Manifold Orientation

An orientation on an -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 , then  is called orientable.

Not all manifolds are orientable, as exemplified by the Möbius strip and the Klein bottle, illustrated above.

However, an -dimensional submanifold of  is orientable iff it has a unit normal vector field. The choice of unit determines the orientation of the submanifold. For example, the sphere  is orientable.

Some types of manifolds are always orientable. For instance, complex manifolds, including varieties, and also symplectic manifolds are orientable. Also, any unoriented manifold has a double cover which is oriented.

A map  between oriented manifolds of the same dimension is called orientation preserving if the volume form on  pulls back to a positive volume form on . Equivalently, the differential  maps an oriented basis in  to an oriented basis in .