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Warped products realizing a certain condition of pseudosymmetry type imposed on the Weyl curvature tensor. (English) Zbl 0817.53008
Let $$(M,g)$$ be a semi-Riemannian manifold, $$\widetilde{R}$$ the curvature operator, $$R = g \circ \widetilde{R}$$ the Riemann-Christoffel curvature tensor and $$\Gamma$$ the operator defined by $$\Gamma_{XY} = g(Y,Z) X - g(X,Z)Y$$. Let then be $$R \cdot R$$ and $$Q(g,R)$$ the tensor fields given by $$(R\cdot R) = D_{\widetilde{R}_{XY}}(R)$$, $$(Q(g,R))_{XY} = - D_{\Gamma_{XY}}(R)$$ where $$D_{\widetilde{R}_{XY}}$$ and $$D_{\Gamma_{XY}}$$ are the derivatives defined by $$R_{XY}$$ and $$\Gamma_{XY}$$ in the tensor algebra of $$M$$. The manifold $$(M,g)$$ is called $$R$$-pseudosymmetric if $$R \cdot R$$ and $$Q(g,R)$$ are linearly dependent at every point of $$M$$. The authors study the warped product manifolds which are $$C$$-pseudosymmetric, where $$C$$ is the Weyl conformal curvature tensor. The 3-dimensional semi-Riemannian manifolds which are $$R$$-pseudo-symmetric are analyzed, too.
Reviewer: V.Cruceanu (Iaşi)

##### MSC:
 53B20 Local Riemannian geometry 53A45 Differential geometric aspects in vector and tensor analysis
##### Keywords:
pseudosymmetric manifold; warped product manifolds