Mathematics

Our talk is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions and Engel distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for geodesic equivalence of (sub-)Riemannian metrics, mentioned above, in terms of divisibility of some polynomials (on the fiber of the cotangent bundle of the ambient manifold) associated with these metrics. In this way first we obtain a new elementary proof of classical Levi-Civita's Theorem about the classification of all Riemannian geodesically equivalent metrics in a neighborhood of so-called regular (stable) point w.r.t. these metrics (in the case of a surface regularity means that the metrics are not proportional at the point). Secondly we prove that sub-Riemannian metrics on contact and Engel distributions are geodesically equivalent iff they are proportional. Then we describe all geodesically equivalent sub-Riemannian metrics on quasi-contact distributions. Finally we make the classification of all pairs of geodesically equivalent Riemannian metrics on a surface, which proportional in an isolated point. This is the simplest case, which was not covered by Levi-Civita's Theorem.