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

Abstract

$(N(k),\xi )$-semi-Riemannian manifolds are defined. Examples and properties of $(N(k),\xi )$-semi-Riemannian manifolds are given. Some relations involving ${\mathcal{T}}_a$-curvature tensor in $(N(k),\xi )$-semi-Riemannian manifolds are proved. $\xi$-${\mathcal{T}}_a$-flat $(N(k),\xi )$-semi-Riemannian manifolds are defined. It is proved that if $M$ is an $n$-dimensional $\xi$-${\mathcal{T}}_a$-flat $(N(k),\xi )$-semi-Riemannian manifold, then it is $\eta$-Einstein under an algebraic condition. We prove that a semi-Riemannian manifold, which is $T$-recurrent or $T$-symmetric, is always $T$-semisymmetric, where $T$ is any tensor of type $(1,3)$. $({\mathcal{T}}_a,{\mathcal{T}}_b)$-semisymmetric semi-Riemannian manifold is defined and studied. The results for ${\mathcal{T}}_a$-semisymmetric, ${\mathcal{T}}_a$-symmetric, ${\mathcal{T}}_a$-recurrent $(N(k),\xi )$-semi-Riemannian manifolds are obtained. The definition of $({\mathcal{T}}_a,S_{{\mathcal{T}}_b})$-semisymmetric semi-Riemannian manifold is given. $({\mathcal{T}}_a,S_{{\mathcal{T}}_b})$-semisymmetric $(N(k),\xi )$-semi-Riemannian manifolds are classified. Some results for ${\mathcal{T}}_a$-Ricci-semisymmetric $(N(k),\xi )$-semi-Riemannian manifolds are obtained.