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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 Schur's lemma (from Riemannian geometry) ： ウィキペディア英語版 Schur's lemma (from Riemannian geometry) Schur's lemma is a result in Riemannian geometry that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. It is essentially a degree of freedom counting argument. == Statement of the Lemma == Suppose $(M^n,g)^$ such that :$\backslash mathrm^$ such that :$\backslash mathrm^\{\}\_\{\}(X\_p)\; =\; f(p)\; X\_p$ for all $X\_p\; \backslash in\; T\_p\; M$ and all $p\; \backslash in\; M,$ :then $f$ is constant, and the manifold is Einstein. The requirement that $n\; \backslash geq\; 3$ cannot be lifted. This result is far from true on two-dimensional surfaces. In two dimensions sectional curvature is always pointwise constant since there is only one two-dimensional subspace $\backslash Pi\_p\; \backslash subset\; T\_p\; M$, namely $T\_p\; M$. Furthermore, in two dimensions the Ricci curvature endomorphism is always a multiple of the identity (scaled by Gauss curvature). On the other hand, certainly not all two-dimensional surfaces have constant sectional (or Ricci) 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「Schur's lemma (from Riemannian geometry)」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.