Abstract
A nilsoliton is a nilpotent Lie algebra \(\mathfrak {g}\) with a metric such that \({{\,\mathrm{Ric}\,}}=\lambda \mathrm {Id}+D\), 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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All code associated with the current submission is available at https://github.com/diego-conti/gleipnir [8].
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We thank the anonymous referee for his/her corrections and suggestions that improved the presentation of the paper.
Funding
The authors acknowledge GNSAGA of INdAM. F.A. Rossi also acknowledges the Young Talents Award of Università degli Studi di Milano-Bicocca joint with Accademia Nazionale dei Lincei.
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Conti, D., Rossi, F.A. Indefinite Nilsolitons and Einstein Solvmanifolds. J Geom Anal 32, 88 (2022). https://doi.org/10.1007/s12220-021-00850-7
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DOI: https://doi.org/10.1007/s12220-021-00850-7