Connections and Curvature in sub-Riemannian geometry

TL;DR

This paper introduces a canonical connection for sub-Riemannian manifolds, generalizes classical notions like normality and curvature, and extends key Riemannian theorems such as Bonnet-Myers to the sub-Riemannian setting.

Contribution

It defines a canonical global connection in sub-Riemannian geometry, generalizes the concept of normality, and explores curvature properties and theorems analogous to Riemannian geometry.

Findings

Established Bianchi identities for sub-Riemannian curvature

Constructed local frames simplifying computations under normality

Extended Bonnet-Myers theorem to sub-Riemannian manifolds

Abstract

For a subRiemannian manifold and a given Riemannian extension of the metric, we define a canonical global connection. This connection coincides with both the Levi-Civita connection on Riemannian manifolds and the Tanaka-Webster connection on strictly pseudoconvex CR manifolds. We define a notion of normality generalizing Tanaka's notion for CR manifolds to the subRiemannian case. Under the assumption of normality, we construct local frames that simplify computations in a manner analogous to Riemannian normal coordinates. We then use these frames to establish Bianchi Identities and symmetries for the associated curvatures. Finally we explore subRiemannian generalizations of the Bonnet-Myers theorem, providing some new results and some new proofs and interpretations of existing results.