A manifold orientation on an 
-dimensional smooth
manifold 
is given by a nowhere vanishing differential n-form.
Alternatively, it is a bundle orientation for
the tangent bundle. If an orientation exists on
,
then 
is called orientable.
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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 vector field of unit
normal vectors. The choice of a vector field
of unit normal vectors determines the orientation
of the submanifold. For example, the sphere 
is orientable.
Some types of manifolds are always orientable. For instance, complex manifolds, including nonsingular complex varieties, and also symplectic manifolds are orientable. Also, any unoriented smooth manifold has a double covering map from an oriented smooth manifold.
A smooth 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 
.
