Global properties of indefinite metrics with parallel Weyl tensor

TL;DR

This paper reviews recent results on ECS manifolds, a class of indefinite pseudo-Riemannian manifolds with parallel Weyl tensor, including classification theorems and existence in higher dimensions.

Contribution

It presents classification theorems for ECS manifolds and demonstrates the existence of compact ECS manifolds in infinitely many dimensions above 4.

Findings

Classification theorems for ECS manifolds

Existence of compact ECS manifolds in infinitely many dimensions

Discussion of properties and open questions about compact ECS manifolds

Abstract

This is an exposition of some recent results on ECS manifolds, by which we mean pseudo-Riemannian manifolds of dimensions greater than 3 that are neither conformally flat nor locally symmetric, and have parallel Weyl tensor. All ECS metrics are indefinite. We state two classification theorems, describing the local structure of ECS manifolds, and outline an argument showing that compact ECS manifolds exist in infinitely many dimensions greater than 4. We also discuss some properties of compact manifolds that admit ECS metrics, and provide a list of open questions about compact ECS manifolds.