Maybe this question is so clear or maybe it is not exact. It is because of my very little knowledge of differential geometry. I am reading some material in this field and I got a question which seems has some meaning, but I am not sure.

Is it true that any simple curve $\gamma$ in the plane is geodesy of a finite volume Riemannian manifold $M$ in which its curvature is not zero? If this is true, is there any strategy to construct at least one manifold for that simple curve?

$\text{Added later by @Anton Petrunin suggestions:}$

$\gamma$ is orthogonal projection to the plane and $M$ is a closed surface.

I think the answer is positive, but I do not have any idea (except imagination) to prove it.

Actually, I did many searches and I did not find any answer in papers or books.