(a) Let $\alpha:(a, b) \rightarrow \mathbb{R}^{2}$ be a regular curve without self intersection given by $\alpha(v)=(f(v), g(v))$ with $f(v)>0$ for $v \in(a, b)$.

Consider the local parametrisation given by

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

where $\phi(u, v)=(f(v) \cos u, f(v) \sin u, g(v))$.

(i) Show that the image $\phi((0,2 \pi) \times(a, b))$ defines a regular surface $S$ in $\mathbb{R}^{3}$.

(ii) If $\gamma(s)=\phi(u(s), v(s))$ is a geodesic in $S$ parametrised by arc length, then show that $f(v(s))^{2} u^{\prime}(s)$ is constant in $s$. If $\theta(s)$ denotes the angle that the geodesic makes with the parallel $S \cap\{z=g(v(s))\}$, then show that $f(v(s)) \cos \theta(s)$ is constant in $s$.

(b) Now assume that $\alpha(v)=(f(v), g(v))$ extends to a smooth curve $\alpha:[a, b] \rightarrow \mathbb{R}^{2}$ such that $f(a)=0, f(b)=0, f^{\prime}(a) \neq 0, f^{\prime}(b) \neq 0$. Let $\bar{S}$ be the closure of $S$ in $\mathbb{R}^{3}$.

(i) State a necessary and sufficient condition on $\alpha(v)$ for $\bar{S}$ to be a compact regular surface. Justify your answer.

(ii) If $\bar{S}$ is a compact regular surface, and $\gamma:(-\infty, \infty) \rightarrow \bar{S}$ is a geodesic, show that there exists a non-empty open subset $U \subset \bar{S}$ such that $\gamma((-\infty, \infty)) \cap U=\emptyset$.