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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 Trace-free Ricci tensor ： ウィキペディア英語版 Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean ''n''-space. The Ricci tensor is defined on any pseudo-Riemannian manifold, as a trace of the Riemann curvature tensor. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold .〔It is assumed that the manifold carries its unique Levi-Civita connection. For a general affine connection, the Ricci tensor need not be symmetric.〕 In relativity theory, the Ricci tensor is the part of the curvature of space-time that determines the degree to which matter will tend to converge or diverge in time (via the Raychaudhuri equation). It is related to the matter content of the universe by means of the Einstein field equation. In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. comparison theorem) with the geometry of a constant curvature space form. If the Ricci tensor satisfies the vacuum Einstein equation, then the manifold is an Einstein manifold, which have been extensively studied (cf. ). In this connection, the Ricci flow equation governs the evolution of a given metric to an Einstein metric; the precise manner in which this occurs ultimately leads to the solution of the Poincaré conjecture. ==Definition== Suppose that $(M,g)$ is an ''n''-dimensional Riemannian manifold, equipped with its Levi-Civita connection $\backslash nabla$. The Riemannian curvature tensor of $M$ is the $(1,3)$ tensor defined by :$R(X,Y)Z\; =\; \backslash nabla\_X\backslash nabla\_Y\; Z\; -\; \backslash nabla\_Y\backslash nabla\_XZ\; -\; \backslash nabla\_Z$ on vector fields $X,Y,Z$. Let $T\_pM$ denote the tangent space of ''M'' at a point ''p''. For any pair of tangent vectors $\backslash xi$ and $\backslash eta$ in $T\_pM$, the Ricci tensor $\backslash operatorname$ evaluated at $(\backslash xi,\; \backslash eta\; )$ is defined to be the trace of the linear map $T\_pM\backslash to\; T\_pM$ given by :$\backslash zeta\; \backslash mapsto\; R(\backslash zeta,\backslash eta)\; \backslash xi.$ In local coordinates (using the Einstein summation convention), one has :$\backslash operatorname\; =\; R\_\backslash ,dx^i\; \backslash otimes\; dx^j$ where :$R\_\; =\; \_.$ In terms of the Riemann curvature tensor and the Christoffel symbols, one has :$$ R_ = _ = \partial_\Gamma^\rho_ + \Gamma^\rho_ \Gamma^\lambda_ - \Gamma^\rho_\Gamma^\lambda_ = 2 \Gamma^_ + 2 \Gamma^\rho_" TITLE="\rho}">\Gamma^\lambda_ . 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 curvature」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.