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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 Gromov's inequality for complex projective space ： ウィキペディア英語版 Gromov's inequality for complex projective space In Riemannian geometry, Gromov's optimal stable 2-systolic inequality is the inequality : $\backslash mathrm\_2(\backslash mathbb^n)$, valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained by the symmetric Fubini–Study metric, providing a natural geometrisation of quantum mechanics. Here $\backslash operatorname$ is the stable 2-systole, which in this case can be defined as the infimum of the areas of rational 2-cycles representing the class of the complex projective line $\backslash mathbb^1\; \backslash subset\; \backslash mathbb^n$ in 2-dimensional homology. The inequality first appeared in as Theorem 4.36. The proof of Gromov's inequality relies on the Wirtinger inequality for exterior 2-forms. ==Projective planes over division algebras $\backslash mathbb$== In the special case n=2, Gromov's inequality becomes $\backslash mathrm\_2^2)$. This inequality can be thought of as an analog of Pu's inequality for the real projective plane $\backslash mathbb^2$. In both cases, the boundary case of equality is attained by the symmetric metric of the projective plane. Meanwhile, in the quaternionic case, the symmetric metric on $\backslash mathbb^2$ is not its systolically optimal metric. In other words, the manifold $\backslash mathbb^2$ admits Riemannian metrics with higher systolic ratio $\backslash mathrm\_4\{\}^2/\backslash mathrm\{vol\}\_8$ than for its symmetric metric . 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「Gromov's inequality for complex projective space」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.