Locally homogeneous pp-waves

Abstract

We show that every n-dimensional locally homogeneous pp-wave is a plane wave, provided it is indecomposable and its curvature operator, when acting on $2$-forms, has rank greater than one. As a consequence we obtain that indecomposable, Ricci-flat locally homogeneous pp-waves are plane waves. This generalises a classical result by Jordan, Ehlers and Kundt in dimension 4. Several examples show that our assumptions on indecomposability and the rank of the curvature are essential.

arXiv commentIn v2 Example 4.3 is added which shows that in the main theorems the assumption on the rank of the curvature is essential

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Cited by in corpus (5)

• Completeness of compact Lorentzian manifolds with Abelian holonomy
• Exact Parallel Waves in General Relativity
• On homogeneous plane waves
• Parallel spinors on globally hyperbolic Lorentzian four-manifolds
• On completeness of foliated structures, and null Killing fields