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Zbl 1047.53040
Deszcz, Ryszard; Hotlós, Marian
On some pseudosymmetry type curvature condition. (English)
[J] Tsukuba J. Math. 27, No. 1, 13-30 (2003). ISSN 0387-4982

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.
[L. Vanhecke (Leuven)]

MSC 2000:: *53C50 Lorentz manifolds, manifolds with indefinite metrics
53B30 Lorentz metrics, indefinite metrics

Keywords: curvature tensor; Weyl tensor; semi-symmetric; warped product

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