I’m given an open set $ U \subset \mathbb{R}^2$ with polar coordinates $(r, \theta)$ and endowed with the first fundamental form \begin{equation} I= dr^2 + c^2 d \theta^2. \end{equation} Here $c$ is a smooth, nonnegative function with domain $[0, \infty)$ depending only on the radial coordinate.
I would like to show that the quantity $ c^2 (dt/t) \theta$ is contant along the geodesics. I computed the Christoffel symbols and the geodesics equations. Hence I tried to take the covariant derivative of the vector field $(0, \sqrt{(dt/t) \theta})$, to show that it is zero. But no success. Any hint?
Thanks!