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

Abstract

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

arXiv admin note: text overlap with arXiv:1202.6138