Let $\alpha: I \rightarrow S$ be a parametrized curve on a smooth embedded surface $S \subset \mathbf{R}^{3}$. Define what is meant by a vector field $V$ along $\alpha$ and the concept of such a vector field being parallel. If $V$ and $W$ are both parallel vector fields along $\alpha$, show that the inner product $\langle V(t), W(t)\rangle$ is constant.

Given a local parametrization $\phi: U \rightarrow S$, define the Christoffel symbols $\Gamma_{j k}^{i}$ on $U$. Given a vector $v_{0} \in T_{\alpha(0)} S$, prove that there exists a unique parallel vector field $V(t)$ along $\alpha$ with $V(0)=v_{0}$ (recall that $V(t)$ is called the parallel transport of $v_{0}$ along $\alpha$ ).

Suppose now that the image of $\alpha$ also lies on another smooth embedded surface $S^{\prime} \subset \mathbf{R}^{3}$ and that $T_{\alpha(t)} S=T_{\alpha(t)} S^{\prime}$ for all $t \in I$. Show that parallel transport of a vector $v_{0}$ is the same whether calculated on $S$ or $S^{\prime}$. Suppose $S$ is the unit sphere in $\mathbf{R}^{3}$ with centre at the origin and let $\alpha:[0,2 \pi] \rightarrow S$ be the curve on $S$ given by

$$\alpha(t)=(\sin \phi \cos t, \sin \phi \sin t, \cos \phi)$$

for some fixed angle $\phi$. Suppose $v_{0} \in T_{P} S$ is the unit tangent vector to $\alpha$ at $P=\alpha(0)=$ $\alpha(2 \pi)$ and let $v_{1}$ be its image in $T_{P} S$ under parallel transport along $\alpha$. Show that the angle between $v_{0}$ and $v_{1}$ is $2 \pi \cos \phi$.

[Hint: You may find it useful to consider the circular cone $S^{\prime}$ which touches the sphere $S$ along the curve $\alpha$.]