Towards a classification of Lorentzian holonomy groups

TL;DR

This paper classifies Lorentzian holonomy groups, showing that under certain conditions, the Riemannian part of the holonomy must be a Riemannian holonomy group, advancing understanding of Lorentzian geometry.

Contribution

It provides a classification of Lorentzian holonomy groups with degenerate invariant subspaces, identifying conditions under which the Riemannian projection is a Riemannian holonomy group.

Findings

Holonomy groups with degenerate invariant subspaces are contained in a specific parabolic group.

Under certain conditions, the Riemannian projection of the holonomy group is a Riemannian holonomy group.

The classification depends on the properties of the SO(n)--projection G, such as being within U(n/2) or having simple irreducible components.

Abstract

If the holonomy representation of an $(n+2)$--dimensional simply-connected Lorentzian manifold $(M,h)$ admits a degenerate invariant subspace its holonomy group is contained in the parabolic group $(\mathbb{R} \times SO(n))\ltimes \mathbb{R}^n$. The main ingredient of such a holonomy group is the SO(n)--projection $G:=pr_{SO(n)}(Hol_p(M,h))$ and one may ask whether it has to be a Riemannian holonomy group. In this paper we show that this is the case if $G\subset U(n/2)$ or if the irreducible acting components of $G$ are simple.