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In mathematics, a space, where is a real number, is a specific type of metric space. Intuitively, triangles in a space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature . In a space, the curvature is bounded from above by . A notable special case is complete spaces are known as Hadamard spaces after the French mathematician Jacques Hadamard. Originally, Alexandrov called these spaces “ domain”. The terminology 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 , let denote the unique simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature . Denote by the diameter of , which is if and is precisely . Let be a triangle in with geodesic segments as its sides. is said to satisfy the inequality if there is a comparison triangle in the model space , with sides of the same length as the sides of , such that distances between points on are less than or equal to the distances between corresponding points on . The geodesic metric space is said to be a space if every geodesic triangle in with perimeter less than satisfies the inequality. A (not-necessarily-geodesic) metric space is said to be a space with curvature if every point of has a geodesically convex neighbourhood. A space with curvature 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
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