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In mathematics, an '''orientation''' of a real [[vector bundle]] is a generalization of an [[orientation of a vector space]]; thus, given a real vector bundle π: ''E'' →''B'', an orientation of ''E'' means: for each fiber ''E''<sub>''x''</sub>, there is an orientation of the vector space ''E''<sub>''x''</sub> and one demands that each trivialization map (which is a bundle map)
:<math>\phi_U :  \pi^{-1}(U) \to U \times \mathbf{R}^n</math>
is fiberwise orientation-preserving, where '''R'''<sup>''n''</sup> is given the [[standard orientation]]. In more concise terms, this says that the structure group of the [[frame bundle]] of ''E'', which is the real [[general linear group]] ''GL''<sub>n</sub>('''R'''), can be reduced to the subgroup consisting of those with positive determinant.

If ''E'' is a real vector bundle of rank ''n'', then a choice of metric on ''E'' amounts to a reduction of the structure group to the [[orthogonal group]] ''O''(''n''). In that situation, an orientation of ''E'' amounts to a reduction from ''O''(''n'') to the [[special orthogonal group]] ''SO''(''n'').

A vector bundle together with an orientation is called an '''oriented bundle'''. A vector bundle that can be given an orientation is called an '''orientable vector bundle'''.

The basic invariant of an oriented bundle is the [[Euler class]]. The multiplication (that is, cup product) by the Euler class of an oriented bundle gives rise to a [[Gysin sequence]].

== Examples ==
A [[complex vector bundle]] is oriented in a canonical way.

The notion of an orientation of a vector bundle generalizes an [[orientation of a manifold]]: an orientation of a manifold is an orientation of the tangent bundle of the manifold. In particular, a manifold is orientable if and only if its tangent bundle is orientable as a vector bundle. (note: as a manifold, a tangent bundle is always orientable.)

== Operations ==
To give an orientation to a real vector bundle ''E'' of rank ''n'' is to give an orientation to the (real) [[determinant bundle]] <math>\operatorname{det} E = \wedge^n E</math> of ''E''. Similarly, to give an orientation to ''E'' is to give an orientation to the [[unit sphere bundle]] of ''E''.

Just as a real vector bundle is classified by the real infinite [[Grassmannian]], oriented bundles are classified by the infinite Grassmannian of oriented real vector spaces.

== Thom space ==
{{main|Thom space}}
From the cohomological point of view, for any ring Λ, a Λ-orientation of a real vector bundle ''E'' of rank ''n'' means a choice (and existence) of a class
:<math>u \in H^n(T(E); \Lambda)</math>
in the cohomology ring of the [[Thom space]] ''T''(''E'') such that ''u'' generates <math>\tilde{H}^*(T(E); \Lambda)</math> as a free <math>H^*(E; \Lambda)</math>-module globally and locally: i.e.,
:<math>H^*(E; \Lambda) \to \tilde{H}^*(T(E); \Lambda), x \mapsto  x \smile u</math>
is an isomorphism (called the [[Thom isomorphism]]), where "tilde" means [[reduced cohomology]], that restricts to each isomorphism
:<math>H^*(\pi^{-1}(U); \Lambda) \to \tilde{H}^*(T(E|_U); \Lambda)</math>
induced by the trivialization <math>\pi^{-1}(U) \simeq U \times \mathbf{R}^n</math>. One can show, with some work,{{citation needed|date=January 2016}} that the usual notion of an orientation coincides with a '''Z'''-orientation.

== See also ==
*The [[integration along the fiber]]
*[[Orientation bundle]] (or [[orientation sheaf]]) - this is used to formulate the Thom isomorphism for non-oriented bundles.

== References ==
*{{citation | last1 = Bott | first1 = Raoul | authorlink = Raoul Bott | last2=Tu |first2= Loring | title = Differential Forms in Algebraic Topology | year = 1982 | publisher = Springer | location = New York | isbn = 0-387-90613-4}}
* J.P. May, ''A Concise Course in Algebraic Topology.'' University of Chicago Press, 1999.
* {{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | last2=Stasheff | first2=James D. |author2-link=Jim Stasheff| title=Characteristic classes | publisher=Princeton University Press; University of Tokyo Press | series=Annals of Mathematics Studies | isbn=978-0-691-08122-9 | year=1974 | volume=76}}

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[[Category:Linear algebra]]
[[Category:Analytic geometry]]
[[Category:Orientation (geometry)]]