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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 Clifton–Pohl torus ： ウィキペディア英語版 Clifton–Pohl torus In geometry, the Clifton–Pohl torus is an example of a compact Lorentzian manifold that is not geodesically complete. While every compact Riemannian manifold is also geodesically complete (by the Hopf–Rinow theorem), this space shows that the same implication does not generalize to pseudo-Riemannian manifolds.〔.〕 It is named after Yeaton H. Clifton and William F. Pohl, who described it in 1962 but did not publish their result.〔.〕 ==Definition== Consider the manifold $\backslash mathrm\; =\; \backslash mathbb^2\; \backslash smallsetminus\; \backslash$ with the metric :$g=\; \backslash frac$ Any homothety is an isometry of $M$, in particular including the map: :$\backslash lambda(x,y)=(2x,2y)$ Let $\backslash Gamma$ be the subgroup of the isometry group generated by $\backslash lambda$. Then $\backslash Gamma$ has a proper, discontinuous action on $M$. Hence the quotient $T\; =\; M/\backslash Gamma,$ which is topologically the torus, is a Lorentz surface that is called the Clifton-Pohl torus.〔 Sometimes, by extension, a surface is called a Clifton-Pohl torus if it is a finite covering of the quotient of $M$ by any homothety of ratio different from $\backslash pm\; 1$. 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「Clifton–Pohl torus」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.