Imagine that we are given a parametrization of a geodesic $\gamma (t)$. I am curious as to whether it is possible to find a given surface or class of surfaces that admit $\gamma (t)$ as its geodesic.
As some motivation, I am curious about a variant on Abel's mechanical problem and I want to see if any curve satisfying the isochrone or tautochrone property under a given potential field will also satisfy the brachistochrone property. Knowing that the cycloid is the brachistochrone of $R^2$ under a potential field pulling in one direction with constant magnitude, it is in some way a geodesic. I was wondering that perhaps modelling this situation as a geodesic on some surface imbedded in a higher dimension would perhaps help me find a way to generalize this problem as a whole.
Also, if this is clearly ill-posed but is related to some branch of research, please let me know! Thanks