Sub-Riemannian Curvature in Contact Geometry

By Andrei A. Agrachev, Davide Barilari, L. Rizzi. Journal of Geometric Analysis, 2015

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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.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/159986656aaf331045695f33c2923f242eb0d9fd