On the full holonomy group of special Lorentzian manifolds

TL;DR

This paper investigates the full holonomy groups of special Lorentzian manifolds, providing classifications, construction methods, and examples of manifolds with various holonomy properties, including disconnected holonomy and parallel spinors.

Contribution

It offers a comprehensive classification of full holonomy groups in Lorentzian geometry and introduces new construction techniques for manifolds with disconnected holonomy.

Findings

Classified full holonomy groups of solvable Lorentzian symmetric spaces

Developed a method to construct manifolds with disconnected holonomy from Riemannian manifolds

Provided examples of globally hyperbolic manifolds with parallel spinors and disconnected holonomy

Abstract

We study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. We prove several results that are based on the classification of the restricted holonomy groups of such manifolds and provide a construction method for manifolds with disconnected holonomy which starts from a Riemannian manifold and a properly discontinuous group of isometries. Most of our examples are quotients of pp-waves with disconnected holonomy and without parallel vector field. Furthermore, we classify the full holonomy groups of solvable Lorentzian symmetric spaces and of Lorentzian manifolds with a parallel null spinor. Finally, we construct examples of globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces, disconnected full holonomy and a parallel spinor.