Sub-Riemannian curvature in contact geometry

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 prove a version of the sub-Riemannian Bonnet-Myers theorem that applies to any contact manifold, with special attention to contact Yang-Mills structures