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.
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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 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 
.
