Model spaces in sub-Riemannian geometry

TL;DR

This paper studies special sub-Riemannian spaces with maximal symmetry, introducing a canonical partial connection that generalizes Riemannian Levi-Civita connections, and discusses invariants characterizing these spaces.

Contribution

It introduces a canonical partial connection on horizontal bundles of symmetric sub-Riemannian spaces, extending concepts from Riemannian geometry and analyzing invariants beyond holonomy.

Findings

Canonical partial connection exists in symmetric sub-Riemannian spaces.

Number of invariants for tangent cones can be greater than one.

Invariants are not necessarily related to holonomy.

Abstract

We consider sub-Riemannian spaces admitting an isometry group that is maximal in the sense that any linear isometry between the horizontal tangent spaces is realized by a global isometry. We will show that these spaces have a canonical choice of partial connection on their horizontal bundle, which is determined by isometries and generalizes the Levi-Civita connection for the special case of Riemannian model spaces. The number of invariants needed to describe model spaces with the same tangent cone is in general greater than one, and these invariants are not necessarily related to the holonomy of the canonical connections.