Curvature properties of some class of warped product manifolds

Abstract

We prove that warped product manifolds with [Formula: see text]-dimensional base, [Formula: see text] satisfy some pseudosymmetry type curvature conditions. These conditions are formed from the metric tensor [Formula: see text], the Riemann–Christoffel curvature tensor [Formula: see text], the Ricci tensor [Formula: see text] and the Weyl conformal curvature [Formula: see text] of the considered manifolds. The main result of the paper states that if [Formula: see text] and the fiber is a semi-Riemannian space of constant curvature (when [Formula: see text] is greater or equal to 5) then the [Formula: see text]-tensors [Formula: see text] and [Formula: see text] of such warped products are proportional to the [Formula: see text]-tensor [Formula: see text] and the tensor [Formula: see text] is a linear combination of some Kulkarni–Nomizu products formed from the tensors [Formula: see text] and [Formula: see text]. 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.