For an oriented surface $S$ in $\mathbb{R}^{3}$, define the Gauss map, the second fundamental form and the normal curvature in the direction $w \in T_{p} S$ at a point $p \in S$.

Let $\tilde{k}_{1}, \ldots, \tilde{k}_{m}$ be normal curvatures at $p$ in the directions $v_{1}, \ldots, v_{m}$, such that the angle between $v_{i}$ and $v_{i+1}$ is $\pi / m$ for each $i=1, \ldots, m-1(m \geqslant 2)$. Show that

$$\tilde{k}_{1}+\ldots+\tilde{k}_{m}=m H$$

where $H$ is the mean curvature of $S$ at $p$.

What is a minimal surface? Show that if $S$ is a minimal surface, then its Gauss $\operatorname{map} N$ at each point $p \in S$ satisfies

$$\left\langle d N_{p}\left(w_{1}\right), d N_{p}\left(w_{2}\right)\right\rangle=\mu(p)\left\langle w_{1}, w_{2}\right\rangle, \quad \text { for all } w_{1}, w_{2} \in T_{p} S,$$

where $\mu(p) \in \mathbb{R}$ depends only on $p$. Conversely, if the identity $(*)$ holds at each point in $S$, must $S$ be minimal? Justify your answer.