Completeness of compact Lorentzian manifolds with Abelian holonomy
| Autor: | Leistner, Thomas, Schliebner, Daniel |
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| Rok vydání: | 2013 |
| Předmět: | |
| Zdroj: | Mathematische Annalen, April 2016, Volume 364, Issue 3, pp 1469-1503 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00208-015-1270-4 |
| Popis: | 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. %the condition that their curvature sends vectors that are orthogonal to the vector field to a multiple of the vector field. 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. Comment: 30 pages, comments welcome; version 2 revised, references and a new result about compact, Ricci-flat pp-waves added. Version 3 is substantially revised with new title. We added Corollary 2 about completeness of indecomposable, compact locally symmetric Lorentzian manifolds |
| Databáze: | arXiv |
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