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Completeness of compact Lorentzian manifolds with abelian holonomy

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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.

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Notes

  1. In fact, in [12] Carrière proved a much more general result for affine manifolds. A direct proof for the flat case was given in [53]. However, this proof has gaps as it was pointed out in [41].

  2. In the following we will consider compact manifolds of this type. We are aware that for compact manifolds the term wave might not be appropriate, but we use this term since it is established in the literature for manifolds with the given curvature properties. Later we will see that an appropriate name would be screen flat, but this term has other obvious problems.

  3. We thank Wolfgang Globke for pointing us to this reference.

  4. One can also argue in the following way: From Proposition 5 we know that \(\widetilde{V}\) is a parallel vector field on the Riemannian manifold \((\widetilde{\mathcal{N}},\hat{h}) \) but also that \(\widetilde{V}\) as a lift of the complete vector field V is complete. Hence, as \(\widetilde{\mathcal{N}}\) is simply connected, the flow of \(\widetilde{V}\) separates a line \(\mathbb {R}\) from \(\widetilde{\mathcal{N}}\) with orthogonal complement being the leaves \(\hat{\mathcal{S}}\) of the integrable distribution \(\hat{\mathbb {S}}\), again proving the lemma.

  5. In fact, during the preparation of the paper we learned that Lemma 8 follows from stronger results by Candela et al. [10, Theorems 1 and 2]. However, for the sake of being self-contained we include a proof of the lemma. For further results and comments see [11, 46].

  6. In [34] we called these Lorentzian manifolds pr-waves for plane fronted with recurrent rays.

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Acknowledgments

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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Authors and Affiliations

  1. School of Mathematical Sciences, University of Adelaide, Adelaide, SA, 5005, Australia

    Thomas Leistner

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Daniel Schliebner

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  1. Thomas Leistner
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Correspondence to Thomas Leistner.

Additional information

This work was supported by the Group of Eight Australia and the German Academic Exchange Service through the Go8-DAAD Joint Research Co-operation Scheme. T. Leistner acknowledges support from the Australian Research Council via the Grants FT110100429 and DP120104582. D. Schliebner was funded by the Berlin Mathematical School.

Dedicated to Helga Baum on the occasion of her 60th birthday.

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Leistner, T., Schliebner, D. Completeness of compact Lorentzian manifolds with abelian holonomy. Math. Ann. 364, 1469–1503 (2016). https://doi.org/10.1007/s00208-015-1270-4

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