Differential Geometry - Gauss Curvature

Question

Let $M$ be a compact (without boundary) orientable surface in $\mathbb R^3$. Suppose $M$ is not diffeomorphic to a sphere. Show that there must exists a point $p$ of $M$ at which the Gauss curvature $K$($p$) < 0.

Any help?

Answer 1

Hint: If the genus of your surface is greater than $1$, apply Gauss-Bonnet. If your surface is a torus, note that any compact surface in $\mathbb{R}^3$ must have a point of positive curvature and again apply Gauss-Bonnet.