A classification of $ N(\kappa)$-contact metric manifolds with $ \mathcal{T} $-curvature tensor

TL;DR

This paper classifies N(κ)-contact metric manifolds based on special flatness conditions of the T-curvature tensor, expanding understanding of their geometric structure under various curvature constraints.

Contribution

It introduces a new classification of N(κ)-contact metric manifolds using flatness conditions on the T-curvature tensor and related curvature conditions.

Findings

Classification of N(κ)-contact metric manifolds under T-flatness conditions

Identification of conditions T(ξ,X).R=0 and T(ξ,X).S=0

Extension of geometric understanding of contact metric manifolds

Abstract

In this paper, we present a classification of $ N(\kappa)$-contact metric manifolds with using some special flatness conditions on $ \mathcal{T} $-curvature tensor. We examine $\mathcal{T}$-flat, quasi-$\mathcal{T}$-flat, $ \xi$-$\mathcal{T}$-flat and $ \varphi $-$\mathcal{T}$-flat $ N(\kappa)$-contact metric manifolds .Also,we consider the conditions $ \mathcal{T}(\xi,X).R=0 $ and $ \mathcal{T}(\xi,X).S=0 $ for the Riemannian curvature tensor $ R $ and Ricci curvature tensor $ S $. Thus, we obtain a classification of $ N(\kappa)$-contact metric manifolds.