Homogeneous structures of $3$-dimensional Lie groups

TL;DR

This paper classifies all homogeneous Riemannian structures on 3-dimensional Lie groups with left invariant metrics, completing the classification of homogeneous 3-spaces and exploring applications in contact and CR geometry.

Contribution

It provides a complete classification of homogeneous Riemannian structures on 3D Lie groups, extending previous work and including applications to contact and CR geometry.

Findings

Complete classification of homogeneous Riemannian structures on 3D Lie groups

Identification of structures relevant to contact Riemannian geometry

Insights into CR geometry applications

Abstract

We give a classification of homogeneous Riemannian structures on (non locally symmetric) $3$-dimensional Lie groups equipped with left invariant Riemannian metrics. This work together with classifications due to previous works yields a complete classification of all the homogeneous Riemannian structures on homogeneous Riemannian $3$-spaces. Two applications of the classification to contact Riemannian geometry and CR geometry are also given.