$\\mathcal{Z^\\ast}$-Tensor on $N(k)$-Contact Metric Manifolds Admitting Ricci Soliton Type Structure

Authors : Abhishek Singh, S. K. Chaubey, Sunil Yadav, Shraddha Patel

Pages : 83-92

Doi:10.32323/ujma.1418496

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Publication Date : 2024-05-23

Article Type : Research

Abstract :The main goal of this manuscript is to investigate the properties of $N(k)$-contact metric manifolds admitting a $\\mathcal{Z^\\ast}$-tensor. We prove the necessary conditions for which $N(k)$-contact metric manifolds endowed with a $\\mathcal{Z^\\ast}$-tensor are Einstein manifolds. In this sequel, we accomplish that an $N(k)$-contact metric manifold endowed with a $\\mathcal{Z^\\ast}$-tensor satisfying $\\mathcal{Z^\\ast}(\\mathcal{G}_{1},\\hat{\\zeta})\\cdot \\mathcal{\\overset{\\star}R}=0$ is either locally isometric to the Riemannian product $E^{n+1}(0)\\times S^{n}(4)$ or an Einstein manifold. We also prove the condition for which an $N(k)$-contact metric manifold endowed with a $\\mathcal{Z^\\ast}$-tensor is a Sasakian manifold. To validate some of our results, we construct a non-trivial example of an $N(k)$-contact metric manifold.
Keywords : $N(k)$-contact metric manifold, Einstein manifold, Ricci soliton, $\\mathcal{Z}^\\star$-recurrent, $\\mathcal{Z^\\ast}$-tensor