On gradient Ricci solitons conformal to a pseudo-Euclidean space
type: Item
instance of: scholarly article
title: On gradient Ricci solitons conformal to a pseudo-Euclidean space
zbMATH Open document ID: 1297.53043
author: Romildo Pina
author: Keti Tenenblat
author: Ezequiel R. Barbosa
published in: Israel Journal of Mathematics
publication date: 2014-06-25T00:00:00Z
review text: The article under review is devoted to particular gradient Ricci solitons. By definition, a semi-Riemannian manifold \((M^n, g)\), \(n\geq 3\), is called a gradient Ricci soliton, if there exists a smooth function \(f:M^n\to \mathbb R\), called a potential function, such that \[ \mathrm{Ric}_g + \mathrm{Hess}_g (f) = \rho g, \] where \(\mathrm{Ric}_g\) is the Ricci tenzor, \(\mathrm{Hess}_g(f)\) is the Hessian of \(f\) with respect to \(g\), and \(\rho\in\mathbb R\) is a constant. The main attention is paid to gradient Ricci solitons, which are steady (\(\rho=0\)) and conformally flat. In the Riemannian case, gradient Ricci solitons which are complete, steady and locally conformally flat, are classified in [\textit{H.-D. Cao} and \textit{Q. Chen}, Trans. Am. Math. Soc. 364, No. 5, 2377--2391 (2012; Zbl 1245.53038)]. In the Lorentzian case the situation is more complicated, non-trivial examples are discussed in [\textit{W. Batat} et al., Bull. Lond. Math. Soc. 43, No. 6, 1219--1227 (2011; Zbl 1266.53049)], [\textit{M. Brozos-Vaźquez} et al, J. Geom. Anal. 23, No. 3, 1196--1212 (2013; Zbl 1285.53059)], [\textit{K. Onda}, Geom. Dedicata 147, 313--322 (2010; Zbl 1203.53044)]. The present article aims at providing an explicit description of steady, conformally flat gradient Ricci solitons, which are invariant under the action of an \((n-1)\)-dimensional translation group. Namely, let \((\mathbb R^n, g)\) be a pseudo-Euclidean space endowed with Cartesian coordinates \((x_1, \dots, x_n)\) such that \(g=\sum\varepsilon_j dx_j^2\), where \(\varepsilon_j=\pm 1\), \(1\leq j\leq n\), with at least one \(\varepsilon_j=1\). Let \(\xi = \sum\alpha_j x_j\) be a basic invariant for an \((n-1)\)-dimensional translation group. The problem is to find smooth functions \(\varphi(\xi)\) and \(f(\xi)\) such that the metric \(\tilde g = g / \varphi^2\) satisfies the equation \[ \mathrm{Ric}_{\tilde g} + \mathrm{Hess}_{\tilde g} (f) = \rho {\tilde g} \] for some constant \(\rho\), which is …
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53B30
zbMATH DE Number: 6308421
zbMATH Keywords: gradient Ricci soliton
zbMATH Keywords: steady gradient Ricci soliton
zbMATH Keywords: conformally flat manifold
reviewed by: Vasyl Gorkaviy
MaRDI profile type: Publication
full work available at URL: s11856-014-0014-6
OpenAlex ID: W2028715966
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DOI: 10.1007/S11856-014-0014-6
description: scientific article; zbMATH DE number 6308421