(a) Let $\gamma:(a, b) \rightarrow \mathbb{R}^{2}$ be a regular curve without self-intersection given by $\gamma(v)=(f(v), g(v))$ with $f(v)>0$ for $v \in(a, b)$ and let $S$ be the surface of revolution defined globally by the parametrisation

$$\phi:(0,2 \pi) \times(a, b) \rightarrow \mathbb{R}^{3}$$

where $\phi(u, v)=(f(v) \cos u, f(v) \sin u, g(v))$, i.e. $S=\phi((0,2 \pi) \times(a, b))$. Compute its mean curvature $H$ and its Gaussian curvature $K$.

(b) Define what it means for a regular surface $S \subset \mathbb{R}^{3}$ to be minimal. Give an example of a minimal surface which is not locally isometric to a cone, cylinder or plane. Justify your answer.

(c) Let $S$ be a regular surface such that $K \equiv 1$. Is it necessarily the case that given any $p \in S$, there exists an open neighbourhood $\mathcal{U} \subset S$ of $p$ such that $\mathcal{U}$ lies on some sphere in $\mathbb{R}^{3}$ ? Justify your answer.