Connectedness criterion for the orientation double cover ($M\text{ orientable}⟺\hat{M}\text{ disconnected}$)

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.