Compact locally homogeneous manifolds with parallel Weyl tensor

Abstract

We construct new examples of compact ECS manifolds, that is, of pseudo-Riemannian manifolds with parallel Weyl tensor that 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. Previously known examples of compact ECS manifolds, in every dimension greater than 4, were all of rank 1, geodesically complete, and none of them locally homogeneous. By contrast, our new examples -- all of them geodesically incomplete -- realize all odd dimensions starting from 5 and are this time of rank 2, as well as locally homogeneous.