The metric structure of compact rank-one ECS manifolds

Pseudo-Riemannian manifolds with nonzero parallel Weyl tensor which are not locally symmetric are known as ECS manifolds. Every ECS manifold carries a distinguished null parallel distribution \(\mathcal {D}\), the rank \(d\in \{1,2\}\) of which is referred to as the rank of the manifold itself. Under a natural genericity assumption on the Weyl tensor, we fully describe the universal coverings of compact rank-one ECS manifolds. We then show that any generic compact rank-one ECS manifold must be translational, in the sense that the holonomy group of the natural flat connection induced on \(\mathcal {D}\) is either trivial or isomorphic to \({\mathbb {Z}}_2\). We also prove that all four-dimensional rank-one ECS manifolds are noncompact, this time without having to assume genericity, as it is always the case in dimension four.