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
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
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