Conformal vector fields on almost Kenmotsu manifolds

In this paper, first we consider that the conformal vector field $\mathbf{X}$ is identical with the Reeb vector field $\varsigma$ and next, assume that $\mathbf{X}$ is pointwise collinear with %the Reeb vector field $\varsigma$, in both cases it is shown that the manifold ${\mathbf{N}}^{2m+1}$ becomes a Kenmotsu manifold and ${\mathbf{N}}^{2m+1}$ is locally a warped product ${\mathbf{N}}^{\mathrm{\prime }}{\times }_f{\mathbf{M}}^{2m}$, where ${\mathbf{M}}^{2m}$ is an almost Kähler manifold, ${\mathbf{N}}^{\mathrm{\prime }}$ is an open interval with coordinate t, and $f=c{e}^t$ for some positive constant c. Beside these, we prove that if a $(\mathtt{\text{k}},\mathit{\mu }{)}^{\mathrm{\prime }}$-almost Kenmotsu manifold admits a Killing vector field $\mathbf{X}$, then either it is locally a warped product of an almost Kähler manifold and an open interval or $\mathbf{X}$ is a strict infinitesimal contact transformation. Furthermore, we also investigate $\mathit{\eta }$-Ricci-Yamabe soliton with conformal vector fields on $(\mathtt{\text{k}},\mathit{\mu }{)}^{\mathrm{\prime }}$-almost Kenmotsu manifolds and finally, we construct an example.