Comparison theorems for conjugate points in sub-Riemannian geometry

D. Barilari (1) , L. Rizzi (2, 3)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 CNRS - Centre National de la Recherche Scientifique
3 IF - Institut Fourier

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 3D left-invariant sub-Riemannian structures.

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]; Optimisation et contrôle [math.OC]

D. Barilari, L. Rizzi. Comparison theorems for conjugate points in sub-Riemannian geometry. ESAIM: Control, Optimisation and Calculus of Variations, 2016, 22 (2), ⟨10.1051/cocv/2015013⟩. ⟨hal-00931840⟩