Indefinite nilsolitons and Einstein solvmanifolds

Abstract

A nilsoliton is a nilpotent Lie algebra g with a metric such that Ric = λId + D, with D a derivation. For indefinite metrics, this determines four different geometries, according to whether λ and D are zero or not. We illustrate with examples the greater flexibility of the indefinite case compared to the Riemannian setting. We determine the algebraic properties that D must satisfy when it is nonzero.For each of the four geometries, we show that under suitable assumptions it is possible to extend the nilsoliton metric to an Einstein solvmanifold of the form g ⋊ R k . Conversely, we introduce a large class of indefinite Einstein solvmanifolds of the form g ⋊ R k that determine a nilsoliton metric on g by restriction. We show with examples that, unlike in the Riemannian case, one cannot establish a correspondence between the full classes of Einstein solvmanifolds and nilsolitons.

New Special Einstein Pseudo-Riemannian Metrics on Solvable Lie Algebras

“…These differences and the wide spectrum of examples that can be constructed make a complete classification of Einstein metrics in the pseudo-Riemannian case very difficult. However, in a previous paper together with D. Conti, we study the Einstein solvable Lie algebras and their relations with pseudo-Riemannian nilsoliton metrics ( [14]). In this section we recall some important facts and we will show how the study of nilsoliton metrics help the construction of Einstein metrics with nonzero scalar curvature.…”

“…In this section we recall some important facts and we will show how the study of nilsoliton metrics help the construction of Einstein metrics with nonzero scalar curvature. The interested reader may refer to [14,15].…”

“…In our paper, we gave examples of nilsoliton metrics for each type (as well as the first example of type (Nil4) with nonsemisimple derivation), but more examples can be found in literature: for example, there are known examples of (Nil4) on the Heisenberg Lie algebra where the derivation D is semisimple [35,24]. It is worth to mention (see [14,Remark 2.6]) that pseudo-Riemannian nilsolitons are not unique, unlike the Riemannian ones, as proved in [25].…”

“…This article aims to go in this direction, showing a concrete strategy for building examples of special pseudo-Riemannian Einstein metrics. In fact, our recent papers [14,15], jointly with D. Conti, have allowed us to construct new examples of Einstein homogeneous manifold with indefinite metrics. The new technique developed may be combined with other tools to address different problems about the construction of special metrics, as we aim to show in the present paper.…”

“…Section 5 contains a brief summary of the results contained in [14], and it explains how to obtain Einstein metrics with nonzero scalar curvature on solvable Lie algebras starting from pseudo-Riemannian nilsoliton metrics.…”