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.

We present some corrections of a part of the original paper [5, Sec. 4, lines 14 9-15 9 ]. In addition, we give examples of manifolds related to the presented corrections. Further, we denote by S and κ the Ricci tensor and the scalar curvature of the product manifold (M × N , g = g × g), respectively. It is obvious that rank S = rank S and κ = κ on U C ⊂ M × N. Now [5, Eq. (24)] yields on this set rank S − κ n − 1 − (n − 2)L C g ≤ 2.

The projective curvature tensor is an invariant under geodesic preserving transformations on semi-Riemannian manifolds. It possesses different geometric properties than other generalized curvature tensors. The main object of the present paper is to study some semisymmetric type and pseudosymmetric type curvature restricted geometric structures due to projective curvature tensor. The reduced pseudosymmetric type structures for various Walker type conditions are deduced and the existence of Venzi space is ensured. It is shown that the geometric structures formed by imposing projective operator on a (0,4)-tensor is different from that for the corresponding (1,3)-tensor. Characterization of various semisymmetric type and pseudosymmetric type curvature restricted geometric structures due to projective curvature tensor are obtained on semi-Riemannian manifolds, and it is shown that some of them reduce to Einstein manifolds for the Riemannian case. Finally, to support our theorems, four suitable examples are presented.