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.
We report on the screening of samples of titanium metal for their radio-purity. The screening process described in this work led to the selection of materials used in the construction of the cryostats for the Large Underground Xenon (LUX) dark matter
experiment. Our measurements establish titanium as a highly desirable material for low background experiments searching for rare events. The sample with the lowest total long-lived activity was measured to contain <0.25 mBq/kg of U-238, <0.2 mBq/kg of Th-232, and <1.2 mBq/kg of K-40. Measurements of several samples also indicated the presence of short-lived (84 day half life) Sc-46, likely produced cosmogenically via muon initiated (n,p) reactions.
