Completeness of compact Lorentzian manifolds with Abelian holonomy

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. %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.

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

References in corpus (7)

• On the full holonomy group of special Lorentzian manifolds
• Completeness of the Trajectories of Particles Coupled to a General Force Field
• Locally homogeneous pp-waves
• Lattices of oscillator groups
• On the Full Holonomy of Lorentzian Manifolds with Parallel Weyl Tensor
• Remarks on the completeness of trajectories of accelerated particles in Riemannian manifolds and plane waves
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Cited by in corpus (15)

• Locally homogeneous pp-waves
• Randers pp-waves
• Exact Parallel Waves in General Relativity
• Hyperbolic evolution equations, Lorentzian holonomy, and Riemannian generalised Killing spinors
• Ehlers-Kundt Conjecture about Gravitational Waves and Dynamical Systems
• New examples of compact Weyl-parallel manifolds
• The metric structure of compact rank-one ECS manifolds
• On homogeneous plane waves
• Completeness of certain compact Lorentzian locally symmetric spaces
• Parallel spinors on globally hyperbolic Lorentzian four-manifolds
• Recurrent Lorentzian Weyl spaces
• On completeness of foliated structures, and null Killing fields
• Lorentzian connections with parallel twistor-free torsion
• On the splitting problem for Lorentzian manifolds with an $\mathbb{R}$-action with causal orbits
• Affine transformations and parallel lightlike vector fields on compact Lorentzian $3$-manifolds