For $S \subset \mathbf{R}^{3}$ a smooth embedded surface, define what is meant by a geodesic curve on $S$. Show that any geodesic curve $\gamma(t)$ has constant speed $|\dot{\gamma}(t)|$.

For any point $P \in S$, show that there is a parametrization $\phi: U \rightarrow V$ of some open neighbourhood $V$ of $P$ in $S$, with $U \subset \mathbf{R}^{2}$ having coordinates $(u, v)$, for which the first fundamental form is

$$d u^{2}+G(u, v) d v^{2}$$

for some strictly positive smooth function $G$ on $U$. State a formula for the Gaussian curvature $K$ of $S$ in $V$ in terms of $G$. If $K \equiv 0$ on $V$, show that $G$ is a function of $v$ only, and that we may reparametrize so that the metric is locally of the form $d u^{2}+d w^{2}$, for appropriate local coordinates $(u, w)$.

[You may assume that for any $P \in S$ and nonzero $\xi \in T_{P} S$, there exists (for some $\epsilon>0)$ a unique geodesic $\gamma:(-\epsilon, \epsilon) \rightarrow S$ with $\gamma(0)=P$ and $\dot{\gamma}(0)=\xi$, and that such geodesics depend smoothly on the initial conditions $P$ and $\xi .]$