The topology of compact rank-one ECS manifolds

Abstract

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.

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Cited by in corpus (4)

• New examples of compact Weyl-parallel manifolds
• The metric structure of compact rank-one ECS manifolds
• Compact locally homogeneous manifolds with parallel Weyl tensor
• Rank-one ECS manifolds of dilational type