Integration over Riemannian Manifolds

Question

Can we integrate over non-orientable riemannian manifold? If so, how do we do it? Some references would be nice. Thank you!

Answer 1

Yes, by using a density. See John Lee's book Introduction to Smooth Manifolds Chapter 16 for a good introduction to the subject.

Answer 2

You must integrate what's called a density, rather than a differential form, over a non-orientable manifold. See, for example, this short discussion, a short article on Wikipedia, or section 8.2 of Marsden-Ratiu-Abraham's Manifolds, Tensor Analysis, and Applications, where they discuss how to generalize Stokes's Theorem to this situation.