Orientation double cover ($\hat{M}=\{(p,o):o\text{ orients }{T}_{p}M\}$)

An oriented coordinate neighborhood of a smooth manifold gives two sheets distinguished by the sign of its orientation. On an overlap, the sheets agree where the two local orientation forms differ by a positive function. These open agreement sets define a two-sheeted smooth covering space, whose projection is locally a diffeomorphism.

For a connected smooth manifold, every component of the orientation double cover maps onto the base by the path lifting theorem. There can therefore be at most two components. A global orientation splits the cover into two copies of the base. Conversely, if there are two components, each meets every fibre once, so its inverse projection gives a smooth global choice of orientation.

At $(p,o)$ in the orientation double cover, use the derivative of the covering projection to transport the chosen orientation $o$ to the tangent space of the cover. In local oriented sheets this prescription is smooth, so it defines a global orientation. The involution $(p,o)\mapsto (p,-o)$ reverses it.