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We construct new examples of compact ECS manifolds, that is, ofpseudo-Riemannian manifolds with parallel Weyl tensor that are neitherconformally flat nor locally symmetric. Every ECS manifold has rank 1 or 2, therank being the dimension of a distinguished null parallel distributiondiscovered by Olszak. Previously known examples of compact ECS manifolds, inevery dimension greater than 4, were all of rank 1, geodesically complete, andnone of them locally homogeneous. By contrast, our new examples -- all of themgeodesically incomplete -- realizing all odd dimensions starting from 5, arestill of rank 1, but this time locally homogeneous.

Compact locally homogeneous manifolds with parallel Weyl tensor