On $(N(k),xi)$-semi-Riemannian manifolds: Pseudosymmetries

Definition of $({cal T}_{a},{cal T}_{b})$-pseudosymmetric semi-Riemannian manifold is given. $({cal T}_{a},{cal T}_{b})$-pseudosy mmetric $(N(k),xi)$-semi-Riemannian manifolds are classified. Some results for ${cal T}_{a}$-pseudosymmetric $(N(k),xi)$-semi-Riemannian manifolds are obtained. $({cal T}_{a},{cal T}_{b},S^{ell})$-pseudosymmetric semi-Riemannian manifolds are defined. $({cal T}_{a},{cal T}_{b},S^{ell})$-pseudosymmetric $(N(k),xi)$-semi-Riemannian manifolds are classified. Some results for $(R,{cal T}_{a},S^{ell})$-pseudosymmetric $(N(k),xi)$-semi-Riemannian manifolds are obtained. In particular, some results for $(R,{cal T}_{a},S)$-pseudosymmetric $(N(k),xi)$-semi-Riemannian manifolds are also obtained. After that, the definition of $({cal T}_{a},S_{{cal T}_{b}})$-pseudosymmetric semi-Riemannian manifold is given. $({cal T}_{a},S_{{cal T}_{b}})$-pseudosymmetric $(N(k),xi)$-semi-Riemannian manifolds are classified. It is proved that a $(R,S_{{cal T}_{a}})$-pseudosymmetric $(N(k),xi)$-semi-Riemannian manifold is either Einstein or $L=k$ under an algebraic condition. Some results for $({cal T}_{a},S)$-pseudosymmetric $(N(k),xi)$-semi-Riemannian manifolds are also obtained. In last, $({cal T}_{a},S_{{cal T}_{b}},S^{ell})$-pseudosymmetric semi-Riemannian manifolds are defined and $({cal T}_{a},S_{{cal T}_{b}},S^{ell})$ -pseudosymmetric $(N(k),xi)$-semi-Riemannian manifolds are classified.