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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 Kulkarni–Nomizu product ： ウィキペディア英語版 Kulkarni–Nomizu product In the mathematical field of differential geometry, the Kulkarni–Nomizu product (named for Ravindra Shripad Kulkarni and Katsumi Nomizu) is defined for two (0,2)-tensors and gives as a result a (0,4)-tensor. If ''h'' and ''k'' are symmetric (0,2)-tensors, then the product is defined via: :$(h\; k)(X\_1,X\_2,X\_3,X\_4)\; :=\; h(X\_1,X\_3)k(X\_2,X\_4)\; +\; h(X\_2,X\_4)k(X\_1,X\_3)\; -\; h(X\_1,X\_4)k(X\_2,X\_3)\; -\; h(X\_2,X\_3)k(X\_1,X\_4)$ where the ''X''''j'' are tangent vectors. Note that $h\; k\; =\; k\; h$. The Kulkarni–Nomizu product is a special case of the product in the graded algebra :$\backslash bigoplus\_^n\; S^2(\backslash Omega^p\; M),$ where, on simple elements, :$(\backslash alpha\backslash cdot\backslash beta)\; (\backslash gamma\backslash cdot\backslash delta)\; =\; (\backslash alpha\backslash wedge\backslash gamma)\backslash cdot(\backslash beta\backslash wedge\backslash delta)$ (the dot denotes the symmetric product). The Kulkarni–Nomizu product of a pair of symmetric tensors has the algebraic symmetries of the Riemann tensor. It is thus commonly used to express the contribution that the Ricci curvature (or rather, the Schouten tensor) and the Weyl tensor each makes to the curvature of a Riemannian manifold. This so-called Ricci decomposition is useful in differential geometry. When there is a metric tensor ''g'', the Kulkarni–Nomizu product of ''g'' with itself is the identity endomorphism of the space of 2-forms, Ω''2''(''M''), under the identification (using the metric) of the endomorphism ring End(Ω''2''(''M'')) with the tensor product Ω''2''(''M'') ⊗ Ω''2''(''M''). A Riemannian manifold has constant sectional curvature ''k'' if and only if the Riemann tensor has the form :$R\; =\; \backslash fracg\; g$ where ''g'' is the metric tensor. ==References== *. * 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「Kulkarni–Nomizu product」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.