In textbooks Stokes' theorem is usually formulated for orientable manifolds (at least I couldn't find any version not using orientability). Is Stokes theorem: $\int\limits_{M}d\omega=\int\limits_{\partial M} \omega$ also true for non-orientable manifolds?
Are there any references, you can tell me?
Regards.
Best Answer
This is just more information on Jose's comment. Look at section 10 (pages 122 to 128) in:
Peter W. Michor: Topics in Differential Geometry. Graduate Studies in Mathematics, Vol. 93 American Mathematical Society, Providence, 2008. (pdf)
On the orientable double cover $\tilde M$ of $M$ (given by $\lbrace \xi\in Or(M): |\xi|=1\rbrace$ using 10.7) there are two kinds differential forms, namely the eigenspaces for $\pm 1$ under the action of the covering map. These are the "formes paire" and "formes impaire" on $M$ in the book of de Rham; for these you formulate Stokes' theorem directly on $M$, and it is the same as the Stokes' theorem on $\tilde M$. Note that you also have the corresponding double cover of the boundary.