A pseudo-Riemannian metric is said to be Einstein if its Ricci tensor is a constant multiple of the metric. Bi-invariant metrics on simple Lie groups are examples; more generally, there are several known constructions to obtain homogeneous Einstein metrics, i.e. invariant under the transitive action of a group.<\/p>\n\n\n\n
A recent result of B\u00f6hm and Lafuente states that every homogeneous, Einstein Riemannian manifold with negative scalar curvature is isometric to a solvmanifold. The structure of Riemannian solvmanifolds is well known, thanks to the work of Lauret and Heber.<\/p>\n\n\n\n
In the case of indefinite signature, this theory can be adapted to construct vast classes of examples, but more flexibility occurs. In particular, it is possible to construct nonflat, Ricci-flat homogeneous metrics, nonsymmetric bi-invariant metrics, and Einstein nilmanifolds with nonzero scalar curvature.<\/p>\n\n\n\n
After introducing the general context, I will illustrate some constructive results for these types of metrics, obtained jointly with Federico A. Rossi and Viviana del Barco.<\/p>\n","protected":false},"excerpt":{"rendered":"
A pseudo-Riemannian metric is said to be Einstein if its Ricci tensor is a constant multiple of the metric. Bi-invariant metrics on simple Lie groups are examples; more generally, there are several known constructions to obtain homogeneous Einstein metrics, i.e. invariant under the transitive action of a group.…<\/p>\n