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Four-dimensional Lorentzian algebraic Ricci solitons

Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey, Ramon Vazquez-Lorenzo

TL;DR

This work provides a comprehensive analysis of four-dimensional Lorentzian algebraic Ricci solitons on simply connected Lie groups, showing that left-invariant Lorentzian metrics yielding Ricci solitons are plentiful and span a wide range of Lie-type geometries, including plane waves, semi-direct extensions of abelian and Heisenberg groups, and direct extensions of non-solvable groups. The authors organize the results by Lie algebra structure, distinguishing solvable and non-solvable cases, and employ polynomial derivation-conditions and Gröbner-basis computations to classify ARS across many families, including almost-Abelian, H^3-extensions, E(2)/Poincaré-type groups, and direct products. A central theme is the interaction between Lorentzian geometry and Lie-algebra structure: plane waves are ARS in many cases but not all, and ARS in higher dimensions often arise from semi-direct/Heisenberg-type constructions rather than simple products. The paper also links ARS to quadratic curvature functionals, showing that four-dimensional Lorentzian ARS are typically S-critical or F[t]-critical with zero energy, and discusses compact steady solitons on nilmanifolds/solvmanifolds, highlighting the richness of the Lorentzian setting compared to the Riemannian one. Overall, the results provide a broad, explicit classification of four-dimensional Lorentzian algebraic Ricci solitons and illuminate the geometric variety of four-dimensional Lorentzian Lie groups as Ricci solitons with strong ties to plane waves and semi-direct Lie-group extensions.

Abstract

We describe four-dimensional Lorentzian algebraic Ricci solitons. In sharp contrast with the Riemannian situation, any connected and simply connected four-dimensional Lie group admits a left-invariant Lorentz metric which is a Ricci soliton.

Four-dimensional Lorentzian algebraic Ricci solitons

TL;DR

This work provides a comprehensive analysis of four-dimensional Lorentzian algebraic Ricci solitons on simply connected Lie groups, showing that left-invariant Lorentzian metrics yielding Ricci solitons are plentiful and span a wide range of Lie-type geometries, including plane waves, semi-direct extensions of abelian and Heisenberg groups, and direct extensions of non-solvable groups. The authors organize the results by Lie algebra structure, distinguishing solvable and non-solvable cases, and employ polynomial derivation-conditions and Gröbner-basis computations to classify ARS across many families, including almost-Abelian, H^3-extensions, E(2)/Poincaré-type groups, and direct products. A central theme is the interaction between Lorentzian geometry and Lie-algebra structure: plane waves are ARS in many cases but not all, and ARS in higher dimensions often arise from semi-direct/Heisenberg-type constructions rather than simple products. The paper also links ARS to quadratic curvature functionals, showing that four-dimensional Lorentzian ARS are typically S-critical or F[t]-critical with zero energy, and discusses compact steady solitons on nilmanifolds/solvmanifolds, highlighting the richness of the Lorentzian setting compared to the Riemannian one. Overall, the results provide a broad, explicit classification of four-dimensional Lorentzian algebraic Ricci solitons and illuminate the geometric variety of four-dimensional Lorentzian Lie groups as Ricci solitons with strong ties to plane waves and semi-direct Lie-group extensions.

Abstract

We describe four-dimensional Lorentzian algebraic Ricci solitons. In sharp contrast with the Riemannian situation, any connected and simply connected four-dimensional Lie group admits a left-invariant Lorentz metric which is a Ricci soliton.
Paper Structure (61 sections, 16 theorems, 188 equations, 2 figures)

This paper contains 61 sections, 16 theorems, 188 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Four-dimensional homogeneous plane waves
  4. Direct extensions of algebraic Ricci solitons
  5. The non-solvable cases $SU(2)\times\mathbb{R}$ and $\widetilde{SL}(2,\mathbb{R})\times\mathbb{R}$
  6. Extensions of the Euclidean and Poincaré Lie groups
  7. Left-invariant Lorentz metrics on the product $\mathcal{H}^3\times\mathbb{R}$
  8. Almost Abelian algebraic Ricci solitons
  9. Algebraic Ricci solitons on semi-direct extensions of the Heisenberg group
  10. Critical metrics for quadratic curvature functionals
  11. Three-dimensional Lorentzian algebraic Ricci solitons
  12. The unimodular case
  13. The non-unimodular case
  14. Four-dimensional Lorentzian Lie groups
  15. Positive definite metrics on $\bm{\mathfrak{k}}$
  16. ...and 46 more sections

Key Result

Theorem 2.2

Let $(\mathbb{R}^3\rtimes\mathbb{R},\langle \cdot,\cdot \rangle)$ be a semi-direct extension of the Abelian Lie group equipped with a left-invariant Lorentzian metric. Then it is a strict algebraic Ricci soliton if and only if it is isomorphically homothetic to one of the following: (R) The restrict Here $\{e_i\}$ is an orthonormal basis of the Lie algebra with $e_4$ timelike. (L) The restriction

Figures (2)

  • Figure 1: Range of the parameter $t$ for homothetic classes of four-dimensional strict algebraic Lorentzian Ricci solitons with $\tau\neq 0$.
  • Figure 2: Range of the parameter $t$ for homothetic classes of three-dimensional irreducible non-Einstein algebraic Lorentzian Ricci solitons.

Theorems & Definitions (64)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 3.1
  • Remark 5.1
  • Remark 5.2
  • Theorem 5.3
  • ...and 54 more