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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 Bishop–Gromov inequality ： ウィキペディア英語版 Bishop–Gromov inequality In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness theorem. ==Statement== Let $M$ be a complete ''n''-dimensional Riemannian manifold whose Ricci curvature satisfies the lower bound : $\backslash mathrm\; \backslash geq\; (n-1)\; K\; \backslash ,$ for a constant $K\backslash in\; \backslash mathbb$. Let $M\_K^n$ be the complete ''n''-dimensional simply connected space of constant sectional curvature $K$ (and hence of constant Ricci curvature $(n-1)K$); thus $M\_K^n$ is the ''n''-sphere of radius $1/\backslash sqrt$ if ''K'' > 0, or ''n''-dimensional Euclidean space if $K=0$, or an appropriately rescaled version of ''n''-dimensional hyperbolic space if . Denote by ''B''(''p'', ''r'') the ball of radius ''r'' around a point ''p'', defined with respect to the Riemannian distance function. Then, for any $p\backslash in\; M$ and $p\_K\backslash in\; M\_K^n$, the function : $\backslash phi(r)\; =\; \backslash frac$ is non-increasing on (0, ∞). As ''r'' goes to zero, the ratio approaches one, so together with the monotonicity this implies that : $\backslash mathrm\; \backslash ,B(p,r)\; \backslash leq\; \backslash mathrm\; \backslash ,\; B(p\_K,r).$ This is the version first proved by Bishop,〔Bishop, R. A relation between volume, mean curvature, and diameter. Amer. Math. Soc. Not. 10 (1963), p. 364.〕〔Bishop R.L., Crittenden R.J. Geometry of manifolds, Corollary 4, p. 256〕 originally assuming the (unnecessary) added hypothesis that $r$ is less than the injectivity radius at $p$. 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「Bishop–Gromov inequality」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.