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.

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

The purpose of the present paper is to introduce the conformal semi-invariant $\xi ^{\perp }$-Riemannian submersions from almost contact metric manifolds onto Riemannian manifolds as a generalization of anti-invariant submersions, semi-invariant submersions, anti-invariant $\xi ^{\perp }$-submersions, semi-invariant $\xi ^{\perp }$% -submersions, conformal anti-invariant $\xi ^{\perp }$-submersions, conformal semi-invariant submersions. We obtain characterizations and investigate the integrability of distributions which are arisen from the definition of conformal semi-invariant $\xi ^{\perp }$-Riemannian submersions. We find out the necessary and sufficient conditions for a conformal semi-invariant $\xi ^{\perp }$-Riemannian submersions to be totally geodesic and harmonic. Finally, examples are also given for conformal semi-invariant submersions with horizontal Reeb vector field.

Shashikant Pandey

Papers

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

Conformal semi-invariant $\xi ^{\perp }$-submersions from almost contact metric manifolds