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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)

53B20 Local Riemannian geometry
53A45 Differential geometric aspects in vector and tensor analysis