$(N(k),xi)$-semi-Riemannian manifolds are defined. Examples and properties of $(N(k),xi)$-semi-Riemannian manifolds are given. Some relations involving ${cal T}_{a}$-curvature tensor in $(N(k),xi)$-semi-Riemannian manifolds are proved. $xi $-${cal T}_{a}$-flat $(N(k),xi)$-semi-Riemannian manifolds are defined. It is proved that if $M$ is an $n$-dimensional $xi $-${cal 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 $(1,3)$. $({cal T}_{a}, {cal T}_{b}) $-semisymmetric semi-Riemannian manifold is defined and studied. The results for ${cal T}_{a}$-semisymmetric, ${cal T}_{a}$-symmetric, ${cal T}_{a}$-recurrent $(N(k),xi)$-semi-Riemannian manifolds are obtained. The definition of $({cal T}_{a},S_{{cal T}_{b}})$-semisymmetric semi-Riemannian manifold is given. $({cal T}_{a},S_{{cal T}_{b}})$-semisymmetric $(N(k),xi)$-semi-Riemannian manifolds are classified. Some results for ${cal T}_{a}$-Ricci-semisymmetric $(N(k),xi)$-semi-Riemannian manifolds are obtained.
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