Indefinite nilsolitons and Einstein solvmanifolds

TL;DR

This paper explores indefinite nilsolitons and Einstein solvmanifolds, revealing greater geometric flexibility than in the Riemannian case and establishing new algebraic and geometric relationships.

Contribution

It classifies indefinite nilsolitons based on the properties of D and constructs Einstein solvmanifolds from them, highlighting differences from the Riemannian setting.

Findings

Indefinite metrics allow four distinct geometries for nilsolitons.

Conditions on D are characterized when D is nonzero.

Constructed examples show the lack of a full correspondence between Einstein solvmanifolds and nilsolitons.

Abstract

A nilsoliton is a nilpotent Lie algebra $\mathfrak{g}$ with a metric such that $\operatorname{Ric}=\lambda \operatorname{Id}+D$, with $D$ a derivation. For indefinite metrics, this determines four different geometries, according to whether $\lambda$ 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 $\mathfrak{g}\rtimes \mathbb{R}^k$. Conversely, we introduce a large class of indefinite Einstein solvmanifolds of the form $\mathfrak{g}\rtimes \mathbb{R}^k$ that determine a nilsoliton metric on $\mathfrak{g}$ 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.