Holonomy algebras of Einstein pseudo-Riemannian manifolds

TL;DR

This paper classifies the holonomy algebras of Einstein pseudo-Riemannian manifolds with arbitrary signature, providing explicit examples and covering various special cases such as Lorentzian and para-quaternionic-Kählerian manifolds.

Contribution

It offers a comprehensive classification of holonomy algebras for Einstein non Ricci-flat pseudo-Riemannian manifolds across different signatures, including explicit metric examples.

Findings

Classification of holonomy algebras for Einstein non Ricci-flat manifolds

Explicit examples of Einstein metrics with various holonomy algebras

Analysis of special cases like Lorentzian and para-quaternionic-Kählerian manifolds

Abstract

The holonomy algebras of Einstein not Ricci-flat pseudo-Riemannian manifolds of arbitrary signature are classified. As illustrating examples, the cases of Lorentzian manifolds, pseudo-Riemannian manifolds of signature $(2,n)$ and the para-quaternionic-K\"ahlerian manifolds with non-zero scalar curvature are considered. Einstein not Ricci-flat metrics of signature $(2,n)$ with all possible holonomy algebras are given.