Mathematische Annalen

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

Completeness of compact Lorentzian manifolds with abelian holonomy

Article

Abstract

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 

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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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