Assume there are two Riemannian metrics on a manifold ( open or closed) with the same set of all geodesics. Are they proportional by a constant? If not in general, what are the affirmative results in this direction?



Can one recover a metric from geodesics?
The short answer is **no**, the geodesics do not determine the metric. For example, in the Cayley-Klein model of hyperbolic geometry the geodesics are straight lines. It is however rather rare for
two Riemannian metrics to have the same geodesics. In two dimensions these metrics were studied by Liouville (see Livre VI of Darboux's *Théorie générale des surfaces*). Their geodesic flow is completely integrable and, in fact, admits an additional integral of motion quadratic in the momenta.


A basic results is that if locally the geodesics are straight lines (or can be mapped to straight lines), the metric has constant curvature (Beltrami's theorem). Other rigidity results of this kind exist. 


For more on this topic you can consult the works of Topalov and Matveev.


