A Universe of Sorts

Let $S \subseteq \mathbb R^3$ be an extrinsic manifold
Let $U \subseteq \mathbb R^2$ be the domain for a parametrization $x: U \to S$ .
The gauss map $N$ is the map which sets each point $p \in S$ to the normal plane at $p$ .
The weingarten map is the map $dN$ which can be identified with

§ Weingarten Map