Conformal equivalence of sub-Riemannian 3D contact structures on Lie groups

TL;DR

This paper classifies three-dimensional left-invariant sub-Riemannian contact structures on Lie groups up to conformal equivalence, revealing a dichotomy between structures conformal to the Heisenberg group and those with metric-based classification.

Contribution

It provides a complete conformal classification of 3D sub-Riemannian contact structures on Lie groups, identifying conditions for conformal flatness and symmetry groups.

Findings

Structures are either conformal to the Heisenberg group or share the same classification as the metric case.

Conformally flat structures have conformal groups isomorphic to SU(2,1).

The classification reveals a dichotomy in the conformal geometry of these structures.

Abstract

In this paper a conformal classification of three dimensional left-invariant sub-Riemannian contact structures is carried out; in particular we will prove the following dichotomy: either a structure is locally conformal to the Heisenberg group $\mathbb H_3$, or its conformal classification coincides with the metric one. If a structure is locally conformally flat, then its conformal group is locally isomorphic to $SU(2,1)$.