Recent Developments in Pseudo-Riemannian Geometry

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pp: 53–96

DOI: 10.4171/051-1/2

Holonomy groups of Lorentzian manifolds: classification, examples, and applications

Anton Galaev[1] and Thomas Leistner

(1) Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37, Brno, Czechia

We review recent developments in the theory of Lorentzian holonomy groups focussing on the classification results. We present the list of indecomposable, non-irreducible Lorentzian holonomy groups, explain the idea of its proof, and describe a method of constructing metrics which realise all the possible groups. This method is then used to construct many examples of metrics. Finally, we give some applications for the existence of parallel spinors and a short outlook on other signatures. As a new result we obtain the holonomy classification for indecomposable, non-irreducible Lorentzian Einstein spaces.

Keywords: Holonomy groups, Lorentzian manifolds, parallel spinors, pp-waves