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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 Riemannian submersion ： ウィキペディア英語版 Riemannian submersion In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces. Let (''M'', ''g'') and (''N'', ''h'') be two Riemannian manifolds and :$f:M\backslash to\; N$ a submersion. Then ''f'' is a Riemannian submersion if and only if the isomorphism :$df\; :\; \backslash mathrm(df)^\; \backslash rightarrow\; TN$ is an isometry. ==Examples== An example of a Riemannian submersion arises when a Lie group $G$ acts isometrically, freely and properly on a Riemannian manifold $(M,g)$. The projection $\backslash pi:\; M\; \backslash rightarrow\; N$ to the quotient space $N\; =\; M\; /G$ equipped with the quotient metric is a Riemannian submersion. For example, component-wise multiplication on $S^3\; \backslash subset\; \backslash mathbb^2$ by the group of unit complex numbers yields the Hopf fibration. 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「Riemannian submersion」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.