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Classification of left invariant Riemannian metrics on nonunimodular 4-dimensional Lie groups

Malika Ait Ben Haddou, Youssef Ayad

TL;DR

This work provides a complete classification of left-invariant Riemannian metrics on four-dimensional nonunimodular Lie groups by reducing inner products on the associated Lie algebras to canonical forms under the automorphism group. Extending the 4D unimodular approach of Van Thuong, the authors employ upper triangular automorphisms to normalize metrics and identify orbit representatives across all decomposable and indecomposable nonunimodular algebras, including corrected automorphism structures from the literature. The main contributions are explicit normal form descriptions for each algebra, together with orbit-based distinctions that yield the moduli spaces of metrics, and the groundwork to determine isometry groups in these settings. The results advance the understanding of geometric structures on nonunimodular Lie groups and have potential applications in geometry and theoretical physics where left-invariant metrics play a central role.

Abstract

We classify, up to automorphism, left invariant Riemannian metrics on 4-dimensional simply connected nonunimodular Lie groups. This is equivalent to classifying, up to automorphism, inner products on 4-dimensional nonunimodular Lie algebras.

Classification of left invariant Riemannian metrics on nonunimodular 4-dimensional Lie groups

TL;DR

This work provides a complete classification of left-invariant Riemannian metrics on four-dimensional nonunimodular Lie groups by reducing inner products on the associated Lie algebras to canonical forms under the automorphism group. Extending the 4D unimodular approach of Van Thuong, the authors employ upper triangular automorphisms to normalize metrics and identify orbit representatives across all decomposable and indecomposable nonunimodular algebras, including corrected automorphism structures from the literature. The main contributions are explicit normal form descriptions for each algebra, together with orbit-based distinctions that yield the moduli spaces of metrics, and the groundwork to determine isometry groups in these settings. The results advance the understanding of geometric structures on nonunimodular Lie groups and have potential applications in geometry and theoretical physics where left-invariant metrics play a central role.

Abstract

We classify, up to automorphism, left invariant Riemannian metrics on 4-dimensional simply connected nonunimodular Lie groups. This is equivalent to classifying, up to automorphism, inner products on 4-dimensional nonunimodular Lie algebras.
Paper Structure (21 sections, 22 theorems, 111 equations)

This paper contains 21 sections, 22 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Decomposable nonunimodular 4-dimensional Lie algebras
  4. The Lie algebra $\operatorname{A}_2 \oplus \operatorname{2A}_1$
  5. The Lie algebra $\operatorname{2A}_2$
  6. The Lie algebra $\operatorname{A}_{3, 2} \oplus \operatorname{A}_1$
  7. The Lie algebra $\operatorname{A}_{3, 3} \oplus \operatorname{A}_1$
  8. The Lie algebra $\operatorname{A}_{3, 5}^{\alpha} \oplus \operatorname{A}_1$
  9. The Lie algebra $\operatorname{A}_{3, 7}^{\alpha} \oplus \operatorname{A}_1$
  10. Indecomposable nonunimodular 4-dimensional Lie algebras
  11. The Lie algebra $\operatorname{A}_{4, 2}^{\alpha}$
  12. The Lie algebra $\operatorname{A}_{4, 2}^{\alpha}$ where $\alpha \neq (0, 1)$
  13. The Lie algebra $\operatorname{A}_{4, 2}^{1}$
  14. The Lie algebra $\operatorname{A}_{4, 3}$
  15. The Lie algebra $\operatorname{A}_{4, 4}$
  16. ...and 6 more sections

Key Result

Proposition 2.1

The following map is bijective

Theorems & Definitions (40)

  • Proposition 2.1
  • proof
  • Corollary 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 3.1
  • proof
  • ...and 30 more