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

TL;DR

This paper introduces and studies $(N(k),\xi)$-semi-Riemannian manifolds, exploring their properties, curvature relations, and various symmetry conditions, including semisymmetry, recurrence, and Ricci-semisymmetry, with classifications and algebraic conditions.

Contribution

It defines new classes of semi-Riemannian manifolds and investigates their geometric properties, relations, and classifications under various symmetry and curvature conditions.

Findings

$(N(k),\xi)$-semi-Riemannian manifolds are characterized and examples provided.

Conditions for $\xi$-${ mf T}_a$-flatness imply $\xi$-Einstein property.

Various symmetry conditions lead to classifications and algebraic characterizations.

Abstract

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