Sub-Riemannian structures on 3D Lie groups

TL;DR

This paper classifies all left-invariant sub-Riemannian structures on three-dimensional Lie groups using differential invariants, connecting to previous results and explicitly identifying isometries between certain Lie groups.

Contribution

It provides a complete classification of sub-Riemannian structures on 3D Lie groups and explicitly finds isometries between nonisomorphic groups.

Findings

Complete classification of structures on 3D Lie groups

Recovery of known curvature-based classifications

Explicit sub-Riemannian isometry between SL(2) and A^+(ty) imes S^1

Abstract

We give the complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. This classifications recovers other known classification results in the literature, in particular the one obtained in [Falbel-Gorodski, 1996] in terms of curvature invariants of a canonical connection. Moreover, we explicitly find a sub-Riemannian isometry between the nonisomorphic Lie groups $SL(2)$ and $A^{+}(\mathbb{R})\times S^1$, where $A^+(\mathbb{R})$ denotes the group of orientation preserving affine maps on the real line.