Bulletin of the
Korean Mathematical Society BKMS

The goal of this paper is to analyze the generalized $m$-quasi-Einstein structure in the context of almost Kenmotsu manifolds. Firstly we showed that a complete Kenmotsu manifold admitting a generalized $m$-quasi-Einstein structure $(g,f,m,\lambda)$ is locally isometric to a hyperbolic space $\mathbb{H}^{2n+1}(-1)$ or a warped product $\widetilde{M}\times_\gamma\mathbb{R}$ under certain conditions. Next, we proved that a $(\kappa,\mu)'$-almost Kenmotsu manifold with $h'\neq0$ admitting a closed generalized $m$-quasi-Einstein metric is locally isometric to some warped product spaces. Finally, a generalized $m$-quasi-Einstein metric $(g,f,m,\lambda)$ in almost Kenmotsu 3-H-manifold is considered and proved that either it is locally isometric to the hyperbolic space $\mathbb{H}^3(-1)$ or the Riemannian product $\mathbb{H}^2(-4)\times\mathbb{R}$.

Keywords: Hyperbolic space, quasi-Einstein, Kenmotsu manifold, Einstein, warped product

MSC numbers: Primary 53C15, 53C25, 53D10

Supported by: The first author is thankful to the Department of Science
and Technology, New Delhi, India for financial support in the form of INSPIRE
Fellowship (DST/INSPIRE Fellowship/2018/IF180830).