Curvature of a non-compact complete surface Assume $\Sigma$ is a non compact, complete surface. Assume the integral $$\int_{\Sigma}K$$ is convergent, where K is the Gauss curvature of $\Sigma$. Is it always true that $$\frac{1}{2\pi}\int_{\Sigma}K$$ is an integer?

Sure, I am going to have to go with the contrapositive on this one: $\int{\Sigma}$div$\gamma$$\nu$ = $\int{\partial\Sigma}< \nu, N> \sigma$ #. In other words Stokes theorem.
