# Uniqueness of quasi-Einstein metrics on 3-dimensional homogeneous manifolds

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The study of 3-dimensional homogeneous Riemannian manifolds is done, in general, according to the dimension of its isometry group $IsoM^3,g$, which can be 3,4 or 6. Following this trend we present here a complete description of $m$-quasi-Einstein metrics, when this manifold is compact or not compact provided $dim\,IsoM^3,g=4$. In addition, we shall show the absence of such gradient structure on $Sol^3,$ which corresponds to $dim\,IsoM^3,g=3.$ When $dim\,IsoM^3,g=6$ it is well known that $M^3$ is a space form. In this case, its canonical structure gives a trivial example. Moreover, we prove that Bergers spheres carry a non-trivial quasi-Einstein structure with non gradient associated vector field, this shows that Perelmans Gradient Theorem can not be extend to quasi-Einstein metrics. Finally, we prove that a 3-dimensional homogeneous manifold carrying a gradient quasi-Einstein structure is either Einstein or $\mathbb{H}^2 {\kappa} \times \mathbb{R}.$

Autor: Abdênago Barros; Ernani Ribeiro Jr; João F. Silva

Fuente: https://archive.org/