Ünal, İnan; Altın, Mustafa; Pandey, Shashikant

In this study, we consider various geometric conditions associated with the vanishing of a generalized curvature tensor, namely the T-tensor, on N(κ)-contact metric manifolds. We define and analyze several types of T-flatness conditions, including T-flat, ξ-T-flat, quasi-T-flat, and φ-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(κ)-contact metric manifolds as either Einstein or η-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.

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