Warped product manifolds with p-dimensional base, p=1,2, satisfy some curvature conditions of pseudosymmetry type. 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 fibre is a semi-Riemannian space of constant curvature, if n is greater or equal to 4, 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 expressed by a linear combination of some Kulkarni-Nomizu products formed from the tensors g and S. Thus these curvature conditions satisfy non-conformally flat non-Einstein warped product spacetimes (p=2, n=4). We also investigate curvature properties of pseudosymmetry type of quasi-Einstein manifolds. In particular, we obtain some curvature property of the Goedel spacetime

The difference tensor R • C − C • R of a semi-Riemannian manifold (M, g), dim M ≥ 4, formed by its Riemann-Christoffel curvature tensor R and the Weyl conformal curvature tensor C, under some assumptions, can be expressed as a linear combination of (0, 6)-Tachibana tensors Q(A, T ), where A is a symmetric (0, 2)-tensor and T a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions. 1

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In the literature, there are two different notions of pseudosymmetric manifolds, one by Chaki [7] and other by Deszcz [16], and there are many papers related to these notions. The object of the present paper is to deduce necessary and sufficient conditions for a Chaki pseudosymmetric [7] (resp. pseudo Ricci symmetric [8]) manifold to be Deszcz pseudosymmetric (resp. Ricci pseudosymmetric). We also study the necessary and sufficient conditions for a weakly symmetric [58] (resp. weakly Ricci symmetric [59]) manifold by Tam\'assy and Binh to be Deszcz pseudosymmetric (resp. Ricci pseudosymmetric). We also obtain the reduced form of the defining condition of weakly Ricci symmetric manifolds by Tam\'assy and Binh [59]. Finally we give some examples to show the independent existence of such types of pseudosymmetry which also ensure the existence of Roter type and generalized Roter type manifolds and the manifolds with recurrent curvature 2-form ([2], [29]) associated to various curvature tensors.Comment: 32 page