Three-dimensional homogeneous Lorentzian Yamabe solitons
type: Item
instance of: scholarly article
title: Three-dimensional homogeneous Lorentzian Yamabe solitons
zbMATH Open document ID: 1275.53065
author: J. Seoane-Bascoy
author: M. E. Vázquez-Abal
author: E. Calviño-Louzao
author: R. Vázquez-Lorenzo
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2013-02-06T00:00:00Z
review text: A Yamabe soliton \((M,g)\) is a (pseudo-)Riemannian manifold \(M\) on which there exists a Yamabe vector field \(X\) whose Lie derivative satisfies \[ L_X g = (\tau-\lambda)g, \] where \(\tau\) is the scalar curvature and \(\lambda\) a real constant. If \(X=\nabla f\) for some smooth function \(f\), then \(M\) is called a gradient Yamabe soliton. A Yamabe soliton is called non-trivial if \(X\) is not a Killing vector field. The authors study 3-dimensional \(M\) with a non-flat Lorentzian homogeneous metric \(g\). Their first main result is that such an \(M\) is a non-trivial Yamabe soliton if and only if \(M\) is locally conformally flat and its Ricci operator is 2-step nilpotent. Such an \(M\) belongs to the class of space-times called homogeneous plane waves, and an earlier result due to Blau and O'Loughlin states that there exist two possible types of homogeneous plane waves. The authors find that the Lorentzian non-trivial Yamabe solitons in dimension three are precisely of this kind. Finally, the authors study left-invariant Yamabe vector fields on Lorentzian Lie groups of dimension three. If the soliton is to be non-trivial, there is only one 3-parameter family of such Lie groups admitting a left-invariant Yamabe vector field. In particular, it follows from the authors' results that there exist more Yamabe solitons than just the left-invariant ones in dimension three.
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53B30
zbMATH DE Number: 6134640
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: Lie groups
zbMATH Keywords: Yamabe solitons
zbMATH Keywords: Lorentz spaces
MaRDI profile type: Publication
full work available at URL: s12188-012-0072-9
OpenAlex ID: W2049040864
Wikidata QID: Q125857087
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DOI: 10.1007/S12188-012-0072-9
description: scientific article; zbMATH DE number 6134640