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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 CAT(k) space ： ウィキペディア英語版 CAT(k) space In mathematics, a $\backslash mathbf$ space, where $k$ is a real number, is a specific type of metric space. Intuitively, triangles in a $\backslash operatorname(k)$ space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature $k$. In a $\backslash operatorname(k)$ space, the curvature is bounded from above by $k$. A notable special case is $k=0$ complete $\backslash operatorname(0)$ spaces are known as Hadamard spaces after the French mathematician Jacques Hadamard. Originally, Alexandrov called these spaces “$\backslash mathfrak\_k$ domain”. The terminology $\backslash operatorname(k)$ was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov (although Toponogov never explored curvature bounded above in publications). ==Definitions== For a real number $k$, let $M\_k$ denote the unique simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature $k$. Denote by $D\_k$ the diameter of $M\_k$, which is $+\backslash infty$ if $k\; \backslash leq\; 0$ and is precisely $d(x,y)$. Let $\backslash Delta$ be a triangle in $X$ with geodesic segments as its sides. $\backslash Delta$ is said to satisfy the $\backslash mathbf$ inequality if there is a comparison triangle $\backslash Delta\text{'}$ in the model space $M\_k$, with sides of the same length as the sides of $\backslash Delta$, such that distances between points on $\backslash Delta$ are less than or equal to the distances between corresponding points on $\backslash Delta\text{'}$. The geodesic metric space $(X,d)$ is said to be a $\backslash mathbf$ space if every geodesic triangle $\backslash Delta$ in $X$ with perimeter less than $2D\_k$ satisfies the $\backslash operatorname(k)$ inequality. A (not-necessarily-geodesic) metric space $(X,\backslash ,d)$ is said to be a space with curvature $\backslash leq\; k$ if every point of $X$ has a geodesically convex $\backslash operatorname(k)$ neighbourhood. A space with curvature $\backslash leq\; 0$ may be said to have non-positive curvature. 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「CAT(k) space」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.