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On some curvature conditions of pseudosymmetry type. (English) Zbl 1374.53030

Summary: It is known that the difference tensor \(R\cdot C-C\cdot R\) and the Tachibana tensor \(Q(S,C)\) of any semi-Riemannian Einstein manifold \((M, g)\) of dimension \(n \geq 4\) are linearly dependent at every point of \(M\). More precisely \(R\cdot C -C \cdot R = (1/(n-1)) Q(S,C)\) holds on \(M\). In the paper we show that there are quasi-Einstein, as well as non-quasi-Einstein semi-Riemannian manifolds for which the above mentioned tensors are linearly dependent. For instance, we prove that every non-locally symmetric and non-conformally flat manifold with parallel Weyl tensor (essentially conformally symmetric manifold) satisfies \(R\cdot C = C \cdot R = Q(S,C) = 0\). Manifolds with parallel Weyl tensor having Ricci tensor of rank two form a subclass of the class of Roter type manifolds. Therefore we also investigate Roter type manifolds for which the tensors \(R \cdot C - C \cdot R\) and \(Q(S,C)\) are linearly dependent. We determine necessary and sufficient conditions for a Roter type manifold to be a manifold having that property.

MSC:

53B20 Local Riemannian geometry
53B25 Local submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53B50 Applications of local differential geometry to the sciences
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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