I have some trouble understanding the following question:

Suppose we have 1st fundamental form $E \, dx^2+2F \, dx \, dy+G \, dy^2$ and we are given that for any $u,v$, the curve given by $x=u, y=v$ are geodesics. Show that ${\partial \over \partial y}\left({F\over \sqrt{G}}\right)={\partial \sqrt{G}\over \partial x}$.

I don't understand what "$x=u, y=v$ are geodesics" mean. So the path is a constant point?? That doesn't make sense!

Can anybody understand what it is saying?

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Remember that $(u,v)$ is a local system of coordinates of a neighborhood of your surface. If you have a first fundamental form given, implicitly the system of local coordinates is given wich is a diffeomorphism. $x=u$ and $y=v$ meaning that you are looking the images of coordinate axis, this images must be geodesics for each one separated. Have not sense see the image of origin.
In this case a local coordinate system is a difeomorfism function $\phi: U\subset \mathbb{R^2} \to S$, the curve given by $x=u$ is the image parametrized $\phi(t,0)$ where $t$ varies in $\mathbb{R}$. It is called sometimes a coordinate curve ;). – Gastón Burrull May 13 '12 at 22:43