Colóquio de Matemática
Geodesic Equivalence of sub-Riemannian metrics: Rigidity and Generalized Levi-Civita Theorem
Prof. Igor Zelenko (Texas A&M University)
The talk is devoted to some aspects of sub-Riuemannian geometry. A sub-Riemannian metric on a manifold is given by a completely nonholonomic distribution and an Euclidean structure on each fiber of the distribution. Two sub-Riemannian metrics are called geodesically equivalent if they have the same geodesics up to a reparametrization. We say that a sub-Riemannian structure is geodesically rigid if sub-Riemannian structures obtained by a multiplication of the original metric by a nonzero constant are the only sub-Riemannian structures which are geodesically equivalent to the original one. In other words, rigid sub-Riemannian metrics are uniquely, up to trivial transformations, determined by their geodesics. In the Riemannian case all nontrivial pairs of locally geodesically equivalent metrics satisfying certain regularity assumption were described by Levi-Civita in 1898. In particular, he showed that geodesic nonrigidity of a Riemannian metric implies the existence of nontrivial integrals quadratic with resprect to velocities. We will describes some progress toward generalizatrion of these results to sub-Riemannian metrics. The talk is based on the recent collaboration with Frederic Jean and Sofya Maslovskaya (ENSTA, Paris) and uses results of my previous collaborations with Boris Kruglikov (Tromso, Norway).
Local: Auditório do Departamento de Matemática(MTM 007)
Dia – Hora: 22/06/2016 – 14:00h