On 3-dimensional homogeneous generalized \(m\)-quasi-Einstein manifolds
type: Item
instance of: scholarly article
title: On 3-dimensional homogeneous generalized \(m\)-quasi-Einstein manifolds
zbMATH Open document ID: 1407.53044
author: Dehe Li
author: Zejun Hu
published in: Kodai Mathematical Journal
publication date: 2018-08-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.kmj/1521424822
review text: For a positive integer \(m\), a Riemannian manifold \((M,g)\) with a potential function \(f\) is said to be \textit{generalized \(m\)-quasi-Einstein} (cf. [\textit{A. Barros} and \textit{E. Ribeiro jun.}, Bull. Braz. Math. Soc. (N.S.) 45, No. 2, 325--341 (2014; Zbl 1301.53041)], [\textit{G. Catino}, Math. Z. 271, No. 3--4, 751--756 (2012; Zbl 1246.53040)]) if there exists a smooth function \(\lambda\) on \(M\) such that the Ricci tensor of \((M,g)\) satisfies \[ \mathrm{Ric} + \nabla^{2}f-\frac{1}{m}df\otimes df = \lambda g. \] In particular, if the function \(\lambda\) is constant \((M,g)\) is called \(m\)-quasi-Einstein, otherwise the generalized \(m\)-quasi-Einstein manifold \((M,g)\) is said to be \textit{proper}. In [\textit{A. Barros} et al., Differ. Geom. Appl. 35, 60--73 (2014; Zbl 1296.53093)] it was proved that if a \(3\)-dimensional homogeneous manifold carries an \(m\)-quasi-Einstein structure then it is either Einstein or the product \(\mathbb{H}^{2}(k)\times \mathbb{R}\), where \(\mathbb{H}^{2}(k)\) denotes the \(2\)-dimensional hyperbolic space with sectional curvature \(k\). In particular, only hyperbolic space and \(\mathbb{H}^{2}(k)\times \mathbb{R}\) can carry a nontrivial \(m\)-quasi-Einstein structure. However, it is known [\textit{A. Barros} and \textit{E. Ribeiro jun.}, Bull. Braz. Math. Soc. (N.S.) 45, No. 2, 325--341 (2014; Zbl 1301.53041)] that every space form can carry a proper generalized \(m\)-quasi-Einstein structure. In the paper under review, the authors give a complete characterization of \(3\)-dimensional homogeneous manifolds carrying a proper generalized \(m\)-quasi-Einstein structure. In fact they prove that space forms are the only \(3\)-dimensional homogeneous manifolds that can carry a proper generalized \(m\)-quasi-Einstein structure.
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 6912399
zbMATH Keywords: generalized \(m\)-quasi-Einstein
zbMATH Keywords: homogeneous 3-manifold
zbMATH Keywords: Einstein manifold
zbMATH Keywords: Bakry-Emery Ricci tensor
reviewed by: Beniamino Cappelletti Montano
MaRDI profile type: Publication
OpenAlex ID: W2793832369
DOI: 10.2996/KMJ/1521424822
Recommended article: Characterizations and integral formulae for generalized \(m\)-quasi-Einstein metrics
Recommended article: On the existence of generalized quasi-Einstein manifolds.
Recommended article: On generalized \(m\)-quasi-Einstein manifolds with constant Ricci curvatures
Recommended article: Q4673310
Recommended article: Generalized \(m\)-quasi-Einstein manifolds admitting a closed conformal vector field
Recommended article: Q3598703
Recommended article: ON SOME CLASSES OF GENERALIZED QUASI-EINSTEIN MANIFOLDS
Recommended article: \(m\)-quasi-Einstein metrics satisfying certain conditions on the potential vector field
Recommended article: Q2990167
Recommended article: Q5737932
description: scientific article; zbMATH DE number 6912399