Translator Disclaimer
may 2022 A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold
Marisa Fernández, Marco Freibert, Jonatan Sánchez
Bull. Belg. Math. Soc. Simon Stevin 28(4): 487-511 (may 2022). DOI: 10.36045/j.bbms.210210

Abstract

We answer in the affirmative the question posed by Conti and Rossi [7,8] on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ of signature $(3, 4)$ on a nilpotent Lie group of dimension 7, such that $g$ is Einstein and not Ricci-flat. We show that the pseudo-metric $g$ cannot be induced by any left-invariant closed $G_2^*$-structure on the Lie group. Moreover, some results on closed and harmonic $G_2^*$-structures on an arbitrary 7-manifold $M$ are given. In particular, we prove that the underlying pseudo-Riemannian metric of a closed and harmonic $G_2^*$-structure on $M$ is not necessarily Einstein, but if it is Einstein then it is Ricci-flat.

Citation

Download Citation

Marisa Fernández. Marco Freibert. Jonatan Sánchez. "A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold." Bull. Belg. Math. Soc. Simon Stevin 28 (4) 487 - 511, may 2022. https://doi.org/10.36045/j.bbms.210210

Information

Published: may 2022
First available in Project Euclid: 11 May 2022

Digital Object Identifier: 10.36045/j.bbms.210210

Subjects:
Primary: 22E25 , 53C15 , 53C25 , 53C50

Keywords: $G_2^*$-structures , Einstein pseudo-Riemannian metrics , nilpotent Lie groups , Ricci curvature

Rights: Copyright © 2022 The Belgian Mathematical Society

JOURNAL ARTICLE
25 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.28 • No. 4 • may 2022
Back to Top