Tangent orientation double cover of a surface with boundary

Let $M$ be a smooth two-dimensional manifold with boundary and let $\det T_{x}M={\mathrm{\Lambda }}^2{T}_{x}M$ be the determinant line of its tangent space. An orientation of $T_{x}M$ is a positive ray $o_x$ in this line. The tangent orientation double cover is the set

$$\tilde{M}=\{(x,o_x):x\in M,o_x\text{ is an orientation of }{T}_{x}M\},\pi (x,o_x)=x\mathrm{.}$$

For a boundary chart $\varphi :U\to V\subseteq {\mathbb{H}}^2$, let $o_{\varphi }(x)$ be the ray obtained by applying $\det (d{\varphi }_x^{-1})$ to the standard positive ray in $\det {\mathbb{R}}^2$. Give $\tilde{M}$ the smooth boundary-chart structure whose two local sheets over $U$ are $\{(x,o_{\varphi }(x)):x\in U\}$ and $\{(x,-o_{\varphi }(x)):x\in U\}$, each with coordinate map $(x,\pm o_{\varphi }(x))\mapsto \varphi (x)$. On an overlap with a second chart $\psi$, the sheet sign changes by $\mathrm{sgn}\det D(\psi \circ {\varphi }^{-1})$; this sign is locally constant, so the sheet transitions are smooth half-space transitions. Thus $\pi$ is the associated two-sheeted smooth covering map, including over the boundary. The involution $\tau (x,o_x)=(x,-o_x)$ exchanges the sheets. The canonical orientation of $\tilde{M}$ at $(x,o_x)$ is the ray $(\det d{\pi }_{(x,o_x)}{)}^{-1}(o_x)$ in $\det T_{(x,o_x)}\tilde{M}$.

Definition of orientations as rays in determinant lines

Definition of a covering map and its sheets

Definition of smooth charts and atlases with boundary

Theorem on smooth tangent bundles at the boundary