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Four-dimensional pp-wave Lie groups and harmonic curvature

Eduardo García-Río, Rosalía Rodríguez-Gigirey, Ramón Vázquez-Lorenzo

TL;DR

This work classifies all four-dimensional Lie groups admitting harmonic curvature, i.e., $div\,W=0$, and, as a key corollary, gives a complete description of four-dimensional left-invariant pp-wave Lorentzian metrics. The authors reduce the problem to a Lie-algebraic classification, examining semi-direct extensions of the 3D unimodular and Heisenberg algebras as well as direct extensions of the non-solvable groups, and they solve curvature constraints via polynomial systems and Gröbner bases. They show that four-dimensional pp-wave Lie groups are plane waves precisely when the curvature is harmonic, and they provide an explicit list of non-flat pp-wave Lie groups, including a unique non-plane-wave example with harmonic curvature tied to $\mathfrak{r}_{4,\mu,-\mu}$ for specific $\mu$. The results highlight a strong dichotomy: harmonic curvature often forces local conformal flatness or simple product structures, with plane waves emerging as the main nontrivial pp-wave by homothety. Overall, the paper advances the understanding of homogeneous Lorentzian geometry in four dimensions and sharpens the link between pp-waves, plane waves, and harmonic curvature in Lie-group settings.

Abstract

We determine all four-dimensional Lie groups which have harmonic curvature. As a consequence, a description of four-dimensional pp-wave Lie groups is obtained.

Four-dimensional pp-wave Lie groups and harmonic curvature

TL;DR

This work classifies all four-dimensional Lie groups admitting harmonic curvature, i.e., , and, as a key corollary, gives a complete description of four-dimensional left-invariant pp-wave Lorentzian metrics. The authors reduce the problem to a Lie-algebraic classification, examining semi-direct extensions of the 3D unimodular and Heisenberg algebras as well as direct extensions of the non-solvable groups, and they solve curvature constraints via polynomial systems and Gröbner bases. They show that four-dimensional pp-wave Lie groups are plane waves precisely when the curvature is harmonic, and they provide an explicit list of non-flat pp-wave Lie groups, including a unique non-plane-wave example with harmonic curvature tied to for specific . The results highlight a strong dichotomy: harmonic curvature often forces local conformal flatness or simple product structures, with plane waves emerging as the main nontrivial pp-wave by homothety. Overall, the paper advances the understanding of homogeneous Lorentzian geometry in four dimensions and sharpens the link between pp-waves, plane waves, and harmonic curvature in Lie-group settings.

Abstract

We determine all four-dimensional Lie groups which have harmonic curvature. As a consequence, a description of four-dimensional pp-wave Lie groups is obtained.
Paper Structure (49 sections, 6 theorems, 70 equations)

This paper contains 49 sections, 6 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. pp-wave and plane wave Lorentzian manifolds
  3. Homogeneous setting
  4. Three-dimensional homogeneous pp-waves
  5. Four-dimensional homogeneous plane waves
  6. Harmonic Weyl tensor
  7. Cotton-flat three-dimensional homogeneous manifolds
  8. Summary of results
  9. Structure of the paper
  10. Semi-direct extensions of the Abelian Lie group
  11. Semi-direct extensions with Riemannian Lie group $\pmb{\mathbb{R}}^{\bm{3}}$
  12. Semi-direct extensions with Lorentzian Lie group $\pmb{\mathbb{R}}^{\bm{3}}$
  13. The self-adjoint part of the derivation is diagonalizable
  14. The self-adjoint part of the derivation has complex eigenvalues
  15. The self-adjoint part of the derivation has a double root of the minimal polynomial
  16. ...and 34 more sections

Key Result

Theorem 1.1

A four-dimensional non-flat left-invariant pp-wave Lie group is isomorphically homothetic to one of the following: Moreover, left-invariant metrics in cases (2), (3), (4) and (5) are plane waves. Here $\{u_i\}$ and $\{v_i\}$ are pseudo-orthonormal bases, with $\langle u_1,u_2\rangle=\langle u_3,u_3\rangle=\langle u_4,u_4\rangle=1$ and $\langle v_1,v_1\rangle=\langle v_2,v_2\rangle=\langle v_3,v_

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 30 more