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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 Minimal volume ： ウィキペディア英語版 Minimal volume In mathematics, in particular in differential geometry, the minimal volume is a number that describes one aspect of a Riemannian manifold's topology. This topological invariant was introduced by Mikhail Gromov. ==Definition== Consider a closed orientable connected smooth manifold $M^n$ with a smooth Riemannian metric $g$, and define $Vol()$ to be the volume of a manifold $M$ with the metric $g$. Let $K\_g$ represent the sectional curvature. The minimal volume of $M$ is a smooth invariant defined as :$MinVol(M):=\backslash inf\_\backslash$ that is, the infimum of the volume of $M$ over all metrics with bounded sectional curvature. Clearly, any manifold $M$ may be given an arbitrarily small volume by selecting a Riemannian metric $g$ and scaling it down to $\backslash lambda\; g$, as $Vol(M,\; \backslash lambda\; g)\; =\backslash lambda^Vol(M,\; g)$. For a meaningful definition of minimal volume, it is thus necessary to prevent such scaling. The inclusion of bounds on sectional curvature suffices, as $\backslash textstyle\; K\_\; =\; \backslash frac\; K\_g$. If $MinVol(M)=0$, then $M^n$ can be "collapsed" to a manifold of lower dimension (and thus one with $n$-dimensional volume zero) by a series of appropriate metrics; this manifold may be considered the Hausdorff limit of the related sequence, and the bounds on sectional curvature ensure that this convergence takes place in a topologically meaningful fashion. 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「Minimal volume」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.