so that for every $v\; \backslash in\; B\_(0)\; \backslash subset\; T\_pM$ the geodesic $\backslash gamma\_$ is defined in 1. So, if $M$ is complete, then, by the Hopf–Rinow theorem, $\backslash exp\_p$ is defined on the whole tangent space. Let $\backslash alpha\; :\; I\backslash rightarrow\; T\_pM$ be a curve differentiable in $T\_pM\backslash$ such that $\backslash alpha(0):=0\backslash$ and $\backslash alpha\text{'}(0):=v\backslash$. Since $T\_pM\backslash cong\; \backslash mathbb\; R^n$, it is clear that we can choose $\backslash alpha(t):=vt\backslash$. In this case, by the definition of the differential of the exponential in $0\backslash$ applied over $v\backslash$, we obtain: :$$ T_0\exp_p(v) = \frac \Bigl(\exp_p\circ\alpha(t)\Bigr)\Big\vert_ = \frac \Bigl(\exp_p(vt)\Bigr)\Big\vert_=\frac \Bigl(\gamma(1,p,vt)\Bigr)\Big\vert_= \gamma'(t,p,v)\Big\vert_=v. So (with the right identification $T\_0\; T\_p\; M\; \backslash cong\; T\_pM$) the differential of $\backslash exp\_p$ is the identity. By the implicit function theorem, $\backslash exp\_p$ is a diffeomorphism on a neighborhood of $0\; \backslash in\; T\_pM$. The Gauss Lemma now tells that $\backslash exp\_p$ is also a radial isometry. 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「Gauss's lemma (Riemannian geometry)」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.