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On some pseudosymmetry type curvature condition. (English) Zbl 1047.53040
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 $\left(M,g\right)$ be a pseudo-Riemannian manifold, $R$ its curvature tensor and $C$ the corresponding Weyl tensor. They study manifolds $\left(M,g\right)$ such that $R·C-C·R$ and the tensor $Q\left(g,R\right)$, 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. $R·R=0$, 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.
##### MSC:
 53C50 Lorentz manifolds, manifolds with indefinite metrics 53B30 Lorentz metrics, indefinite metrics
##### Keywords:
curvature tensor; Weyl tensor; semi-symmetric; warped product