Towards a classification of Lorentzian holonomy groups. Part II: Semisimple, non-simple weak-Berger algebras

Abstract

The holonomy group of an (n+2)-dimensional simply-connected, indecomposable but non-irreducible Lorentzian manifold (M,h) is contained in the parabolic group $(\mathbb{R}\times SO(n))⋉{\mathbb{R}}^n$. The main ingredient of such a holonomy group is the SO(n)--projection $G:=p{r}_{SO(n)}(Ho{l}_p(M,h))$ and one may ask whether it has to be a Riemannian holonomy group. In this paper we show that this is always the case, completing our results of the first part math.DG/0305139. We draw consequences for the existence of parallel spinors on Lorentzian manifolds.

13 pages

References in corpus (1)

• The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds

Cited by in corpus (11)

• Metrics that realize all Lorentzian holonomy algebras
• Screen bundles of Lorentzian manifolds and some generalisations of pp-waves
• The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds
• Parallal Spinors on Pseudo-Riemannian SpinC Manifolds
• Holonomy groups and special geometric structures of pseudo-Kählerian manifolds of index 2
• Lightlike foliations on Lorentzian manifolds with weakly irreducible holonomy algebra
• Classification of connected holonomy groups of pseudo-Kählerian manifolds of index 2
• Irreducibly acting subgroups of $Gl(n,\rr)$
• Holonomy representations which are a diagonal direct sum of two faithful representations
• On the Holonomy of Kaluza-Klein metrics
• Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds