Gradient Ricci almost solitons on two classes of almost Kenmotsu manifolds
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Let $(M^{2n+1},\phi,\xi,\eta,g)$ be a $(k,\mu)'$-almost Kenmotsu manifold with $k<-1$ which admits a gradient Ricci almost soliton $(g,f,\lambda)$, where $\lambda$ is the soliton function and $f$ is the potential function. In this paper, it is proved that $\lambda$ is a constant and this implies that $M^{2n+1}$ is locally isometric to a rigid gradient Ricci soliton $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^n$, and the soliton is expanding with $\lambda=-4n$. Moreover, if a three dimensional Kenmotsu manifold admits a gradient Ricci almost soliton, then either it is of constant sectional curvature $-1$ or the potential vector field is pointwise colinear with the Reeb vector field.
Abstract
Let $(M^{2n+1},\phi,\xi,\eta,g)$ be a $(k,\mu)'$-almost Kenmotsu manifold with $k<-1$ which admits a gradient Ricci almost soliton $(g,f,\lambda)$, where $\lambda$ is the soliton function and $f$ is the potential function. In this paper, it is proved that $\lambda$ is a constant and this implies that $M^{2n+1}$ is locally isometric to a rigid gradient Ricci soliton $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^n$, and the soliton is expanding with $\lambda=-4n$. Moreover, if a three dimensional Kenmotsu manifold admits a gradient Ricci almost soliton, then either it is of constant sectional curvature $-1$ or the potential vector field is pointwise colinear with the Reeb vector field.
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