On some pseudosymmetry type curvature condition. (Q1411733)

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 \((M, g)\) be a pseudo-Riemannian manifold, \(R\) its curvature tensor and \(C\) the corresponding Weyl tensor. They study manifolds \((M,g)\) such that \(R\cdot C-C\cdot R\) and the tensor \(Q(g,R)\), which they defined in earlier papers, are linearly dependent at any point of \(M\). Here \(R\) and \(C\) act as derivations. Their main result is that such manifolds must be semi-symmetric, i.e. \(R\cdot R= 0\), 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.