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Comparison theorems in sub-Riemannian geometry

Luca Rizzi
Friday, October 11, 2013 - 11:00 to 12:30

The typical Riemannian comparison theorem is a result in which a local curvature-type bound (e.g. Ric > k) implies a global comparison between some property on the actual manifold (e.g. its diameter) and the same property on a constant curvature model. The generalization of these results to the sub-Riemannian setting is not straightforward, the main difficulty being the lack of a proper theory of Jacobi fields, an analytic definition of curvature and, a fortiori, constant curvature models. In this talk, we propose a theory of Jacobi fields valid for any sub-Riemannian manifold, in which the Riemannian sectional curvature is generalized by the curvature introduced by Agrachev and his students. This allows to extend a wide range of comparison theorems to the sub-Riemannian setting. In particular, we discuss the sub-Riemannian Bonnet-Myers theorem and the generalized Measure Contraction Property for sub-Riemannian manifolds with bounded canonical Ricci curvatures. In this setting, the models with constant curvature are represented by Linear-Quadratic optimal control problems with constant potential. (This is a joint project with D. Barilari.)

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