Pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces

Abstract

The aim of this work is to study homogeneous pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces. First, we deduce a formula for Ricci tensor of a homogeneous pseudo-Riemannian manifold with compact isotropy subgroup. Based on this formula, we establish a one-to-one correspondence between homogeneous pseudo-Riemannian Einstein metrics on noncompact homogeneous spaces and homogeneous Riemannian Einstein metrics on compact homogeneous spaces. As an application, we prove that every noncompact connected simple Lie group except SL(2) admits at least two nonproportional left invariant pseudo-Riemannian Einstein metrics. Furthermore, we study left invariant pseudo-Riemannian Einstein metrics on solvable Lie groups. By showing that if a nilpotent Lie group admits a left invariant Riemannian Ricci soliton, then it admits a left invariant pseudo-Riemannian Ricci soliton as well, we construct a left invariant pseudo-Riemannian Einstein metric on any given Riemannian standard Einstein solvmanifold.