Classification of homogeneous Einstein metrics on pseudo-hyperbolic spaces

TL;DR

This paper classifies homogeneous Einstein metrics on pseudo-hyperbolic spaces by analyzing Lie group actions, revealing that such metrics are either canonical or derived from Hopf pseudo-Riemannian submersions.

Contribution

It provides a complete classification of Einstein metrics on pseudo-hyperbolic spaces under specific Lie group action assumptions, extending previous results to various pseudo-Riemannian contexts.

Findings

Homogeneous Einstein metrics are homothetic to canonical or Hopf-derived metrics.

Effective and transitive Lie group actions are classified on hyperboloids.

Results apply to real, complex, quaternionic, and para-analogues of pseudo-hyperbolic spaces.

Abstract

We classify the effective and transitive actions of a Lie group $G$ on an n-dimensional non-degenerate hyperboloid (also called real pseudo-hyperbolic space), under the assumption that $G$ is a closed, connected Lie subgroup of $SO_0(n-r,r+1)$, the connected component of the indefinite special orthogonal group. Assuming additionally that $G$ acts completely reducible on $\mathbb R^{n+1}$, we also obtain that any $G$-homogeneous Einstein pseudo-Riemannian metric on a real, complex or quaternionic pseudo-hyperbolic space, or on a para-complex or para-quaternionic projective space is homothetic to either the canonical metric or the Einstein metric of the canonical variation of a Hopf pseudo-Riemannian submersion.