Sub-Riemannian curvature in contact geometry

Andrei Agrachev 1 Davide Barilari 2 Luca Rizzi 3, 4
2 Géométrie et dynamique
IMJ - Institut de Mathématiques de Jussieu
4 GECO - Geometric Control Design
Inria Saclay - Ile de France, X - École polytechnique, CNRS - Centre National de la Recherche Scientifique : UMR7641
Abstract : We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Riemannian distance. We explicitly compute their expressions in terms of the standard tensors of contact geometry. As an application of these results, we obtain a sub-Riemannian version of the Bonnet-Myers theorem that applies to any contact manifold.
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Andrei Agrachev, Davide Barilari, Luca Rizzi. Sub-Riemannian curvature in contact geometry. Journal of Geometric Analysis, 2016, ⟨10.1007/s12220-016-9684-0⟩. ⟨hal-01160901v3⟩

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