Let $\eta$ be a smooth curve in the $x z$-plane $\eta(s)=(f(s), 0, g(s))$, with $f(s)>0$ for every $s \in \mathbb{R}$ and $f^{\prime}(s)^{2}+g^{\prime}(s)^{2}=1$. Let $S$ be the surface obtained by rotating $\eta$ around the $z$-axis. Find the first fundamental form of $S$.

State the equations for a curve $\gamma:(a, b) \rightarrow S$ parametrised by arc-length to be a geodesic.

A parallel on $S$ is the closed circle swept out by rotating a single point of $\eta$. Prove that for every $n \in \mathbb{Z}_{>0}$ there is some $\eta$ for which exactly $n$ parallels are geodesics. Sketch possible such surfaces $S$ in the cases $n=1$ and $n=2$.

If every parallel is a geodesic, what can you deduce about $S$ ? Briefly justify your answer.