Invariant characterization of Liouville metrics and polynomial integrals
arXiv:0709.0423 · doi:10.1016/j.geomphys.2008.03.005
Abstract
A criterion in terms of differential invariants for a metric on a surface to be Liouville is established. Moreover, in this paper we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals does the geodesic flow of a given metric possess? The method is also applied to recognition of other polynomial integrals of geodesic flows.