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.
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 .