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Isometry groups of three-dimensional Lie groups

Ana Cosgaya, Silvio Reggiani

TL;DR

This work determines the full isometry group of any left-invariant metric on a simply connected, non-unimodular $3$-dimensional Lie group by combining Singer's description of the isotropy with a case-by-case analysis of the Lie group structures $G_I$ and $G_c$. It provides a complete computation of the index of symmetry for all such metrics, showing nontrivial symmetry only in specific loci and that symmetric (Einstein) cases yield index $3$. The authors also relate the moduli space of left-invariant metrics to its singularities, proving that the singular set lies inside the maximal-index subset and describing the topology of the moduli space, including notable phenomena where non-isometric groups share isometric metrics. Overall, the paper advances the classification of homogeneous spaces by giving explicit isometry groups, symmetry indices, and moduli-space geometry for all non-unimodular $3$-dimensional cases, with potential implications for geometric structures and physical models on low-dimensional solvable groups.

Abstract

We compute the full isometry group of any left invariant metric on a simply connected, non-unimodular Lie group of dimension three. As an application, we determine the index of symmetry of such metrics and prove that the singularities of the moduli space of left-invariant metrics, up to isometric automorphism, is contained in the subspace of classes of metrics with maximal index of symmetry.

Isometry groups of three-dimensional Lie groups

TL;DR

This work determines the full isometry group of any left-invariant metric on a simply connected, non-unimodular -dimensional Lie group by combining Singer's description of the isotropy with a case-by-case analysis of the Lie group structures and . It provides a complete computation of the index of symmetry for all such metrics, showing nontrivial symmetry only in specific loci and that symmetric (Einstein) cases yield index . The authors also relate the moduli space of left-invariant metrics to its singularities, proving that the singular set lies inside the maximal-index subset and describing the topology of the moduli space, including notable phenomena where non-isometric groups share isometric metrics. Overall, the paper advances the classification of homogeneous spaces by giving explicit isometry groups, symmetry indices, and moduli-space geometry for all non-unimodular -dimensional cases, with potential implications for geometric structures and physical models on low-dimensional solvable groups.

Abstract

We compute the full isometry group of any left invariant metric on a simply connected, non-unimodular Lie group of dimension three. As an application, we determine the index of symmetry of such metrics and prove that the singularities of the moduli space of left-invariant metrics, up to isometric automorphism, is contained in the subspace of classes of metrics with maximal index of symmetry.

Paper Structure

This paper contains 14 sections, 8 theorems, 50 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $3$-dimensional, non-unimodular, metric Lie algebras
  4. A theorem by Singer on the Lie algebra of the isometry group
  5. The index of symmetry of a homogeneous manifold
  6. The full isometry groups
  7. The case of $G_I$
  8. The case of $G_0$
  9. The case of $G_1$
  10. The case of $G_c$, $c < 0$
  11. The case of $G_c$, $0 < c < 1$
  12. The case of $G_c$, $1 < c$
  13. Computation of the index of symmetry
  14. The geometric meaning of the subset of metrics with maximal index of symmetry inside the moduli space of left invariant metrics

Key Result

Theorem 2.1

Let $M$ be a homogeneous Riemannian space of dimension $n$. Let $p \in M$ and $A \in \mathfrak{so}(T_pM)$. Then, there exists a Killing vector field $X$ on $M$ such that $X_p = 0$ and $(\nabla X)_p = A$ if and only if $A \cdot (\nabla^s R)_p = 0$ for all $0 \le s < \frac{n (n - 1)}{2}$, where $R$ de

Theorems & Definitions (17)

  • Theorem 2.1: Singer singer-1960
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • proof
  • ...and 7 more