Holonomy of Einstein Lorentzian manifolds

TL;DR

This paper classifies all holonomy algebras of Einstein and vacuum Einstein Lorentzian manifolds, providing explicit constructions and analyzing Ricci-isotropic cases, thereby advancing the understanding of Lorentzian holonomy structures.

Contribution

It offers a complete classification of holonomy algebras for Einstein and vacuum Einstein Lorentzian manifolds, including explicit examples and curvature tensor descriptions.

Findings

All Einstein and vacuum Einstein holonomy algebras are realizable.

Holonomy algebras of Ricci-isotropic Lorentzian manifolds are classified.

Complete description of curvature tensor spaces for these holonomies.

Abstract

The classification of all possible holonomy algebras of Einstein and vacuum Einstein Lorentzian manifolds is obtained. It is shown that each such algebra appears as the holonomy algebra of an Einstein (resp., vacuum Einstein) Lorentzian manifold, the direct constructions are given. Also the holonomy algebras of totally Ricci-isotropic Lorentzian manifolds are classified. The classification of the holonomy algebras of Lorentzian manifolds is reviewed and a complete description of the spaces of curvature tensors for these holonomies is given.