Conformal vector fields on almost Kenmotsu manifolds

Abstract

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 $\mathbf{N}' \times_{f} \mathbf{M}^{2m}$, where $\mathbf{M}^{2m}$ is an almost K\"ahler manifold, $\mathbf{N}'$ is an open interval with coordinate t, and $f = ce^{t}$ for some positive constant c. Beside these, we prove that if a $(\verb"k",\boldsymbol{\mu})'$-almost Kenmotsu manifold admits a Killing vector field $\mathbf{X}$, then either it is locally a warped product of an almost K\"ahler manifold and an open interval or $\mathbf{X}$ is a strict infinitesimal contact transformation. Furthermore, we also investigate $\boldsymbol{\eta}$-Ricci-Yamabe soliton with conformal vector fields on $(\verb"k",\boldsymbol{\mu})'$-almost Kenmotsu manifolds and finally, we construct an example.