Recent developments in pseudo-Riemannian holonomy theory

Abstract

We review recent results in the theory of holonomy groups of pseudo-Riemannian manifolds, i.e., manifolds with indefinite metrics. First we present the classification of Lorentzian holonomy groups, that is a list of possible groups and metrics which realise these as holonomy groups. This is followed by applications and some remarks about holonomy related structures on Lorentzian manifolds. Then we review partial results in signature (2, n + 2), in particular the classification of unitary holonomy groups, again presenting the groups and the realising metrics. Then we turn to results in neutral signature (n, n), focussing on the situation of a para-Kähler structure. Finally, the classification in signature (2, 2) obtained by Bérard-Bergery and Ikemakhen is presented. As a new result we prove the existence of metrics in cases for which the realisation as holonomy group was left open in their article.