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= $

=$\{4\pi,$ 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 dont' even know how to start.

Thank you in advance

  • $\begingroup$ Have you done the Gauss-Bonnet Theorem yet? $\endgroup$ – Alan Apr 21 '14 at 16:54
  • $\begingroup$ Hi, can you include the definition for "spherical type" and "toric type", and also the calculation of $K$? $\endgroup$ – user99914 Apr 22 '14 at 3:00
  • $\begingroup$ @Alan I cant use the Gauss-Bonnet theorem $\endgroup$ – Trian Apr 22 '14 at 11:45
  • $\begingroup$ @John, with spherical type I mean that the surface is diffeomorphic to a sphere and with toric type that it is diffeomorphic to a torus. On the other hand $K$ is the Gauss curvature $\endgroup$ – Trian Apr 22 '14 at 11:47
  • $\begingroup$ @Trian: Ok, so you can express the surfaces as $x(s, \theta)$, so what is $K(s, \theta)$ in terms of $f$ and $h$? $\endgroup$ – user99914 Apr 22 '14 at 11:59

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