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

作者：Barros A; Ribeiro E Jr^*; Silva Filho J

来源：Differential Geometry and its Applications, 2014, 35: 60-73.

DOI：10.1016/j.difgeo.2014.05.007

• Quasi-Einstein metrics
• Homogeneous manifolds
• Isometry group
• Static metrics

The purpose of this article is to study the existence and uniqueness of quasi-Einstein structures on 3-dimensional homogeneous Riemannian manifolds. To this end, we use the eight model geometries for 3-dimensional manifolds identified by Thurston. First, we present here a complete description of quasi-Einstein metrics on 3-dimensional homogeneous manifolds with isometry group of dimension 4. In addition, we shall show the absence of such gradient structure on Sol(3), which has 3-dimensional isometry group. Moreover, we prove that Berger's spheres carry a non-trivial quasi-Einstein structure with non-gradient associated vector field, this shows that a theorem due to Perelman cannot 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 H-k(2) x R.

• 出版日期2014-8