On 3-dimensional homogeneous generalized \(m\)-quasi-Einstein manifolds
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.
scientific article; zbMATH DE number 6912399