I have been reading Riemannian Geometry and Geometric Analysis by Jürgen Jost. He uses integration over Riemannian manifolds (without boundary) which are not assumed to be orientable. While I know that there are many ways to define a natural notion of integration on such manifolds, e.g., here (the author seems to be using the fourth definition mentioned there), I need several things to hold for this situation:
- I need some form of Stokes' theorem. At least I need the fact that the Laplacian is self-adjoint, and also Green's identities.
- I need Sobolev spaces over such manifolds. In particular, I need the Sobolev embedding theorem and the Rellich–Kondrachov theorem.
Are these actually correct in the non-orientable situation? And what is a reference for such matters?
If you lift up to the orientable double cover all this follows.
If you can't use that, you still know $M-\operatorname{Cut}(p)$ is diffeomorphic to a ball, so you can appeal to results on $\mathbb{R}^n$ for that.