Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Left-invariant Codazzi tensors and harmonic curvature on Lie groups endowed with a left invariant Lorentzian metric

Ilyes Aberaouze, Mohamed Boucetta

Abstract

A Lorentzian Lie group is a Lie group endowed with a left invariant Lorentzian metric. We study left-invariant Codazzi tensors on Lorentzian Lie groups. We obtain new results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator. In contrast to the Riemannian case, the Ricci operator of a let-invariant Lorentzian metric can be of four types: diagonal, of type $\{n-2,z\bar{z}\}$, of type $\{n,a2\}$ and of type $\{n,a3\}$. We first describe Lorentzian Lie algebras with a non-diagonal Codazzi operator and with these descriptions in mind, we study three classes of Lorentzian Lie groups with harmonic curvature. Namely, we give a complete description of the Lie algebra of Lorentzian Lie groups having harmonic curvature and where the Ricci operator is non-diagonal and its diagonal part consists of one real eigenvalue $α$.

Left-invariant Codazzi tensors and harmonic curvature on Lie groups endowed with a left invariant Lorentzian metric

Abstract

A Lorentzian Lie group is a Lie group endowed with a left invariant Lorentzian metric. We study left-invariant Codazzi tensors on Lorentzian Lie groups. We obtain new results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator. In contrast to the Riemannian case, the Ricci operator of a let-invariant Lorentzian metric can be of four types: diagonal, of type , of type and of type . We first describe Lorentzian Lie algebras with a non-diagonal Codazzi operator and with these descriptions in mind, we study three classes of Lorentzian Lie groups with harmonic curvature. Namely, we give a complete description of the Lie algebra of Lorentzian Lie groups having harmonic curvature and where the Ricci operator is non-diagonal and its diagonal part consists of one real eigenvalue .
Paper Structure (18 sections, 24 theorems, 109 equations)

This paper contains 18 sections, 24 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Lorentzian Lie algebras having a symmetric Codazzi operator: preliminary study
  3. Characterization of Lorentzian Lie algebras having a symmetric Codazzi operator of type non diagonal
  4. Codazzi operator of type $\{n-2,z\bar{z}\}$
  5. Codazzi operator of type $\{n,a2\}$
  6. Codazzi operator of type $\{n,a3\}$
  7. A complete description of Lorentzian Lie algebras whose curvature is harmonic and the Ricci operator is of type $\{n-2,z\bar{z}\}$ with one real Ricci direction
  8. Proof of Theorem \ref{['mainbiszz']}
  9. A complete description of Lorentzian Lie algebras having harmonic curvature and the Ricci operator is of type $\{n,a2,\alpha\}$ with $a\not=\alpha$
  10. Proof of Theorem \ref{['mainbisa2']}
  11. A complete description of Lorentzian Lie algebras having harmonic curvature and the Ricci operator is of type $\{n,a3,\alpha\}$ with $a\not=\alpha$
  12. Proof of Theorem \ref{['mainbisa3']}
  13. Appendix
  14. The conditions for $(l)$ to hold
  15. The conditions for $(hl0)$ to hold
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $f$ be a symmetric operator of a Lorentzian vector space $(V, \langle \;,\;\rangle)$ of dimension $n\geq3$. Then there exists a basis $\hbox{B}$ of $V$ such that the matrices of $f$ and $\langle \;,\;\rangle$ have one of the following forms:

Theorems & Definitions (38)

  • Conjecture 1
  • Theorem 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • ...and 28 more