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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 Splitting theorem ： ウィキペディア英語版 Splitting theorem The splitting theorem is a classical theorem in Riemannian geometry. It states that if a complete Riemannian manifold ''M'' with Ricci curvature :$(M)\; \backslash ge\; 0$ has a straight line, i.e., a geodesic γ such that :$d(\backslash gamma(u),\backslash gamma(v))=|u-v|$ for all :$u,\; v\backslash in\backslash mathbb,$ then it is isometric to a product space :$\backslash mathbb\backslash times\; L,$ where $L$ is a Riemannian manifold with :$(L)\; \backslash ge\; 0.$ ==History== For the surfaces, the theorem was proved by Stephan Cohn-Vossen.〔S. Cohn-Vossen, “Totalkrümmung und geodätische Linien auf einfachzusammenhängenden offenen vollständigen Flächenstücken”, Матем. сб., 1(43):2 (1936), 139–164〕 Victor Andreevich Toponogov generalized it to manifolds with non-negative sectional curvature.〔Toponogov, V. A. Riemannian spaces containing straight lines. (Russian) Dokl. Akad. Nauk SSSR 127 1959 977–979.〕 Jeff Cheeger and Detlef Gromoll proved that non-negative Ricci curvature is sufficient. Later the splitting theorem was extended to Lorentzian manifolds with nonnegative Ricci curvature in the time-like directions.〔Eschenburg, J.-H. The splitting theorem for space-times with strong energy condition. J. Differential Geom. 27 (1988), no. 3, 477–491.〕 〔Galloway, Gregory J.(1-MIAM) The Lorentzian splitting theorem without the completeness assumption. J. Differential Geom. 29 (1989), no. 2, 373–387.〕 〔Newman, Richard P. A. C. A proof of the splitting conjecture of S.-T. Yau. J. Differential Geom. 31 (1990), no. 1, 163–184.〕 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「Splitting 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.