These investigations also lead to deeper insights into the geometric structure and local isometry types of the manifolds under study.
📌 3 Key Takeaways
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Objective & Hypothesis: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.
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Key Findings & Data: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.
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Conclusion & Impact:These investigations also lead to deeper insights into the geometric structure and local isometry types of the manifolds under study.
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.
Ünal, İ., Altın, M., Pandey, S. (2025). A Classification of $ N(\kappa)$-contact Metric Manifolds with $ \mathcal{T} $-curvature Tensor. Turkish Journal of Mathematics and Computer Science, Volume: 17 Issue: 2. https://doi.org/10.47000/tjmcs.1632911
Barman, A., On N(k)-contact metric manifolds admitting a type of semi-symmetric non-metric connection, Acta Math. Univ. Comenianae, 86(1)(2017), 81–90.
Blair, D.E., Kim, J.S., Tripathi, M.M., On the concircular curvature tensor of a contact metric manifold, J. Korean Math. Soc., 42(2005), 883–892.
Blair, D.E., Koufogiorgos, T., Papantoniou, B.J., Contact metric manifolds satisfying a nullity condition, Israel J. Math., 91(1–3)(1995), 189–214.
Boeckx, E., A full classification of contact metric (k, μ)-spaces, Illinois J. Math., 44(1)(2000), 212–219.
De, U.C., Certain results on N(k)-contact metric manifolds, Tamkang J. Math., 49(3)(2018), 205–220.
De, U.C., Ghosh, S., E-Bochner curvature tensor on N(k)-contact metric manifolds, Hacet. J. Math. Stat., 43(3)(2014), 365–374.
De, U.C., Yildiz, A., Ghosh, S., On a class of N(k)-contact metric manifolds, Math. Reports, 16(2014), 66–74.
Ingalahalli, G., Anil, S., Bagewadi, C., Certain results on N(k)-contact metric manifold, Asian J. Math. Comput. Res., (2019), 123–130.
Ishii, Y., On conharmonic transformations, Tensor N.S., 7(1957), 73–80.
Majhi, P., De, U.C., Classifications of N(k)-contact metric manifolds satisfying certain curvature conditions, Acta Math. Univ. Comenianae, 84(1)(2015), 167–178.
Pokhariyal, G.P., Mishra, R.S., Curvature tensors and their relativistic significance (II), Tensor, 22(1971), 41–48.
Pokhariyal, G.P., Mishra, R.S., Curvature tensors and their relativistic significance, Tensor, 21(1970), 254–260.
Pokhariyal, G.P., Relativistic significance of curvature tensors, Int. J. Math. Math. Sci., 5(1)(1982), 133–139.
Prasad, B., A pseudo-projective curvature tensor on a Riemannian manifold, Bull. Cal. Math. Soc., 94(2002), 163–166.
Tripathi, M.M., Gupta, P., On T-curvature tensor in K-contact and Sasakian manifolds, Int. Electron. J. Geom., 4(1)(2011), 32–47.
Tripathi, M.M., Gupta, P., T-curvature tensor on a semi-Riemannian manifold, J. Adv. Math. Stud., 4(1)(2011), 117–129.
Ünal, İ., Altin, M., N(κ)-contact metric manifolds with generalized Tanaka-Webster connection, Filomat, 35(4)(2021), 1383–1392.
Yano, K., Curvature and Betti Numbers, Princeton University Press, 2016.
Yano, K., Sawaki, S., Riemannian manifolds admitting a conformal transformation group, J. Differential Geom., 2(2)(1968), 161–184.
Yildiz, A., De, U.C., Murathan, C., Arslan, K., On the Weyl projective curvature tensor of an N(k)-contact metric manifold, Math. Panonica, 21(1)(2010), 129–142.