We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and MikeÅ¡, we show that such metrics correspond precisely to suitably positiveâ€¦ (More)

Two metrics g and á¸¡ are geodesically equivalent if they share the same (unparameterized) geodesics. We introduce two constructions that allow one to reduce many natural problems related toâ€¦ (More)

Two pseudo-Riemannian metrics g and á¸¡ are geodesically equivalent if they share the same (unparameterized) geodesics. We give a complete local description of such metrics which solves the naturalâ€¦ (More)

We suggest a construction that, given a trajectorial diffeomorphism between two Hamiltonian systems, produces integrals of them. As the main example we treat geodesic equivalence of metrics. We showâ€¦ (More)

We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric g of nonconstant curvature is geodesically rigid: if any other complete metric á¸¡ has the same (unparametrized) geodesicsâ€¦ (More)

We prove the classical Yano-Obata conjecture by showing that the connected component of the group of holomorph-projective transformations of a closed, connected Riemannian KÃ¤hler manifold consists ofâ€¦ (More)

We give a complete list of normal forms for the 2-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutuallyâ€¦ (More)

We extend the Gallot-Tanno Theorem to closed pseudo-Riemannian manifolds. It is done by showing that if the cone over a manifold admits a parallel symmetric (0, 2)âˆ’tensor then it is Riemannian.â€¦ (More)

As we recall in Section 2.1, the set of metrics geodesically equivalent to a given one (say, g) is in one-to-one correspondence with the nondegenerate solutions of the equation (9). Since theâ€¦ (More)

We present a new Liouville-integrable natural Hamiltonian system on the (cotangent bundle of the) sphere S2. The second integral is cubic in the momenta. MSC2000: 37J35, 58F07, 58F17, 70H06, 70E40