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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 Generalized Gauss–Bonnet theorem ： ウィキペディア英語版 Generalized Gauss–Bonnet theorem In mathematics, the generalized Gauss–Bonnet theorem (''also called Chern–Gauss–Bonnet theorem'') presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature. It is a direct generalization of the Gauss–Bonnet theorem (named after Carl Friedrich Gauss and Pierre Ossian Bonnet) to higher dimensions. Let ''M'' be a compact orientable 2''n''-dimensional Riemannian manifold without boundary, and let $\backslash Omega$ be the curvature form of the Levi-Civita connection. This means that $\backslash Omega$ is an $\backslash mathfrak\; s\backslash mathfrak\; o(2n)$-valued 2-form on ''M''. So $\backslash Omega$ can be regarded as a skew-symmetric 2''n'' × 2''n'' matrix whose entries are 2-forms, so it is a matrix over the commutative ring $\backslash wedge^(\backslash Omega)$, which turns out to be a 2''n''-form. The generalized Gauss–Bonnet theorem states that :$\backslash int\_M\; \backslash mbox(\backslash Omega)=(2\backslash pi)^n\backslash chi(M)\backslash$ where $\backslash chi(M)$ denotes the Euler characteristic of ''M''. ==Example: dimension 4== In dimension $n=4$, for a compact oriented manifold, we get :$\backslash chi(M)=\backslash frac\backslash int\_M\backslash left(|Rm|^2-4|Rc|^2+R^2\backslash right)d\backslash mu$ where $Rm$ is the full Riemann curvature tensor, $Rc$ is the Ricci curvature tensor, and $R$ is the scalar 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「Generalized Gauss–Bonnet theorem」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.