Compact pseudo-Riemannian homogeneous Einstein manifolds of low dimension

TL;DR

This paper classifies low-dimensional compact pseudo-Riemannian Einstein manifolds with transitive Lie group actions, revealing that such spaces are either associated with nilpotent groups or are products of Einstein manifolds from semisimple groups.

Contribution

It provides a classification of low-dimensional homogeneous Einstein manifolds, showing that in certain cases they are related to nilpotent or semisimple Lie groups, and introduces conjectures linking to transcendental number theory.

Findings

Compact quotients exist only for nilpotent groups in dimensions ≤7.

Semisimple cases lead to products of Einstein manifolds.

Conjecture: all such compact quotients are nilpotent in all dimensions.

Abstract

Let $M$ be pseudo-Riemannian homogeneous Einstein manifold of finite volume, and suppose a connected Lie group $G$ acts transitively and isometrically on $M$. In this situation, the metric on $M$ induces a bilinear form $\langle\cdot,\cdot\rangle$ on the Lie algebra $\mathfrak{g}$ of $G$ which is nil-invariant, a property closely related to invariance. We study such spaces $M$ in three important cases. First, we assume $\langle\cdot,\cdot\rangle$ is invariant, in which case the Einstein property requires that $G$ is either solvable or semisimple. Next, we investigate the case where $G$ is solvable. Here, $M$ is compact and $M=G/\Gamma$ for a lattice $\Gamma$ in $G$. We show that in dimensions less or equal to $7$, compact quotients $M=G/\Gamma$ exist only for nilpotent groups $G$. We conjecture that this is true for any dimension. In fact, this holds if Schanuel's Conjecture on transcendental numbers is true. Finally, we consider semisimple Lie groups $G$, and find that $M$ is covered by a pseudo-Riemannian product of Einstein manifolds corresponding to the compact and the non-compact factors of $G$.