Compact plane waves with parallel Weyl curvature

TL;DR

This paper reviews recent findings on essentially conformally symmetric pseudo-Riemannian manifolds with parallel Weyl curvature, focusing on their structure, examples, and classification, especially in the rank-one case.

Contribution

It summarizes recent joint work on the structure, construction, and classification of compact rank-one ECS manifolds, highlighting their bundle structure over the circle.

Findings

Pseudo-Riemannian ECS manifolds exist in all dimensions ≥ 4.

Every compact rank-one ECS manifold is a bundle over S^1 with specific fiber structure.

Classification results for compact rank-one ECS manifolds are discussed.

Abstract

This is an exposition of recent results -- obtained in joint work with Andrzej Derdzinski -- on essentially conformally symmetric (ECS) manifolds, that is, those pseudo-Riemannian manifolds with parallel Weyl curvature which are not locally symmetric or conformally flat. In the 1970s, Roter proved that while Riemannian ECS manifolds do not exist, pseudo-Riemannian ones do exist in all dimensions $n\geq 4$, and realize all indefinite metric signatures. The local structure of ECS manifolds is known, and every ECS manifold carries a distinguished null parallel distribution $\mathcal{D}$, whose rank is always equal to $1$ or $2$. We review basic facts about ECS manifolds, briefly discuss the construction of compact examples, and outline the proof of a topological structure result: outside of the locally homogeneous case and up to a double covering, every compact rank-one ECS manifold is a bundle over $\mathbb{S}^1$ whose fibers are the leaves of $\mathcal{D}^\perp$. Finally, we mention some classification results for compact rank-one ECS manifolds.