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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 Ricci flow ： ウィキペディア英語版 Ricci flow In differential geometry, the Ricci flow () is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric. The Ricci flow, named after Gregorio Ricci-Curbastro, was first introduced by Richard Hamilton in 1981 and is also referred to as the Ricci–Hamilton flow. It is the primary tool used in Grigori Perelman's solution of the Poincaré conjecture,〔Perelman, Grisha (2002), "The entropy formula for the Ricci flow and its geometric applications" (ArXiv )〕 as well as in the proof of the differentiable sphere theorem by Simon Brendle and Richard Schoen. == Mathematical definition == Given a Riemannian manifold with metric tensor $g\_$, we can compute the Ricci tensor $R\_$, which collects averages of sectional curvatures into a kind of "trace" of the Riemann curvature tensor. If we consider the metric tensor (and the associated Ricci tensor) to be functions of a variable which is usually called "time" (but which may have nothing to do with any physical time), then the Ricci flow may be defined by the geometric evolution equation〔 〕 :$\backslash partial\_t\; g\_=-2\; R\_.$ The normalized Ricci flow makes sense for compact manifolds and is given by the equation :$\backslash partial\_t\; g\_=-2\; R\_\; +\backslash frac\; R\_\backslash mathrm\; g\_$ where $R\_\backslash mathrm$ is the average (mean) of the scalar curvature (which is obtained from the Ricci tensor by taking the trace) and $n$ is the dimension of the manifold. This normalized equation preserves the volume of the metric. The factor of −2 is of little significance, since it can be changed to any nonzero real number by rescaling ''t''. However, the minus sign ensures that the Ricci flow is well defined for sufficiently small positive times; if the sign is changed, then the Ricci flow would usually only be defined for small negative times. (This is similar to the way in which the heat equation can be run forwards in time, but not usually backwards in time.) Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions. 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「Ricci flow」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.