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In this paper, we present a classification of $ N(\kappa)$-contact metricmanifolds 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)$-contactmetric 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 $ andRicci curvature tensor $ S $. Thus, we obtain a classification of $N(\kappa)$-contact metric manifolds.

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