I'm relatively green in the differential geometry area, so my apologies if what I'm asking is ill-posed and/or not research-level.

I have a situation where I know the shortest path between any two points in the plane. Is there a way to reconstruct a corresponding 2D-manifold such that the shortest path between points on the manifold with respect to the Euclidean metric projects to the given shortest path on the plane? I'm only interested (for the moment) in the nicest cases (i.e. dense, locally compact, analytic, etc.).

I can only think of the answer in the trivial example: if the shortest path is always a straight line, then the corresponding manifold is just the plane. But what is the calculation that shows this must be true?

And what if the shortest path between two points is given by the unique exponential curve $y=C_1+C_2e^x$ through those points? What if it's the unique monic parabola through the points?