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Register a new userIndex of quasi-conformally symmetric semi-Riemannian manifolds
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We find the index of $widetilde{ abla}$-quasi-conformally symmetric and $widetilde{ abla}$-concircularly symmetric semi-Riemannian manifolds, where $widetilde{ abla}$ is metric connection.
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We introduce a class of null hypersurfaces of a semi-Riemannian manifold, namely, screen quasi-conformal hypersurfaces, whose geometry may be studied through the geometry of its screen distribution. In particular, this notion allows us to extend some results of previous works to the case in which the sectional curvature of the ambient space is different from zero. As applications, we study umbilical, isoparametric and Einstein null hypersurfaces in Lorentzian space forms and provide several classification results.
We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a $pp$-wave or a warped product.
In this paper, we prove that the deformed Riemannian extension of any affine Szabo manifold is a Szabo pseudo-Riemannian metric and vice-versa. We proved that the Ricci tensor of an affine surface is skew-symmetric and nonzero everywhere if and only if the affine surface is Szabo. We also find the necessary and sufficient condition for the affine Szabo surface to be recurrent. We prove that for an affine Szabo recurrent surface the recurrence covector of a recurrence tensor is not locally a gradient.
$(N(k),xi)$-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.
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.
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