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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 decomposition ： ウィキペディア英語版 Ricci decomposition In semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties. This decomposition is of fundamental importance in Riemannian- and pseudo-Riemannian geometry. ==The pieces appearing in the decomposition== The decomposition is :$R\_=\; \backslash ,\; S\_+E\_+C\_.$ The three pieces are: # the ''scalar part'', the tensor $S\_$ # the ''semi-traceless part'', the tensor $E\_$ # the ''fully traceless part'', the Weyl tensor $C\_$ Each piece possesses all the algebraic symmetries of the Riemann tensor itself, but has additional properties. The decomposition can have different signs, depending on the Ricci curvature convention, and only makes sense if the dimension satisfies $n>2$. The scalar part :$S\_\; =\; \backslash frac\; \backslash ,\; H\_$ is built using the scalar curvature $R\; =\; \_m$, where $R\_=\_$ is the Ricci curvature, and a tensor constructed algebraically from the metric tensor $g\_$, :$H\_\; =\; g\_\; \backslash ,\; g\_\; -\; g\_\; \backslash ,\; g\_\; =\; 2g\_"\; TITLE="c\}">\backslash ,\; g\_.$ The semi-traceless part :$E\_\; =\; \backslash frac\; \backslash ,\; \backslash left(\; g\_\; \backslash ,\; S\_\; -\; g\_\; \backslash ,\; S\_\; +\; g\_\; \backslash ,\; S\_\; -\; g\_\; \backslash ,\; S\_\; \backslash right)\; =$ \frac \, \left( g_" TITLE="c}">\, S_ - g_" TITLE="c}">\, S_ \right) is constructed algebraically using the metric tensor and the ''traceless part'' of the Ricci tensor :$S\_\; =\; R\_\; -\; \backslash frac\; \backslash ,\; g\_\; \backslash ,\; R$ where $g\_$ is the metric tensor. The Weyl tensor $C\_$ or ''conformal curvature tensor'' is completely traceless, in the sense that taking the trace, or contraction, over any pair of indices gives zero. Hermann Weyl showed that this tensor measures the deviation of a semi-Riemannian manifold from ''conformal flatness''; if it vanishes, the manifold is (locally) conformally equivalent to a flat manifold. No additional differentiation is needed anywhere in this construction. In the case of a Lorentzian manifold, $n=4$, the Einstein tensor $G\_\; =\; R\_\; -\; 1/2\; \backslash ,\; g\_\; R$ has, by design, a trace which is just the negative of the Ricci scalar, and one may check that the traceless part of the Einstein tensor agrees with the traceless part of the Ricci tensor. :$S\_\; =\; R\_\; -\; \backslash frac\; \backslash ,\; g\_\; \backslash ,\; R\; =\; G\_\; -\; \backslash frac\; \backslash ,\; g\_\; \backslash ,\; G$ ''Terminological note:'' the notation $R\_,\; \backslash ,\; C\_$ is standard in the modern literature, the notations $S\_,\; \backslash ,\; E\_$ are commonly used but not standardized, and there is no standard notation for the scalar part. 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 decomposition」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.