Let $S \subset \mathbb{R}^{N}$ be a manifold and let $\alpha:[a, b] \rightarrow S \subset \mathbb{R}^{N}$ be a smooth regular curve on $S$. Define the total length $L(\alpha)$ and the arc length parameter $s$. Show that $\alpha$ can be reparametrized by arc length.

Let $S \subset \mathbb{R}^{3}$ denote a regular surface, let $p, q \in S$ be distinct points and let $\alpha:[a, b] \rightarrow S$ be a smooth regular curve such that $\alpha(a)=p, \alpha(b)=q$. We say that $\alpha$ is length minimising if for all smooth regular curves $\tilde{\alpha}:[a, b] \rightarrow S$ with $\tilde{\alpha}(a)=p, \tilde{\alpha}(b)=q$, we have $L(\tilde{\alpha}) \geqslant L(\alpha)$. By deriving a formula for the derivative of the energy functional corresponding to a variation of $\alpha$, show that a length minimising curve is necessarily a geodesic. [You may use the following fact: given a smooth vector field $V(t)$ along $\alpha$ with $V(a)=V(b)=0$, there exists a variation $\alpha(s, t)$ of $\alpha$ such that $\left.\left.\partial_{s} \alpha(s, t)\right|_{s=0}=V(t) .\right]$

Let $\mathbb{S}^{2} \subset \mathbb{R}^{3}$ denote the unit sphere and let $S$ denote the surface $\mathbb{S}^{2} \backslash(0,0,1)$. For which pairs of points $p, q \in S$ does there exist a length minimising smooth regular curve $\alpha:[a, b] \rightarrow S$ with $\alpha(a)=p$ and $\alpha(b)=q$ ? Justify your answer.