Projectively equivalent Riemannian metrics and compatible Poisson
structure
Two Riemannian metrics given on the same manifold M are called projectively equivalent if they have the same geodesics (considered as unparametrised curves on M). It is well known that the geodesic flows of such metrics admit "many" additional integrals and are often completely integrable. We discuss this relationship between the integrability and projective equivalence (Topalov-Tabachnikov construction and applications) and show that geodesic flows of projectively equivalent metrics are in fact bi-hamiltonian, i.e. admit an additional invariant Poisson structure compatible with the standard one.