Curvature properties of some class of warped product manifolds
Résumé
We prove that warped product manifolds with p-dimensional base, p=1,2, satisfy some pseudosymmetry type curvature conditions. These conditions are formed from the metric tensor g, the Riemann–Christoffel curvature tensor R, the Ricci tensor S and the Weyl conformal curvature C of the considered manifolds. The main result of the paper states that if p=2 and the fiber is a semi-Riemannian space of constant curvature (when n is greater or equal to 5) then the (0,6)-tensors R⋅R−Q(S,R) and C⋅C of such warped products are proportional to the (0,6)-tensor Q(g,C) and the tensor C is a linear combination of some Kulkarni–Nomizu products formed from the tensors g and S. We also obtain curvature properties of this kind of quasi-Einstein and 2-quasi-Einstein manifolds, and in particular, of the Goedel metric, generalized spherically symmetric metrics and generalized Vaidya metrics.
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