Uniqueness of quasi-einstein metrics on 3-dimensional homogeneous riemannian manifold

TLDR: In this article, a complete description of m-quasi-Einstein metrics on 3-dimensional homogeneous Riemannian manifold is presented, when this manifold compact or not compact provided dim Iso(M3, g) = 4.

Abstract: One of the motivation to study m-quasi-Einstein metrics on a Riemannian manifold (Mn, g) is its closed relation with warped product Einstein metrics, see e.g. [3]. For instance, when m is a positive integer, m-quasi-Einstein metrics correspond to exactly those n-dimensional manifolds which are the base of an (n + m)-dimensional Einstein warped product. It is important to detach that gradient 1-quasi-Einstein metrics satisfying ∆e−f + λe−f = 0 are more commonly called static metrics with cosmological constant λ. These static metrics have been studied extensively because their connection with scalar curvature, the positive mass theorem and general relativity, for more details see e.g. [1] and [4]. The study of 3-dimensional homogeneous Riemannian manifolds is done, in general, according to the dimension of its isometry group Iso(M3, 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 compact or not compact provided dim Iso(M3, g) = 4. In addition, we shall show the absence of such structure on Sol3, which corresponds to dim Iso(M3, g) = 3. When dim Iso(M3, g) = 6 it is well known that M3 is a space form. In this case, its canonical structure gives a trivial example. In particular, we shall prove that Berger’s spheres carry naturally a non trivial structure of quasi-Einstein metrics. Since they have constant scalar curvature, their associated vector fields can not be gradient, this shows that Perelman’s Theorem can not be extend to quasi-Einstein metrics. Moreover, these examples show that Theorem 4.6 of [5] can not be extended for a non gradient vector field. Finally, we prove that if (M3, g, X, λ) is a non compact 3-dimensional homogeneous Riemannian manifold such that g is a m-quasi-Einstein metric, then, either M3 is a space form or M3 is E3(κ, τ) such as our Example obtained in this work. 1ernani@mat.ufc.br Departamento de Matematica-Universidade Federal do Ceara UFC, 60455-760-Fortaleza-CE-BR.