Mathematics

Two sub-Riemannian metrics are called projectively equivalent if they have the same geodesics up to a reparametrization and affinely equivalent if they have the same geodesics up to affine reparametrization. In the Riemannian case both equivalence problems are classical: local classifications of projectively and affinely equivalent Riemannian metrics were established by Levi-Civita in 1898 and Eisenhart in 1923, respectively.In particular, a Riemannian metric admitting a nontrivial (i.e. non-constant proportional) affinely equivalent metric must be a product of two Riemannian metrics i.e. certain separation of variable occur, while for the analogous property in the projectively equivalent case a more involved (``twisted") product structure is necessary. The latter is also related to the existence of sufficiently many commuting nontrivial integrals quadratic with respect to velocities for the corresponding geodesic flow.  We will describes some progress toward generalization of these classical results to sub-Riemannian metrics. The talk is based on the collaboration with Frederic Jean and Sofya Maslovskaya (ENSTA, Paris).