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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 Complex vector bundle ： ウィキペディア英語版 Complex vector bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be promoted to a complex vector bundle, the complexification :$E\; \backslash otimes\; \backslash mathbb$; whose fibers are ''E''''x'' ⊗R C. Any complex vector bundle over a paracompact space admits a hermitian metric. The basic invariant of a complex vector bundle is a Chern class. == Complex structure == A complex vector bundle can be thought of as a real vector bundle with an additional structure, the complex structure. By definition, a complex structure is a bundle map between a real vector bundle ''E'' and itself: :$J:\; E\; \backslash to\; E$ such that ''J'' acts as the square root ''i'' of -1 on fibers: if $J\_x:\; E\_x\; \backslash to\; E\_x$ is the map on fiber-level, then $J\_x^2\; =\; -1$ as a linear map. If ''E'' is a complex vector bundle, then the complex structure ''J'' can be defined by setting $J\_x$ to be the scalar multiplication by $i$. Conversely, if ''E'' is a real vector bundle with a complex structure ''J'', then ''E'' can be turned into a complex vector bundle by setting: for any real numbers ''a'', ''b'' and a real vector ''v'' in a fiber ''E''''x'', :$(a\; +\; ib)\; v\; =\; a\; v\; +\; J(b\; v).$ Example: A complex structure on the tangent bundle of a real manifold ''M'' is usually called an almost complex structure. A theorem of Newlander and Nirenberg says that an almost complex structure ''J'' is "integrable" in the sense it is induced by a structure of a complex manifold if and only if a certain tensor involving ''J'' vanishes. 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「Complex vector 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.