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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 :; 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: : such that ''J'' acts as the square root ''i'' of -1 on fibers: if is the map on fiber-level, then as a linear map. If ''E'' is a complex vector bundle, then the complex structure ''J'' can be defined by setting to be the scalar multiplication by . 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'', : 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
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