Paper 3, Section II, I

Differential Geometry
Part II, 2017

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

Let SR3S \subset \mathbb{R}^{3} denote a regular surface, let p,qSp, q \in S be distinct points and let α:[a,b]S\alpha:[a, b] \rightarrow S be a smooth regular curve such that α(a)=p,α(b)=q\alpha(a)=p, \alpha(b)=q. We say that α\alpha is length minimising if for all smooth regular curves α~:[a,b]S\tilde{\alpha}:[a, b] \rightarrow S with α~(a)=p,α~(b)=q\tilde{\alpha}(a)=p, \tilde{\alpha}(b)=q, we have L(α~)L(α)L(\tilde{\alpha}) \geqslant L(\alpha). By deriving a formula for the derivative of the energy functional corresponding to a variation of α\alpha, show that a length minimising curve is necessarily a geodesic. [You may use the following fact: given a smooth vector field V(t)V(t) along α\alpha with V(a)=V(b)=0V(a)=V(b)=0, there exists a variation α(s,t)\alpha(s, t) of α\alpha such that sα(s,t)s=0=V(t).]\left.\left.\partial_{s} \alpha(s, t)\right|_{s=0}=V(t) .\right]

Let S2R3\mathbb{S}^{2} \subset \mathbb{R}^{3} denote the unit sphere and let SS denote the surface S2\(0,0,1)\mathbb{S}^{2} \backslash(0,0,1). For which pairs of points p,qSp, q \in S does there exist a length minimising smooth regular curve α:[a,b]S\alpha:[a, b] \rightarrow S with α(a)=p\alpha(a)=p and α(b)=q\alpha(b)=q ? Justify your answer.