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Type: Journal article
Title: Completeness of compact Lorentzian manifolds with abelian holonomy
Author: Leistner, T.
Schliebner, D.
Citation: Mathematische Annalen, 2016; 364(3-4):1469-1503
Publisher: Springer-Verlag
Issue Date: 2016
ISSN: 0025-5831
Statement of
Thomas Leistner, Daniel Schliebner
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 timelike 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.
Rights: © Springer-Verlag Berlin Heidelberg 2015
RMID: 0030033381
DOI: 10.1007/s00208-015-1270-4
Grant ID:
Appears in Collections:Mathematical Sciences publications

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