The topology of compact rank-one ECS manifolds

• Published in: arXiv.org 2023-06
• Main Authors: Derdzinski, Andrzej, Terek, Ivo
• Format: Article
• Language: English
• Subjects: Manifolds (mathematics); Riemann manifold; Tensors; Topology
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2210.09195

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.