Curvature of sub-Riemannian spaces

TL;DR

This paper introduces a way to define and classify curvature in sub-Riemannian spaces using metric profiles and rectifiability, providing a new framework for understanding their geometric structure.

Contribution

It develops a classification of curvature classes in sub-Riemannian spaces by analyzing homogeneous metric spaces, especially for contact 3-manifolds.

Findings

Classification of curvature classes for contact 3-manifolds

Identification of a 3-dimensional family of homogeneous contact manifolds

Discovery of a 2-dimensional family of non-group-structured contact manifolds

Abstract

To any metric spaces there is an associated metric profile. The rectifiability of the metric profile gives a good notion of curvature of a sub-Riemannian space. We shall say that a curvature class is the rectifiability class of the metric profile. We classify then the curvatures by looking to homogeneous metric spaces. The classification problem is solved for contact 3 manifolds, where we rediscover a 3 dimensional family of homogeneous contact manifolds, with a distinguished 2 dimensional family of contact manifolds which don't have a natural group structure.