On $3$-dimensional $left(varepsilon right)$-para Sasakian manifold

The purpose of the present paper is to study the globally and locally $varphi $-${cal T}$-symmetric $left( varepsilon right) $-para Sasakian manifold in dimension $3$. The globally $varphi $-$ {cal T}$-symmetric $3$-dimensional $left( varepsilon right) $-para Sasakian manifold is either Einstein manifold or has a constant scalar curvature. The necessary and sufficient condition for Einstein manifold to be globally $varphi $-${cal T}$ -symmetric is given. A $3$-dimensional $% left( varepsilon right) $ -para Sasakian manifold is locally $varphi $-$ {cal T}$-symmetric if and only if the scalar curvature $r$ is constant. A $3 $-dimensional $left( varepsilon right) $-para Sasakian manifold with $% eta $-parallel Ricci tensor is locally $varphi $-${cal T}$-symmetric. In the last, an example of $3$-dimensional locally $varphi $-${cal T}$-symmetric $left( varepsilon right) $-para Sasakian manifold is given.

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.

Index of quasi-conformally symmetric semi-Riemannian manifolds

We find the index of $widetilde{ abla}$-quasi-conformally symmetric and $widetilde{ abla}$-concircularly symmetric semi-Riemannian manifolds, where $widetilde{ abla}$ is metric connection.

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

$(N(k),xi)$-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.