The topology of compact rank-one ECS manifolds

TL;DR

This paper characterizes the global topology of compact rank-one ECS manifolds, showing they are fiber bundles over a circle with specific geometric structures, extending understanding of their global geometry.

Contribution

It proves that non-homogeneous compact rank-one ECS manifolds are fiber bundles over the circle, and describes their universal cover as a product space, advancing classification of these manifolds.

Findings

Compact rank-one ECS manifolds are bundles over the circle.

Universal covers decompose as a product space.

Results extend to locally homogeneous cases under certain conditions.

Abstract

Pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as ECS manifolds, have a natural local invariant, the rank, which equals 1 or 2, and is the dimension of a certain distinguished null parallel distribution $\,\mathcal{D}$. All known examples of compact ECS manifolds are of rank one and have dimensions greater than 4. We prove that a compact rank-one ECS manifold, if not locally homogeneous, replaced when necessary by a two-fold isometric covering, must be a bundle over the circle with leaves of $\,\mathcal{D}^\perp$ serving as the fibres. The same conclusion holds in the locally-homogeneous case if one assumes that $\,\mathcal{D}^\perp$ has at least one compact leaf. We also show that in the pseudo-Riemannian universal covering space of any compact rank-one ECS manifold the leaves of $\,\mathcal{D}^\perp$ are the factor manifolds of a global product decomposition.