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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 Cut locus ： ウィキペディア英語版 Cut locus (Riemannian manifold) In Riemannian geometry, the cut locus of a point $p$ in a manifold is roughly the set of all other points for which there are multiple minimizing geodesics connecting them from $p$, but it may contain additional points where the minimizing geodesic is unique, under certain circumstances. The distance function from ''p'' is a smooth function except at the point ''p'' itself and the cut locus. == Definition == Fix a point $p$ in a complete Riemannian manifold $(M,g)$, and consider the tangent space $T\_pM$. It is a standard result that for sufficiently small $v$ in $T\_p\; M$, the curve defined by the Riemannian exponential map, $\backslash gamma(t)\; =\; \backslash exp\_p(tv)$ for $t$ belonging to the interval $()$ is a minimizing geodesic, and is the unique minimizing geodesic connecting the two endpoints. Here $\backslash exp\_p$ denotes the exponential map from $p$. The cut locus of $p$ in the tangent space is defined to be the set of all vectors $v$ in $T\_pM$ such that $\backslash gamma(t)=\backslash exp\_p(tv)$ is a minimizing geodesic for $t\; \backslash in\; ()$ but fails to be minimizing for $t\; \backslash in\; [0,1\; +\; \backslash epsilon)$ for each $\backslash epsilon\; >\; 0$. The cut locus of $p$ in $M$ is defined to be image of the cut locus of $p$ in the tangent space under the exponential map at $p$. Thus, we may interpret the cut locus of $p$ in $M$ as the points in the manifold where the geodesics starting at $p$ stop being minimizing. The least distance from ''p'' to the cut locus is the injectivity radius at ''p''. On the open ball of this radius, the exponential map at ''p'' is a diffeomorphism from the tangent space to the manifold, and this is the largest such radius. The global injectivity radius is defined to be the infimum of the injectivity radius at ''p'', over all points of the manifold. 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「Cut locus (Riemannian manifold)」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.