A Classification of $ N(\kappa)$-contact Metric Manifolds with $ \mathcal{T} $-curvature Tensor

Year 2025, Volume: 17 Issue: 2, 550 - 561, 30.12.2025

İnan Ünal , Mustafa Altın , Shashıkant Pandey

https://doi.org/10.47000/tjmcs.1632911

Abstract

In this study, we consider various geometric conditions associated with the vanishing of a generalized curvature tensor, namely the $\mathcal{T}$-tensor, on $N(\kappa)$-contact metric manifolds. We define and analyze several types of $\mathcal{T}$-flatness conditions, including $\mathcal{T}$-flat, $\xi$-$\mathcal{T}$-flat, quasi-$\mathcal{T}$-flat, and $\varphi$-$\mathcal{T}$-flat structures. By applying these flatness conditions, we obtain algebraic constraints on the curvature parameters, particularly involving the Ricci tensor. The resulting characterizations allow for the classification of $N(\kappa)$-contact metric manifolds as either Einstein or $\eta$-Einstein, depending on the specific values of the structure constants. These investigations also lead to deeper insights into the geometric structure and local isometry types of the manifolds under study.

Keywords

$ N(\kappa)$- contact metric manifolds , $\mathcal{T}$-curvature tensor , $ \eta- $Einstein manifolds

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