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Type: Journal article
Title: On the full holonomy group of Lorentzian manifolds
Author: Baum, H.
Lärz, K.
Leistner, T.
Citation: Mathematische Zeitschrift, 2014; 277(3-4):797-828
Publisher: Springer-Verlag
Issue Date: 2014
ISSN: 0025-5874
Statement of
Helga Baum, Kordian Lärz, Thomas Leistner
Abstract: The classification of restricted holonomy groups of n-dimensional Lorentzian manifolds was obtained about ten years ago. However, up to now, not much is known about the structure of the full holonomy group. In this paper we study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. Based on the classification of the restricted holonomy groups of such manifolds, we prove several structure results about the full holonomy. We establish a construction method for manifolds with disconnected holonomy starting from a Riemannian manifold and a properly discontinuous group of isometries. This leads to a variety of examples, most of them being quotients of pp-waves with disconnected holonomy, including a non-flat Lorentzian manifold with infinitely generated holonomy group. 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.
Keywords: Lorentzian manifolds; Holonomy groups; Isometry groups; Parallel spinor fields Globally hyperbolic manifolds, pp-waves
Rights: © Springer-Verlag Berlin Heidelberg 2014
RMID: 0030008035
DOI: 10.1007/s00209-014-1279-5
Grant ID:
Appears in Collections:Mathematical Sciences publications

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