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

TL;DR

This paper introduces and classifies various types of pseudosymmetry conditions in $(N(k),\xi)$-semi-Riemannian manifolds, expanding the understanding of their geometric properties and relationships.

Contribution

It defines new pseudosymmetry concepts in semi-Riemannian manifolds and provides classifications and results, including conditions under which manifolds are Einstein or satisfy specific algebraic relations.

Findings

Classified $(N(k),\xi)$-semi-Riemannian manifolds with various pseudosymmetry conditions.

Proved that certain pseudosymmetric manifolds are either Einstein or satisfy $L=k$.

Established relationships between different pseudosymmetry types and their geometric implications.

Abstract

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.