Locally homogeneous pp-waves

TL;DR

This paper proves that under certain conditions, all locally homogeneous pp-waves are plane waves, extending classical results to higher dimensions and Ricci-flat cases, with examples illustrating the necessity of these conditions.

Contribution

It generalizes the classical 4-dimensional result to higher dimensions, establishing that indecomposable, locally homogeneous pp-waves with certain curvature properties are necessarily plane waves.

Findings

Indecomposable, locally homogeneous pp-waves are plane waves under specified conditions.

Ricci-flat, locally homogeneous pp-waves are necessarily plane waves.

Examples demonstrate the importance of the assumptions on indecomposability and curvature rank.

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.