Hyperbolic manifolds are geodesically rigid
Since the time of Beltrami, it is known that all geodesics of two non-proportional metrics on the sphere can coincide (as unparameterized curves). We show that it is not always the case: some manifolds are geodesically rigid, in the sense that any two Riemannian metrics share the same geodesics (considered as unparameterized curves) if and only if they are proportional.
I will try to explain why:
1. A closed surface is geodesically rigid if and only if its Euler characteristic is negative (this is a joint result with P. Topalov);
2. A closed 3-manifold is geodesically rigid if and only if it is homeomorphic neither to a Lens space nor to a Seifert manifold with zero Euler number;
3. A sufficient condition for geodesic rigidity of a closed $n$-dimensional manifold is the existence of a metrics of negative sectional curvature.