Sub-Riemannian curvature in contact geometry

TL;DR

This paper introduces canonical curvature notions in contact sub-Riemannian geometry, linking them to the asymptotic expansion of the sub-Riemannian distance and deriving a contact version of the Bonnet-Myers theorem.

Contribution

It defines canonical curvatures via the sub-Riemannian Jacobi equation and relates them to distance derivatives, providing explicit formulas in contact geometry.

Findings

Canonical curvatures are encoded in the asymptotic expansion of horizontal derivatives.

Explicit expressions for curvatures in terms of contact geometry tensors.

A sub-Riemannian Bonnet-Myers theorem for contact manifolds.

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.