Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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D. Conti - F. A. Rossi

Indefinite nilsolitons and Einstein solvmanifolds

created by rossi on 20 May 2021
modified on 27 Apr 2022


Published Paper

Inserted: 20 may 2021
Last Updated: 27 apr 2022

Journal: J. Geom. Anal.
Volume: 32.3
Pages: 88
Year: 2022

ArXiv: 2105.09209 PDF


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.

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