The Gauss–Bonnet theorem says that: If $\Sigma \subset M =\mathbb{R}^3$ is a compact 2-dimensional Riemannian manifold without boundary, then $$ \int_{\Sigma} K = 2\pi\chi_{\Sigma}$$

where $K$ is the Gaussian curvature of $\Sigma$ and $\chi_{\Sigma}$ is the Euler characteristic of $\Sigma$.

In Proposition 7 of http://arxiv.org/abs/0909.1665, $M^3$ is compact (thus $M\ne \mathbb{R}^3$), but they use Gauss-Bonnet. Why?

Moreover, in this case, is $K$ the curvature sectional, or is it the product of principal curvatures of the surface?

Can someone suggest a reference? Thank you!