A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold

TL;DR

This paper constructs a 7-dimensional nilpotent Lie group with a non-Ricci-flat Einstein pseudo-metric of signature (3,4), answering a question about the existence of such metrics and exploring related $G_2^*$-structures.

Contribution

It provides the first explicit example of a non-Ricci-flat Einstein pseudo-metric on a 7-dimensional nilpotent Lie group and analyzes the properties of $G_2^*$-structures in this context.

Findings

Existence of a 7D nilpotent Lie group with a non-Ricci-flat Einstein pseudo-metric.

The constructed pseudo-metric cannot be induced by any closed $G_2^*$-structure.

If a closed and harmonic $G_2^*$-structure's metric is Einstein, then it must be Ricci-flat.

Abstract

We answer in the affirmative the question posed by Conti and Rossi 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.