Orientability - Orientation of Vector Bundles

Orientation of Vector Bundles

A real vector bundle, which a priori has a GL(n) structure group, is called orientable when the structure group may be reduced to, the group of matrices with positive determinant. For the tangent bundle, this reduction is always possible if the underlying base manifold is orientable and in fact this provides a convenient way to define the orientability of a smooth real manifold: a smooth manifold is defined to be orientable if its tangent bundle is orientable (as a vector bundle). Note that as a manifold in its own right, the tangent bundle is always orientable, even over nonorientable manifolds.