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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 Berger's sphere ： ウィキペディア英語版 Berger's sphere In Riemannian geometry, a Berger sphere, named after Marcel Berger, is a standard 3-sphere with Riemannian metric from a one-parameter family, which can be obtained from the standard metric by shrinking along fibers of a Hopf fibration. It is interesting in that it is one of the simplest examples of Gromov collapse.〔. See in particular (p. 122 ).〕 More precisely, one first considers the Lie algebra spanned by generators ''x''1, ''x''2, ''x''3 with Lie bracket () = −2ε''ijk''''x''''k''. This is well known to correspond to the simply connected Lie group ''S''3. Then, taking the product ''S''3×R, extending the Lie bracket so that the generator ''x''4 is left invariant under the operation of the Lie group, and taking the quotient by α''x''1+β''x''4, where α2+β2 = 1, we finally obtain the Berger spheres ''B''(β).〔.〕 There are also higher-dimensional analogues of Berger spheres. ==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「Berger's sphere」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.