Mathematische Annalen

, Volume 364, Issue 3–4, pp 1469–1503 | Cite as

Completeness of compact Lorentzian manifolds with abelian holonomy

  • Thomas LeistnerEmail author
  • Daniel Schliebner


We address the problem of finding conditions under which a compact Lorentzian manifold is geodesically complete, a property, which always holds for compact Riemannian manifolds. It is known that a compact Lorentzian manifold is geodesically complete if it is homogeneous, or has constant curvature, or admits a time-like conformal vector field. We consider certain Lorentzian manifolds with abelian holonomy, which are locally modelled by the so called pp-waves, and which, in general, do not satisfy any of the above conditions. We show that compact pp-waves are universally covered by a vector space, determine the metric on the universal cover, and prove that they are geodesically complete. Using this, we show that every Ricci-flat compact pp-wave is a plane wave.

Mathematics Subject Classification

53C50 53C29 53C12 53C22 



We would like to thank Helga Baum and Miguel Sánchez for helpful discussions and comments on the first draft of the paper. We also thank the referees for valuable comments and Ines Kath for alerting us to the implications our result has for locally symmetric spaces.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

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

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