Rank-one ECS manifolds of dilational type

TL;DR

This paper investigates the structure and classification of rank-one ECS manifolds, especially focusing on generic compact examples, their translational or locally homogeneous nature, and their relation to explicit model manifolds.

Contribution

It establishes conditions under which generic compact rank-one ECS manifolds are translational or locally homogeneous, linking them to specific explicit model manifolds.

Findings

Generic compact rank-one ECS manifolds are either translational or locally homogeneous.

All known compact ECS manifolds of dimension greater than 4 are rank 1, translational, and non-homogeneous.

The paper connects these manifolds to explicit model manifolds whose universal covers include all such ECS manifolds.

Abstract

We study ECS manifolds, that is, pseudo-Riemannian manifolds with parallel Weyl tensor which are neither conformally flat nor locally symmetric. Every ECS manifold has rank 1 or 2, the rank being the dimension of a distinguished null parallel distribution discovered by Olszak, and a rank-one ECS manifold may be called translational or dilational, depending on whether the holonomy group of a natural flat connection in the Olszak distribution is finite or infinite. Some such manifolds are in a natural sense generic, which refers to the algebraic structure of the Weyl tensor. Known examples of compact ECS manifolds, in every dimension greater than 4, are all of rank 1 and translational, some of them generic, none of them locally homogeneous. As we show, generic compact rank-one ECS manifolds must be translational or locally homogeneous, provided that they arise as isometric quotients of a specific class of explicitly constructed "model" manifolds. This result is relevant since the clause starting with "provided that" may be dropped: according to a theorem which we prove in a forthcoming paper, the models just mentioned include the isometry types of the pseudo-Riemannian universal coverings of all generic compact rank-one ECS manifolds.