Let $S \subset \mathbb{R}^{3}$ be a connected oriented surface.

(a) Define the Gauss map $N: S \rightarrow S^{2}$ of $S$. Given $p \in S$, show that the derivative of $N$,

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

is self-adjoint.

(b) Show that if $N$ is a diffeomorphism, then the Gaussian curvature is positive everywhere. Is the converse true?