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
Abstract. In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann-Christoffel curvature tensor, the same type structures given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various geometric structures obtained by same restriction imposing on different curvature tensors. In this purpose we present a tensor by combining Riemann-Christoffel curvature tensor, Ricci tensor, the metric tensor and scalar curvature which describe various curvature tensors as its particular cases. Then with the help of this generalized tensor and using algebraic classification we prove the equivalency of different geometric structures (see, Theorem 6.3 -6.7, Table 2 and Table 3).Mathematics Subject Classication (2010). 53C15, 53C21, 53C25, 53C35.
The main aim of this paper is to investigate the geometric structures admitting by the Gödel spacetime which produces a new class of semi-Riemannian manifolds. We also consider some extension of Gödel metric.
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