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Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group

Giovanni Calvaruso, Lorenzo Pellegrino, Amirhesam Zaeim

TL;DR

The paper identifies and analyzes the exceptional Lorentzian homogeneous 3-manifold with a four-dimensional isometry group, realized by a left-invariant metric on $G=\\widetilde{SL}(2,\\mathbb{R})$. It provides an explicit global-coordinate model for this metric, including its Levi-Civita connection, curvature, and Ricci tensor. The main result is a complete solution of the Ricci soliton equation for this family, yielding expanding solitons with $\\lambda = -\\tfrac{1}{2}\\mu^2$ and a four-parameter soliton field $X$; these solitons are shown to be non-gradient, with explicit formulas depending on the sign of $\\varepsilon\\mu$. This work completes the classification of Ricci solitons on 3D Lorentzian Lie groups with a 4D isometry group outside the BCV and plane-wave classes, providing concrete non-gradient examples and detailed geometric data.

Abstract

In the framework of the study of homogeneous Lorentzian three-manifolds, we consider here the only class of examples which admit a four-dimensional group of isometries but are neither Lorentzian Bianchi-Cartan-Vranceanu spaces nor plane waves. We obtain an explicit description in global coordinates of these special homogeneous Lorentzian manifolds. We then prove that all such examples are non-gradient expanding Ricci solitons.

Ricci solitons of special Lorentzian Lie groups with a four-dimensional isometry group

TL;DR

The paper identifies and analyzes the exceptional Lorentzian homogeneous 3-manifold with a four-dimensional isometry group, realized by a left-invariant metric on . It provides an explicit global-coordinate model for this metric, including its Levi-Civita connection, curvature, and Ricci tensor. The main result is a complete solution of the Ricci soliton equation for this family, yielding expanding solitons with and a four-parameter soliton field ; these solitons are shown to be non-gradient, with explicit formulas depending on the sign of . This work completes the classification of Ricci solitons on 3D Lorentzian Lie groups with a 4D isometry group outside the BCV and plane-wave classes, providing concrete non-gradient examples and detailed geometric data.

Abstract

In the framework of the study of homogeneous Lorentzian three-manifolds, we consider here the only class of examples which admit a four-dimensional group of isometries but are neither Lorentzian Bianchi-Cartan-Vranceanu spaces nor plane waves. We obtain an explicit description in global coordinates of these special homogeneous Lorentzian manifolds. We then prove that all such examples are non-gradient expanding Ricci solitons.

Paper Structure

This paper contains 3 sections, 2 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Explicit coordinates and geometry of the exceptional example
  3. The Ricci soliton equation for the exceptional example

Key Result

Theorem 1.1

For any choice of the real parameter $\mu \neq 0$, the unimodular Lie group $G=\widetilde{SL}(2,\mathbb R)$, equipped with the left-invariant Lorentzian metric corresponding to glie, is isometric to $\mathbb R^3$ equipped with the metric Every metric g is an expanding Ricci soliton, satisfying the Ricci soliton equation solit with $\lambda=-\frac{1}{2} \mu ^2$ and $X=X^i \partial_i$, where if $\

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Remark 1
  • Remark 2