(i) State and prove the Theorema Egregium.

(ii) Define the notions principal curvatures, principal directions and umbilical point.

(iii) Let $S \subset \mathbb{R}^{3}$ be a connected compact regular surface (without boundary), and let $D \subset S$ be a dense subset of $S$ with the following property. For all $p \in D$, there exists an open neighbourhood $\mathcal{U}_{p}$ of $p$ in $S$ such that for all $\theta \in[0,2 \pi), \psi_{p, \theta}\left(\mathcal{U}_{p}\right)=\mathcal{U}_{p}$, where $\psi_{p, \theta}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ denotes rotation by $\theta$ around the line through $p$ perpendicular to $T_{p} S$. Show that $S$ is in fact a sphere.