The metric structure of compact rank-one ECS manifolds

Abstract

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 assuming genericity, as it is always the case in dimension four.

References updated, typos fixed, and other stylistic improvements

References in corpus (3)

• The topology of compact rank-one ECS manifolds
• New examples of compact Weyl-parallel manifolds
• Rank-one ECS manifolds of dilational type

Cited by in corpus (2)

• Compact locally homogeneous manifolds with parallel Weyl tensor
• Rank-one ECS manifolds of dilational type