Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics

We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture.Comment: 33 pages, 7 pictures. This paper replaces arXiv:0911.3521v1 and arXiv:1002.013

There exist no locally symmetric Finsler spaces of positive or negative flag curvature

We show that the results of Foulon (1997 an 2002) and Kim (2007) (independently, Deng and Hou (2007)) about the nonexistence of locally symmetric Finsler metrics of positive or negative flag curvature are in fact local

Closed manifolds admitting metrics with the same geodesics

The goal of this survey is to give a list of resent results about topology of manifolds admitting different metrics with the same geodesics. We emphasize the role of the theory of integrable systems in obtaining these results.Comment: Submitted to Conference Proceedings SPT2004: Symmetry and perturbation theory (Cala Gonone

Strictly non-proportional geodesically equivalent metrics have htop(g)=0h_\text{top}(g)=0

Suppose the Riemannian metrics $g$ and $\bar g$ on a closed connected manifold $M^n$ are geodesically equivalent and strictly non-proportional at least at one point. Then the topological entropy of the geodesic flow of $g$ vanishes.Comment: This is a slightly extended version of the paper submitted to ETDS. 16 pages, one .eps figur