# Integration on manifolds with singular points, corners

I'm looking for interesting examples of application of Stokes theorem for manifolds with singularities/corners.

The theorem was mentioned here: https://mathoverflow.net/questions/12920/stokes-theorem-for-manifolds-with-corners

Here's a version of the theorem presented in John Lee's Introduction to Smooth Manifolds:

Let $M$ be a smooth $n$-manifold with corners, and let $\omega$ be a compactly supported $(n − 1)$-form on $M$. Then $$\int_M \text{d} \omega = \int_{\partial M} \omega$$

And here's a version of the theorem presented in G. Stolzenberg's notes (page 10).

Could you provide me with some nice examples of applications of this theorem? Especially those using the fact that a manifold has corners, singularities.

I will appreciate all your help.

As someone who is currently teaching a "calculus on manifolds" course, there has been two times now that I had to handwave something which could have been done rigorously if I had access to manifolds with corners. To me, handwaving is better than working for the extra rigor, but your tastes might differ:

Stokes theorem on a cube: Stokes' theorem is easy to prove on $[0,1]^n$. For the sake of exposition, take $n=2$ and consider a $1$-form $\omega = f dx + g dy$. Then $$\int_{[0,1]^2} d \omega = \int_{[0,1]^2} \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) dx dy =$$ $$\int_{y=0}^1 \left(g(1,y) - g(0,y) \right) dy - \int_{x=0}^1 \left(f(x,1) - f(x,0) \right) dx =\int_{\partial\left( [0,1]^2 \right)} \omega$$ where, at the line break, we have used Fubini and the Fundamental Theorem of Calculus.

This is probably the simplest example of Stokes, and I presented it before getting into the general case. However, as several students pointed out, $[0,1]^2$ is not a manifold with boundary! The vocabulary of manifolds of corners would have allowed me to fix this issue rigorously.

This is issue is more than the issue of losing a nice example: Proofs of Stokes' theorem usually work by reduction to simpler and simpler cases. It is nice for the cube to be the final case where we do the detailed check, because it is the easiest case.

Dividing manifolds up into polyhedral pieces Suppose we want to integrate a form on the sphere $S^2$. We might want to divide $S^2$ up into its intersections with the $8$ octants in $\mathbb{R}^3$ and do the integral on each piece. But the intersection of $S^2$ with $\mathbb{R}_{\geq 0}^3$ is not a manifold with boundary! Similarly, we might want to cut the torus $(S^1)^2$ up into $4$ square pieces. Again, these squares are not manifolds with boundary!

As I said, I decided to handwave these issues rather than introduce all the terminology of manifolds with corners -- I thought that manifolds with boundary was enough new vocabulary for my students. Moreover, as someone who does polyhedral combinatorics in my research life, I didn't like the standard axiomatics for manifolds with corners. For example, let $P$ be the square pyramid $\{ (x,y,z) : 0 \leq z \leq 1,\ |x|, |y| \leq 1-z \}$ in $\mathbb{R}^3$. To my mind, any theory that works hard enough to incorporate the cube should cover $P$ as well, but $P$ is not a manifold with corners!

But I hope this gives an idea of where the issues are.