I am currently reading the proof of the Chern-Gauß-Bonnet theorem by Chern. In one of the last steps I don't understand how we obtain the Euler characteristic with Stokes. I know that he uses the Poincaré-Hopf theorem to obtain the Euler characteristic from the index of the vector field, but where do we get the mapping degree that gives the index?
So, we have a closed, riemannian manifold $M$ with $dim(M)=n=2p$ and the unit tangent bundle $\pi:UT(M)\to M$. Furthermore we have the integrand of the Gauß-Bonnet integral $pf(\Omega/2\pi)$. This is a closed form on $M$ and we can pull this back to $UT(M)$ to get a exact form $\pi^*(pf(\Omega/2\pi))=d\Pi$. Now Chern takes a continuous unit vector field $X$ on $M$ with a singular point in $0\in M$. This vector field defines a n-dimensional submanifold $V^n$ in $UT(M)$, I guess by taking the graph of the vector field.
First Question: He says that $\partial V^n=\chi(M) \mathcal{S}^{n-1}_0 $, where $\mathcal{S}^{n-1}_0 $ is the fiber over the singularity of the vector field and $\chi(M)$ is the Euler characteristic of $M$. Why is this true?
Second Question: In the calculation we have \begin{align*} \int_M pf(\Omega/2\pi)=\int_{V^n}\pi|_{V^n}^*(pf(\Omega/2\pi))=\int_{V^n}d\Pi\overset{?}{=}\chi(M)\int_{\mathcal{S}_0^{n-1}}\Pi. \end{align*} In the equality with the (?) he uses Stokes. But why do we get $\chi(M)$? From Poincaré-Hopf we know that $\chi(M)=ind_X(0)=deg(X)$. But \begin{align*} deg(X)\int_{\mathcal{S}_0^{n-1}}\Pi=\int_{\partial\overline{B_r(0)}}X^*(\Pi) \end{align*} if I understood the mapping degree correct. So I wonder where the RHS is hidden in the above computation.
Can someone give me a hint as to what I am missing?
Although it's certainly not necessary to do so, Chern takes the vector field $X$ to have its only singularity at $0\in M$. (You could do the standard proof of taking a generic vector field with singularities at $p_1,\dots,p_k$, remove small balls centered at each of the $p_i$, and then apply Stokes's Theorem to the corresponding remaining part of $V$.) Then, either way, the answer to your first question is to apply the Poincaré-Hopf Theorem.
The equality in your second question is a direct application of the statement in your first. Your argument with the degree theorem is, indeed, a justification of his original assertion.