Compact Weyl-parallel manifolds

TL;DR

This paper studies compact ECS manifolds, which are pseudo-Riemannian with parallel Weyl tensor but not conformally flat or locally symmetric, revealing their structure as bundles over the circle in certain cases.

Contribution

It proves that non-locally homogeneous compact rank-one ECS manifolds are, up to a cover, fiber bundles over the circle, advancing understanding of their global structure.

Findings

Existence of compact ECS manifolds for all dimensions n ≥ 5.

Compact rank-one ECS manifolds are either locally homogeneous or fiber bundles over the circle.

Non-homogeneous examples are characterized as total spaces of certain bundles.

Abstract

By ECS manifolds one means pseudo-Riemannian manifolds of dimensions $\,n\ge4\,$ which have parallel Weyl tensor, but not for one of the two obvious reasons: conformal flatness or local symmetry. As shown by Roter [10, 2], they exist for every $\,n\ge4$, and their metrics are always indefinite. The local structure of ECS manifolds has been completely described [3]. Every ECS manifold has an invariant called rank, equal to 1 or 2. Known examples of compact ECS manifolds [4, 6], representing every dimension $\,n\ge5$, are of rank 1. When $\,n\,$ is odd, some further, recently found examples are locally homogeneous [7]. We outline the proof of the author's result, joint with Ivo Terek [5], which states that a compact rank-one ECS manifold, if not locally homogeneous, replaced if necessary by a two-fold isometric covering, must be the total space of a bundle over the circle.