Comparison theorems for conjugate points in sub-Riemannian geometry

Barilari, D.; Rizzi, L.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 2, pp. 439-472.

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.

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