Define the Gauss map $N$ for an oriented surface $S \subset \mathbb{R}^{3}$. Show that at each $p \in S$ the derivative of the Gauss map

$$d N_{p}: T_{p} S \rightarrow T_{N(p)} S^{2}=T_{p} S$$

is self-adjoint. Define the principal curvatures $k_{1}, k_{2}$ of $S$.

Now suppose that $S$ is compact (and without boundary). By considering the square of the distance to the origin, or otherwise, prove that $S$ has a point $p$ with $k_{1}(p) k_{2}(p)>0$.

[You may assume that the intersection of $S$ with a plane through the normal direction at $p \in S$ contains a regular curve through $p$.]