Geodesic Equivalence in sub-Riemannian Geometry

Sub-Riemannian geometry is an intensively developing field of Mathematics lying at the intersection of Differential Geometry, Control Theory with application to Robotics, Hamil- tonian dynamics and PDEs. Our research is devoted to the geodesic equivalence of sub-Riemannian metrics, when one wants to study the metrics not up to isometries but up to the group of transformations preserving all their geodesics considered as unparametrized curves. In Riemannian geometry this equivalence problem is well understood thanks to the classical works of Beltrami, Dini, Levi-Civita. The existence of nontrivial pairs of geodesically equivalent metrics is related to the Liouville integrability of the corresponding geodesic flows with integrals of special type and the separability of the corresponding Hamilton-Jacobi equations. For proper sub-Riemannian metrics only few classification results are known up to now, mainly concerning sub-Riemannian metrics on generic corank 1 distributions. However, there is strong evidence that a general classification theorem on geodesic equivalence of sub- Riemannian metrics defined on a very general class of distributions exists and it includes the classical Levi-Civita theorem as a particular case. The presented research is a step forward to discovering such a theorem