I recently came across the idea of a pseudo differential form (and "pseudo" objects in a manifold in general).
Let $\Psi M$ is the real line bundle on $M$ which changes sign between two different coordinate patches if those patches have opposite orientations. We can define a pseudo-$k$-form on $M$ to be a section of the vector bundle $(\Lambda^k(T^*) \otimes \Psi) M$.
Alternatively, we can define a pseudo $k$-form $\alpha$ in a local coordinate patch as a sum of expressions of the form $f_I(x) dx^{I_1} dx^{I_2} \cdots dx^{I_k} \Psi$ where $I$ is a degree-$k$ multiindex. When given a new coordinate system $(y_1, y_2 \dots y_n)$, let $\beta$ be the sum of expressions of the form $g_I(x) dy^{I_1} dy^{I_2} \cdots dy^{I_k} \Psi$ you would get if you interpreted $\alpha$ as a normal differential form (e.g. by substituting $dx^i = \frac{\partial x^i}{\partial y^j} dy^j$). Define $\alpha$ in this new coordinate system to be $+\beta$ if the transition map between coordinate systems is orientation-preserving and $-\beta$ if this transition map is orientation-reversing.
Either way, if $M$ is a $n$-dimensional manifold, then a pseudo $n$-form is precisely the geometric object which can be integrated over $M$ in a well-defined manner. My question is whether Stokes's theorem for differential forms also holds for pseudo-forms, which might hold even for non-orientable manifolds. Specifically, I would want, for any compactly supported $(n-1)$ pseudo-form $\omega$, we have:
$$\int_{\partial M} \omega = \int_M d\omega$$
The only roadblock to formulating such a theorem clearly is that I don't know if there is a good way to define the restriction of a pseudo-differential form to the boundary of $M$.
If this theorem is true (and well-defined), then is it also true if $\omega$ is a $(k-1)$-dimensional form and we integrate over an arbitrary embedded submanifold of $M$?
Thanks!
