The topology of compact rank-one ECS manifolds

• Published in: Proceedings of the Edinburgh Mathematical Society 2023-08, Vol.66 (3), p.789-809
• Main Authors: Derdzinski, Andrzej, Terek, Ivo
• Format: Article
• Language: English
• Subjects: Neighborhoods; Riemann manifold; Tensors; Topology
• ISSN: 0013-0915; 1464-3839
• DOI: 10.1017/S0013091523000408

Pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as essentially conformally symmetric (ECS) manifolds, have a natural local invariant, the rank, which equals 1 or 2, and is the rank 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 twofold 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.