Assume that $ T $ is a two dimenional compact smooth torus in $ \mathbb{R}^3 $. How to show that there must be some point $ p\in T $ such that $ K(p)>0 $ or $ K(p)<0 $, where $ K(p) $ means the Gauss curvature on $ T $ at the point $ p $.
I want to use Gauss-Bonnet formula to solve this problem. Since the Euler characteristic number for $ T $ is $ 0 $, then
$$
\int_{T}KdS=0.
$$
However there is no contradiction. I do not how to go on. Can you give me some hints or references?
Suppose that $K$ is identically $0$. Then we know that $T$ is a cylinder. $K$ is thus not identically $0$ and we can pick a point $p$ such that $K(p)>0$ or $K(p)<0$. For the torus in particular there must then exist a point $q$ such that $K(q)$ has the opposite sign as $K(p)$, because the integral is $0$.
In conclusion there exists both a point such that $K(p)>0$ and another such that $K(q)<0$.