Let γ : [ a , b ] → S \gamma:[a, b] \rightarrow S γ : [ a , b ] → S be a curve on a smoothly embedded surface S ⊂ R 3 S \subset \mathbf{R}^{3} S ⊂ R 3 . Define the energy of γ \gamma γ . Show that if γ \gamma γ is a stationary point for the energy for proper variations of γ \gamma γ , then γ \gamma γ satisfies the geodesic equations
d d t ( E γ ˙ 1 + F γ ˙ 2 ) = 1 2 ( E u γ ˙ 1 2 + 2 F u γ ˙ 1 γ ˙ 2 + G u γ ˙ 2 2 ) d d t ( F γ ˙ 1 + G γ ˙ 2 ) = 1 2 ( E v γ ˙ 1 2 + 2 F v γ ˙ 1 γ ˙ 2 + G v γ ˙ 2 2 ) \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} d t d ( E γ ˙ 1 + F γ ˙ 2 ) d t d ( F γ ˙ 1 + G γ ˙ 2 ) = 2 1 ( E u γ ˙ 1 2 + 2 F u γ ˙ 1 γ ˙ 2 + G u γ ˙ 2 2 ) = 2 1 ( E v γ ˙ 1 2 + 2 F v γ ˙ 1 γ ˙ 2 + G v γ ˙ 2 2 )
where γ = ( γ 1 , γ 2 ) \gamma=\left(\gamma_{1}, \gamma_{2}\right) γ = ( γ 1 , γ 2 ) in terms of a smooth parametrization ( u , v ) (u, v) ( u , v ) for S S S , with first fundamental form E d u 2 + 2 F d u d v + G d v 2 E d u^{2}+2 F d u d v+G d v^{2} E d u 2 + 2 F d u d v + G d v 2 .
Now suppose that for every c , d c, d c , d the curves u = c , v = d u=c, v=d u = c , v = d are geodesics.
(i) Show that ( F / G ) v = ( G ) u (F / \sqrt{G})_{v}=(\sqrt{G})_{u} ( F / G ) v = ( G ) u and ( F / E ) u = ( E ) v (F / \sqrt{E})_{u}=(\sqrt{E})_{v} ( F / E ) u = ( E ) v .
(ii) Suppose moreover that the angle between the curves u = c , v = d u=c, v=d u = c , v = d is independent of c c c and d d d . Show that E v = 0 = G u E_{v}=0=G_{u} E v = 0 = G u .