Towards a classification of Lorentzian holonomy groups

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.