This is Problem 11 of this year's Miklos Schweitzer.
(a) Consider an ellipse in the plane. Prove that there exists a Riemannian metric which is defined on the whole plane, and with respect to which the ellipse is a geodesic. Prove that the Gaussian curvature of any such Riemannian metric takes a positive value.
(b) Consider two nonintersecting, simple closed smooth curves in the plane. Prove that if there is a Riemmanian metric defined on the whole plane and the two curves are geodesics of that metric, then the Gaussian curvature of the metric vanishes somewhere.
I do not know how this can be approached, and my knowledge of Riemmanian geometry is limited. I posted this question because I'm curious what do we need to do in order to find the solution.