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

Year 2024, Volume: 7 Issue: 2, 83 - 92, 23.05.2024

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

https://doi.org/10.32323/ujma.1418496

Cited By: 1

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

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