  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.
