# Ricci-flat and Einstein pseudoriemannian nilmanifolds

## TL;DR

This paper reviews pseudoriemannian Einstein metrics on nilpotent Lie groups, introduces a new criterion for diagonal Einstein metrics, and classifies Ricci-flat metrics in dimensions up to 8, highlighting open questions.

## Contribution

It provides a new criterion for diagonal Einstein metrics on nice nilpotent Lie groups and classifies Ricci-flat metrics in low dimensions.

## Key findings

- New criterion for diagonal Einstein metrics
- Classification of Ricci-flat metrics up to dimension 8
- Presentation of open questions in the field

## Abstract

This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-flat metrics on nilpotent Lie groups of dimension $\leq8$ are obtained. Some related open questions are presented.