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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 tangent bundle ： ウィキペディア英語版 tangent bundle In differential geometry, the tangent bundle of a differentiable manifold $M$ is a manifold $TM$, which assembles all the tangent vectors in $M$. As a set, it is given by the disjoint union〔The disjoint union assures that for any two points ''x''1 and ''x''2 of manifold $M$ the tangent spaces T''1 and ''T''2 have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle ''S''1, see Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle.〕 of the tangent spaces of ''M''. That is, :$TM\; =\backslash bigsqcup\_T\_xM=\backslash bigcup\_\; \backslash left\backslash \backslash times\; T\_xM$ =\bigcup_ \left\. where $T\_x\; M$ denotes the tangent space to $M$ at the point $x$. So, an element of $TM$ can be thought of as a pair $(x,v)$, where $x$ is a point in $M$ and $v$ is a tangent vector to $M$ at $x$. There is a natural projection :$\backslash pi\; :\; TM\; \backslash twoheadrightarrow\; M$ defined by $\backslash pi(x,\; v)\; =\; x$. This projection maps each tangent space $T\_xM$ to the single point $x$. The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (a fiber bundle whose fibers are vector spaces). A section of $TM$ is a vector field on $M$, and the dual bundle to $TM$ is the cotangent bundle, which is the disjoint union of the cotangent spaces of $M$. By definition, a manifold $M$ is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold ''M'' is framed if and only if the tangent bundle ''TM'' is stably trivial, meaning that for some trivial bundle ''E'' the Whitney sum is trivial. For example, the ''n''-dimensional sphere ''Sn'' is framed for all ''n'', but parallelizable only for ''n''=1,3,7 (by results of Bott-Milnor and Kervaire). ==Role== One of the main roles of the tangent bundle is to provide a domain and range for the derivative of a smooth function. Namely, if ''f'' : ''M'' → ''N'' is a smooth function, with ''M'' and ''N'' smooth manifolds, its derivative is a smooth function ''Df'' : ''TM'' → ''TN''. 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「tangent bundle」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.