Sub-Riemannian structures and non-transitive Cartan geometries via Lie groupoids

TL;DR

This paper develops a framework linking sub-Riemannian geometry with non-transitive Cartan geometries using Lie groupoids, providing new tools for analyzing symmetries and structures in these manifolds.

Contribution

It introduces a non-transitive Cartan connection associated with sub-Riemannian manifolds and explores the properties of the symmetry groupoid and its relation to the geometry.

Findings

The type map characterizes when sub-Riemannian symbols are constant.

The symmetry groupoid's properties depend on the type map.

A non-transitive Cartan connection can be constructed from the symmetry groupoid.

Abstract

In this paper we discuss how to associate a suitable non-transitive version of a Cartan connection to sub-Riemannian manifolds of corank 1 (including contact and quasi-contact sub-Riemannian manifolds) with non-necessarily constant sub-Riemannian symbols. In particular, we recast the variation of the sub-Riemannian symbols into a suitable "type" map, which is constant if and only if the symbols are constant. We then consider the (non-transitive) groupoid of sub-Riemannian symmetries and investigate its smoothness, properness, regularity, and other properties in relation with the type map. Last, we describe how to build a "non-transitive" analogue of a Cartan connection on top of such (Lie) groupoid, obtained as the sum of a tautological form with a multiplicative Ehresmann connection. We conclude by illustrating our results on concrete examples in dimension 5.