I have to prove that a compact surface of revolution is diffeomorphic to a sphere or to a torus. And show that
$\int_{S} K dA=\cases{4\pi&if $S$ is spherical type \\ 0&if $S$ is toric type}$, where $K$ is the Gauss curvature
Let $S_\alpha$ be a surface of revolution by rotating $\alpha(s)=(f(s),0,h(s))$. Then, $S_\alpha$ can be parametrized by $\mathbb x(s,\theta)=(f(s)\cos\theta, f(s)\sin\theta, h(s))$
Could you help me please? I don’t even know how to start.
Thank you in advance