In more concise terms, this says that the structure group of the frame bundle of E, which is the real general linear group GLn(R), can be reduced to the subgroup consisting of those with positive determinant.
In that situation, an orientation of E amounts to a reduction from O(n) to the special orthogonal group SO(n).
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.
A complex vector bundle is oriented in a canonical way.
To give an orientation to a real vector bundle E of rank n is to give an orientation to the (real) determinant bundle
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 in the cohomology ring of the Thom space T(E) such that u generates
One can show, with some work,[citation needed] that the usual notion of an orientation coincides with a Z-orientation.