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?

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.

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.

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 Riemannian manifold $M$? If this is true, is there any strategy to construct at least one manifold for that simple curve?

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.

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?

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.