Orientation double cover
= Orientation double cover
{title2=$\widehat M=\{(p,o):o\text{ orients }T_pM\}$}
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>.