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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 Parallelizable manifold ： ウィキペディア英語版 Parallelizable manifold In mathematics, a differentiable manifold $\backslash scriptstyle\; M$ of dimension ''n'' is called parallelizable if there exist smooth vector fields :$\backslash$ on the manifold, such that at any point $\backslash scriptstyle\; p$ of $\backslash scriptstyle\; M$ the tangent vectors :$\backslash$ provide a basis of the tangent space at $\backslash scriptstyle\; p$. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a section on $\backslash scriptstyle\; M.$ A particular choice of such a basis of vector fields on $\backslash scriptstyle\; M$ is called a parallelization (or an absolute parallelism) of $\backslash scriptstyle\; M$. ==Examples== *An example with ''n'' = 1 is the circle: we can take ''V''1 to be the unit tangent vector field, say pointing in the anti-clockwise direction. The torus of dimension ''n'' is also parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take ''n'' = 2, and construct a torus from a square of graph paper with opposite edges glued together, to get an idea of the two tangent directions at each point. More generally, any Lie group ''G'' is parallelizable, since a basis for the tangent space at the identity element can be moved around by the action of the translation group of ''G'' on ''G'' (any translation is a diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points in ''G''). *A classical problem was to determine which of the spheres ''S''''n'' are parallelizable. The zero-dimensional case ''S''0 is trivially parallelizable. The case ''S''1 is the circle, which is parallelizable as has already been explained. The hairy ball theorem shows that ''S''2 is not parallelizable. However ''S''3 is parallelizable, since it is the Lie group SU(2). The only other parallelizable sphere is ''S''7; this was proved in 1958, by Michel Kervaire, and by Raoul Bott and John Milnor, in independent work. The parallelizable spheres correspond precisely to elements of unit norm in the normed division algebras of the real numbers, complex numbers, quaternions, and octonions, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, and requires algebraic topology. *The product of parallelizable manifolds is parallelizable. *Any orientable three-dimensional manifold is parallelizable. 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「Parallelizable 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.