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Mathematische Zeitschrift

, Volume 277, Issue 3–4, pp 797–828 | Cite as

On the full holonomy group of Lorentzian manifolds

  • Helga Baum
  • Kordian Lärz
  • Thomas LeistnerEmail author
Article
First Online:

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 

Mathematics Subject Classification (2010)

Primary 53C29 53C50 Secondary 53C27 
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Notes

Acknowledgments

We thank the anonymous referees for valuable comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

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