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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 Calabi–Eckmann manifold ： ウィキペディア英語版 Calabi–Eckmann manifold In complex geometry, a part of mathematics, a Calabi–Eckmann manifold (or, often, Calabi–Eckmann space), named after Eugenio Calabi and Beno Eckmann, is a complex, homogeneous, non-Kähler manifold, homeomorphic to a product of two odd-dimensional spheres of dimension ≥ 3. The Calabi–Eckmann manifold is constructed as follows. Consider the space $^n\backslash backslash\; 0\; \backslash times\; ^m\backslash backslash\; 0$, ''m,n'' > 1, equipped with an action of a group $$: : $t\backslash in\; ,\; \backslash \; (x,y)\backslash in\; ^n\backslash backslash\; 0\; \backslash times\; ^m\backslash backslash\; 0\backslash \; \backslash \; |\backslash \; \backslash \; t(x,y)=\; (e^tx,\; e^y)$ where $\backslash alpha\backslash in\; \backslash backslash$ is a fixed complex number. It is easy to check that this action is free and proper, and the corresponding orbit space ''M'' is homeomorphic to ''S''2''n''−1 × ''S''2''m''−1. Since ''M'' is a quotient space of a holomorphic action, it is also a complex manifold. It is obviously homogeneous, with a transitive holomorphic action of $GL(n,)\; \backslash times\; GL(m,\; )$ A Calabi–Eckmann manifold ''M'' is non-Kähler, because $H^2(M)=0$. It is the simplest example of a non-Kähler manifold which is simply connected (in dimension 2, all simply connected compact complex manifolds are Kähler). The natural projection : $^n\backslash backslash\; 0\; \backslash times\; ^m\backslash backslash\; 0\backslash mapsto\; P^\backslash times\; P^$ induces a holomorphic map from the corresponding Calabi–Eckmann manifold ''M'' to $P^\backslash times\; P^$. The fiber of this map is an elliptic curve ''T'', obtained as a quotient of $\backslash Bbb\; C$ by the lattice $+\; \backslash alpha\backslash cdot$. This makes ''M'' into a principal ''T''-bundle. Calabi and Eckmann discovered these manifolds in 1953.〔E. Calabi and B. Eckmann: A class of compact complex manifolds which are not algebraic. Annals of Mathematics, 58, 494–500 (1953)〕 ==Notes== 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「Calabi–Eckmann manifold」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.