4.II.12G

Geometry
Part IB, 2008

Let γ:[a,b]S\gamma:[a, b] \rightarrow S be a curve on a smoothly embedded surface SR3S \subset \mathbf{R}^{3}. Define the energy of γ\gamma. Show that if γ\gamma is a stationary point for the energy for proper variations of γ\gamma, then γ\gamma satisfies the geodesic equations

ddt(Eγ˙1+Fγ˙2)=12(Euγ˙12+2Fuγ˙1γ˙2+Guγ˙22)ddt(Fγ˙1+Gγ˙2)=12(Evγ˙12+2Fvγ˙1γ˙2+Gvγ˙22)\begin{aligned} \frac{d}{d t}\left(E \dot{\gamma}_{1}+F \dot{\gamma}_{2}\right) &=\frac{1}{2}\left(E_{u} \dot{\gamma}_{1}^{2}+2 F_{u} \dot{\gamma}_{1} \dot{\gamma}_{2}+G_{u} \dot{\gamma}_{2}^{2}\right) \\ \frac{d}{d t}\left(F \dot{\gamma}_{1}+G \dot{\gamma}_{2}\right) &=\frac{1}{2}\left(E_{v} \dot{\gamma}_{1}^{2}+2 F_{v} \dot{\gamma}_{1} \dot{\gamma}_{2}+G_{v} \dot{\gamma}_{2}^{2}\right) \end{aligned}

where γ=(γ1,γ2)\gamma=\left(\gamma_{1}, \gamma_{2}\right) in terms of a smooth parametrization (u,v)(u, v) for SS, with first fundamental form Edu2+2Fdudv+Gdv2E d u^{2}+2 F d u d v+G d v^{2}.

Now suppose that for every c,dc, d the curves u=c,v=du=c, v=d are geodesics.

(i) Show that (F/G)v=(G)u(F / \sqrt{G})_{v}=(\sqrt{G})_{u} and (F/E)u=(E)v(F / \sqrt{E})_{u}=(\sqrt{E})_{v}.

(ii) Suppose moreover that the angle between the curves u=c,v=du=c, v=d is independent of cc and dd. Show that Ev=0=GuE_{v}=0=G_{u}.