Z^-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure

The main goal of this manuscript is to investigate the properties of $N(k)$-contact metric manifolds admitting a Z^-tensor. We prove the necessary conditions for which $N(k)$-contact metric manifolds endowed with a Z^-tensor are Einstein manifolds. In this sequel, we accomplish that an $N(k)$-contact metric manifold endowed with a Z^-tensor satisfying Z⁽G₁,ζ̂)· R=0 is either locally isometric to the Riemannian product Eⁿ⁺¹(0)× Sⁿ(4) or an Einstein manifold. We also prove the condition for which an $N(k)$-contact metric manifold endowed with a Z^-tensor is a Sasakian manifold. To validate some of our results, we construct a non-trivial example of an $N(k)$-contact metric manifold.