Rank-one ECS manifolds of dilational type

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. Various examples of compact rank-one ECS manifolds are known: translational ones (both generic and nongeneric) in every dimension $n\ge 5$, as well as odd-dimensional nongeneric dilational ones, some of which are 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 another paper, the models just mentioned include the isometry types of the pseudo-Riemannian universal coverings of all generic compact rank-one ECS manifolds. Consequently, all generic compact rank-one ECS manifolds are translational.