2008-18.md

course	course_year	question_number	tags	title	year
Geometry	IB	18	IB 2008 Geometry	4.II.12G	2008

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

$$\begin{aligned} \frac{d}{d t}\left(E \dot{\gamma}{1}+F \dot{\gamma}{2}\right) &=\frac{1}{2}\left(E_{u} \dot{\gamma}{1}^{2}+2 F{u} \dot{\gamma}{1} \dot{\gamma}{2}+G_{u} \dot{\gamma}{2}^{2}\right) \ \frac{d}{d t}\left(F \dot{\gamma}{1}+G \dot{\gamma}{2}\right) &=\frac{1}{2}\left(E{v} \dot{\gamma}{1}^{2}+2 F{v} \dot{\gamma}{1} \dot{\gamma}{2}+G_{v} \dot{\gamma}_{2}^{2}\right) \end{aligned}$$

where $\gamma=\left(\gamma_{1}, \gamma_{2}\right)$ in terms of a smooth parametrization $(u, v)$ for $S$, with first fundamental form $E d u^{2}+2 F d u d v+G d v^{2}$.

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

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

(ii) Suppose moreover that the angle between the curves $u=c, v=d$ is independent of $c$ and $d$. Show that $E_{v}=0=G_{u}$.