The topology of compact rank-one ECS manifolds

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 know...

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Bibliographic Details
Published in:arXiv.org 2023-06
Main Authors: Derdzinski, Andrzej, Terek, Ivo
Format: Article
Language:English
Subjects:
Manifolds (mathematics)
Riemann manifold
Tensors
Topology
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2210.09195