Let S⊂RN be a manifold and let α:[a,b]→S⊂RN be a smooth regular curve on S. Define the total length L(α) and the arc length parameter s. Show that α can be reparametrized by arc length.
Let S⊂R3 denote a regular surface, let p,q∈S be distinct points and let α:[a,b]→S be a smooth regular curve such that α(a)=p,α(b)=q. We say that α is length minimising if for all smooth regular curves α~:[a,b]→S with α~(a)=p,α~(b)=q, we have L(α~)⩾L(α). By deriving a formula for the derivative of the energy functional corresponding to a variation of α, show that a length minimising curve is necessarily a geodesic. [You may use the following fact: given a smooth vector field V(t) along α with V(a)=V(b)=0, there exists a variation α(s,t) of α such that ∂sα(s,t)∣s=0=V(t).]
Let S2⊂R3 denote the unit sphere and let S denote the surface S2\(0,0,1). For which pairs of points p,q∈S does there exist a length minimising smooth regular curve α:[a,b]→S with α(a)=p and α(b)=q ? Justify your answer.