On locally homogeneous pseudo-Riemannian compact einstein manifolds

TL;DR

This paper investigates the classification of locally homogeneous compact pseudo-Riemannian Einstein manifolds, showing that standard compact Clifford-Klein forms admit Einstein metrics and proposing a conjecture on their uniqueness.

Contribution

It demonstrates that standard compact Clifford-Klein forms of simple non-compact Lie groups admit Einstein metrics and formulates a conjecture on the exclusivity of such manifolds.

Findings

Standard compact Clifford-Klein forms admit Einstein metrics

Conjecture that these are the only locally homogeneous Einstein pseudo-Riemannian compact manifolds

Provides a new perspective on the structure of such manifolds

Abstract

We ask a general question: what are locally homogeneous compact pseudo-Riemannian Einstein manifolds? We show that any standard compact Clifford-Klein form of a simple non-compact Lie group admits at least one Einstein metric. We conjecture that these are basically the only possible locally homogeneous Einstein pseudo-Riemannian compact manifolds using T. Kobayashi's conjecture as a guiding principle.