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Homogeneous structures of $3$-dimensional Lie groups

Jun-ichi Inoguchi, Yu Ohno

TL;DR

The paper provides a complete classification of homogeneous Riemannian structures on 3-dimensional Lie groups with left-invariant metrics, covering both unimodular and non-unimodular cases and removing previous left-invariance assumptions. It employs the Ambrose–Singer framework, moving frames, and canonical Lie-group connections to derive explicit $S$-tensors (including $S=\nabla^{(-)}-\nabla$ and one-parameter families $S^{(r)}$) and their coset representations. The results unify and extend prior work (Abe, CFG, Ohno) and yield concrete descriptions of when homogeneous structures are left-invariant, as well as when they arise from more general Lie-group actions, with explicit isotropy and holonomy analyses. The work further links to contact and CR geometry by characterizing homogeneous structures in non-Sasakian $(\kappa,\mu)$-spaces and Sasakian 3-manifolds, highlighting practical implications for geometric models and curvature behavior in dimension three.

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.

Homogeneous structures of $3$-dimensional Lie groups

TL;DR

The paper provides a complete classification of homogeneous Riemannian structures on 3-dimensional Lie groups with left-invariant metrics, covering both unimodular and non-unimodular cases and removing previous left-invariance assumptions. It employs the Ambrose–Singer framework, moving frames, and canonical Lie-group connections to derive explicit -tensors (including and one-parameter families ) and their coset representations. The results unify and extend prior work (Abe, CFG, Ohno) and yield concrete descriptions of when homogeneous structures are left-invariant, as well as when they arise from more general Lie-group actions, with explicit isotropy and holonomy analyses. The work further links to contact and CR geometry by characterizing homogeneous structures in non-Sasakian -spaces and Sasakian 3-manifolds, highlighting practical implications for geometric models and curvature behavior in dimension three.

Abstract

We give a classification of homogeneous Riemannian structures on (non locally symmetric) -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 -spaces. Two applications of the classification to contact Riemannian geometry and CR geometry are also given.
Paper Structure (27 sections, 20 theorems, 175 equations, 3 tables)

This paper contains 27 sections, 20 theorems, 175 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Homogeneous Riemannian spaces
  3. Ambrose and Singer connections
  4. The eight classes of homogeneous Riemannian structures
  5. Canonical connections on Lie groups
  6. Moving frames
  7. Connection forms
  8. Curvature forms
  9. Homogeneous Riemannian structures
  10. Three dimensional Lie groups
  11. Unimodularity
  12. Unimodular Lie groups
  13. Non-unimodular Lie groups
  14. The homogeneous Riemannian structures on $3$-dimensional unimodular Lie groups
  15. Connection forms
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $G$ be a $3$-dimensional unimodular Lie group admitting a unimodular basis of the Lie algebra $\mathfrak{g}$ satisfying $c_1=c_2\not=c_3$ and $c_3\not=0$. Then

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 1
  • Proposition 2.1
  • Theorem 2.1: AS
  • Definition 2.3
  • Theorem 2.2: TV
  • ...and 24 more