Comparison theorems for conjugate points in sub-Riemannian geometry

Barilari, D.¹  ; Rizzi, L.² 

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

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

Résumé

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.

Détail

Reçu le : 2014-09-06
DOI : 10.1051/cocv/2015013

Classification : 53C17, 53C21, 53C22, 49N10
Keywords: Sub-Riemannian geometry, curvature, comparison theorems, conjugate points

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