(i) Suppose that $C$ is a curve in the Euclidean $(\xi, \eta)$-plane and that $C$ is parameterized by its arc length $\sigma$. Suppose that $S$ in Euclidean $\mathbb{R}^{3}$ is the surface of revolution obtained by rotating $C$ about the $\xi$-axis. Take $\sigma, \theta$ as coordinates on $S$, where $\theta$ is the angle of rotation.

Show that the Riemannian metric on $S$ induced from the Euclidean metric on $\mathbb{R}^{3}$ is

$$d s^{2}=d \sigma^{2}+\eta(\sigma)^{2} d \theta^{2}$$

(ii) For the surface $S$ described in Part (i), let $e_{\sigma}=\partial / \partial \sigma$ and $e_{\theta}=\partial / \partial \theta$. Show that, along any geodesic $\gamma$ on $S$, the quantity $g\left(\dot{\gamma}, e_{\theta}\right)$ is constant. Here $g$ is the metric tensor on $S$.

[You may wish to compute $\left[X, e_{\theta}\right]=X e_{\theta}-e_{\theta} X$ for any vector field $X=A e_{\sigma}+B e_{\theta}$, where $A, B$ are functions of $\sigma, \theta$. Then use symmetry to compute $D_{\dot{\gamma}}\left(g\left(\dot{\gamma}, e_{\theta}\right)\right)$, which is the rate of change of $g\left(\dot{\gamma}, e_{\theta}\right)$ along $\gamma$.]