Orientation of a vector bundle

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_x, there is an orientation of the vector space E_x and one demands that each trivialization map (which is a bundle map)

${\displaystyle \phi _{U}:\pi ^{-1}(U)\to U\times \mathbf {R} ^{n}}$

is fiberwise orientation-preserving, where Rⁿ 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_n(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.