Completeness of compact Lorentzian manifolds with Abelian holonomy

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.

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