Bundle Orientation

A real vector bundle  has an orientation if there exists a covering by trivializations  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, . A bundle is called orientable if there exists an orientation. Hence a bundle  of bundle rank  is orientable iff  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.