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?

The short answer is no, the geodesics do not determine the metric. For example, in the CayleyKlein 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. 


A simple counterexample is the space $\mathbb R^n$, with a metric $g_{ab}$ independent on the point. As examples, the Euclidean space $\mathbb R^n$, but also the Minkowski spacetime (which is semiRiemannian, but has the same geodesics as $\mathbb R^4$). The geodesics are the lines in $\mathbb R^n$, no matter how we choose the constant metric $g_{ab}$. Hence, different metrics can give the same set of geodesics. 


I attended today a talk at a conference, where Graham Hall presented some very interesting results about recovering the metric from geodesics, and mentioned some previous results too. Here are some of his papers about this: http://www.sciencedirect.com/science/article/pii/S0393044011002269 http://link.springer.com/article/10.2478/s1153301200876 


I hope that my ``answer'' will not be understood solely as a propaganda of my survey http://arxiv.org/abs/1101.2069 where I discussed (1) how, given geodesics, to reconstruct a connection (in both cases: when geodesics are parameterised and not parameterised) (2) how, given a class od projectively related connections, to reconstruct a metric, and what are advantages of additional curvature assumptions on the metric (3) what is the freedom in reconstructing the metric by geodesics  in particular I proved the statement wellknown to experts and mentionen by Robert Bryant in his comment that for generic metric the geodesics, even unparameterized, determine the metric. 


The example of Cristi Stoica is isometric to Euclidean space $\mathbb R^n$ if the $g_{ab}$ defines a Riemannian metric. 

