Comparison theorems for conjugate points in sub-Riemannian geometry

¹ Université Paris Diderot – Paris 7, Institut de Mathematique de Jussieu, UMR CNRS 7586 – UFR de Mathématiques  
davide.barilari@imj-prg.fr
² CNRS, CMAP École Polytechnique and Équipe INRIA GECO Saclay Île-de-France, Paris, France

Received: 6 September 2014

Abstract

We prove sectional and Ricci-type comparison theorems for the existence of conjugate points along sub-Riemannian geodesics. In order to do that, we regard sub-Riemannian structures as a special kind of variational problems. In this setting, we identify a class of models, namely linear quadratic optimal control systems, that play the role of the constant curvature spaces. As an application, we prove a version of sub-Riemannian Bonnet−Myers theorem and we obtain some new results on conjugate points for three dimensional left-invariant sub-Riemannian structures.