In several papers the authors and some of their collaborators published already a series of results concerning pseudo-Riemannian manifolds satisfying some “pseudo-symmetry” curvature condition. In this paper they continue this research. Let
be a pseudo-Riemannian manifold,
its curvature tensor and
the corresponding Weyl tensor. They study manifolds
and the tensor
, which they defined in earlier papers, are linearly dependent at any point of
act as derivations. Their main result is that such manifolds must be semi-symmetric, i.e.
, a condition which provided the starting point of their research on this type of conditions. Furthermore, they provide some examples of semi-symmetric warped products which satisfy the relation mentioned above and which illustrate their search for a possible inverse of their main result.