Geometry Seminar

Date: March 22, 2019

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Igor Zelenko, TAMU

Title: Projective and affine equivalence of sub-Riemannian metrics: generic rigidity and separation of variables conjecture.

Abstract: Two sub-Riemannian metrics are called projectively equivalent if they have the same geodesics up to a reparameterization and affinely equivalent if they have the same geodesics up to affine reparameterization. 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 describe the recent progress toward the generalization of these classical results to sub-Riemannian metrics. In particular, we will discuss genericity of metrics that do not admit non-constantly proportional affinely/projectively equivalent metrics and the separation of variables on the level of linearization of geodesic flows (i.e. on the level of Jacobi curves) for metrics that admit non-constantly proportional affinely equivalent metrics. The talk is based on the collaboration with Frederic Jean (ENSTA, Paris) and Sofya Maslovskaya (INRIA, Sophya Antipolis).