Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 24H / a / Solution

The global Gauss-Bonnet theorem for the induced Riemannian metric isHere $K$ is the Gaussian curvature, $dA$ the induced surface-area element, and $\chi (X)$ the Euler characteristic. For a connected orientable compact surface of genus $g$, $\chi (X)=2-2g$. For several components, sum their contributions. There is no boundary-curvature term because $X$ has no boundary.