Calculus of Variations and Geometric Measure Theory

D. Barilari - L. Rizzi

Comparison theorems for conjugate points in sub-Riemannian geometry

created by rizzi1 on 14 Jan 2014
modified on 15 May 2017


Published Paper

Inserted: 14 jan 2014
Last Updated: 15 may 2017

Journal: Control, Optimisation and Calculus of Variations
Volume: 22
Number: 2
Pages: 439 - 472
Year: 2014
Doi: 10.1051/cocv/2015013

ArXiv: 1401.3193v5 PDF


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.

Keywords: sub-Riemannian geometry, Curvature, comparison theorems, conjugate points