A real vector bundle π:E->M has an orientation if there exists a covering by trivializations U_i×R^k such that the transition functions are vector space orientation-preserving. Alternatively, there exists a section of the projectivization of the top exterior power of the bundle, P_R(⋀^k E). A bundle is called orientable if there exists an orientation. Hence a bundle E of bundle rank k is orientable iff ⋀^k E is a trivial line bundle. An orientation of the tangent bundle is equivalent to an orientation on the base manifold. Not all bundles are orientable, as can be seen by the tangent bundle of the Möbius strip. The nontrivial line bundle on the circle is also not orientable.

bundle | manifold orientation | vector bundle | vector space orientation