The typical Riemannian comparison theorem is a result in which a local curvaturetype 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 subRiemannian 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 subRiemannian 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 subRiemannian setting. In particular, we discuss the subRiemannian BonnetMyers theorem and the generalized Measure Contraction Property for subRiemannian manifolds with bounded canonical Ricci curvatures. In this setting, the models with constant curvature are represented by LinearQuadratic optimal control problems with constant potential. (This is a joint project with D. Barilari.)
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Comparison theorems in subRiemannian geometry
Research Group:
Luca Rizzi
Institution:
SISSA
Location:
A133
Schedule:
Friday, October 11, 2013  11:00 to 12:30
Abstract:
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