An orientation on an n-dimensional manifold is given by a nowhere vanishing differential n-form. Alternatively, it is an bundle orientation for the tangent bundle. If an orientation exists on M, then M is called orientable. Not all manifolds are orientable, as exemplified by the Möbius strip and the Klein bottle, illustrated above. However, an (n - 1)-dimensional submanifold of R^n is orientable iff it has a unit normal vector field. The choice of unit determines the orientation of the submanifold. For example, the sphere S^2 is orientable.

bundle orientation | differential k-form | orientable manifold | vector space orientation