It is proved the non-existence of Hopf hypersurfaces in $G_{2}({Bbb C}^{m+2})$, $m geq 3$, whose normal Jacobi operator is semi-parallel, if the principal curvature of the Reeb vector field is non-vanishing and the component of the Reeb vector field
in the maximal quaternionic subbundle ${frak D}$ or its orthogonal complement ${frak D}^{bot}$ is invariant by the shape operator.
In this paper, we obtain some sufficient conditions for a 3-dimensional compact trans-Sasakian manifold of type $(alpha ,beta)$ to be homothetic to a Sasakian manifold. A characterization of a 3-dimensional cosymplectic manifold is also obtained.
Einstein like $(varepsilon)$-para Sasakian manifolds are introduced. For an $(varepsilon) $-para Sasakian manifold to be Einstein like, a necessary and sufficient condition in terms of its curvature tensor is obtained. The scalar curvature of an Eins
tein like $(varepsilon) $-para Sasakian manifold is obtained and it is shown that the scalar curvature in this case must satisfy certain differential equation. A necessary and sufficient condition for an $(varepsilon) $-almost paracontact metric hypersurface of an indefinite locally Riemannian product manifold to be $(varepsilon) $-para Sasakian is obtained and it is proved that the $(varepsilon) $-para Sasakian hypersurface of an indefinite locally Riemannian product manifold of almost constant curvature is always Einstein like.
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.
We find the index of $widetilde{ abla}$-quasi-conformally symmetric and $widetilde{ abla}$-concircularly symmetric semi-Riemannian manifolds, where $widetilde{ abla}$ is metric connection.
$(N(k),xi)$-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.
