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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 (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·C-C·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·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:
53C50Lorentz manifolds, manifolds with indefinite metrics
53B30Lorentz metrics, indefinite metrics