The orientation cover, or orientation double cover, of a positive-dimensional smooth manifold 
is the two-sheeted covering map 
whose fiber
over 
consists of the two orientations of 
. A coordinate
chart identifies the two sheets over its domain by
the orientations induced on the tangent
spaces. Changes of coordinate charts interchange
the sheets precisely when the Jacobian determinant is
negative (Zinger 2011, p. 70).
The manifold 
has a canonical manifold
orientation, and 
has a bundle section iff 
is an orientable
manifold. For connected 
, the orientation cover is connected iff 
is nonorientable. When 
is orientable, the
orientation cover is the disjoint union of two
copies of 
(Hatcher 2002, pp. 234-235).
