Could someone please explain what this question is asking? 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?

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.
