Comparison theorems for conjugate points in sub-Riemannian geometry
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
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.
Mathematics Subject Classification: 53C17 / 53C21 / 53C22 / 49N10
Key words: Sub-Riemannian geometry / curvature / comparison theorems / conjugate points
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