Almost Kenmotsu manifolds admitting certain vector fields

Abstract

The object of the present paper is to characterize almost Kenmotsu manifolds admitting holomorphically planar conformal vector (in short, HPCV) fields. It is shown that an almost Kenmotsu manifold $M^{2n+1}$ admitting a non-zero HPCV field $V$ such that $V$ is pointwise collinear with the Reeb vector field $xi$ is locally a warped product of an almost Kaehler manifold and an open interval. Further, if an almost Kenmotsu manifold with constant $xi$-sectional curvature admits a non-zero HPCV field $V$, then $M^{2n+1}$ is locally a warped product of an almost Kaehler manifold and an open interval. Moreover, a $(k,mu)'$-almost Kenmotsu manifold admitting a HPCV field $V$ such that $phi V neq 0$ is either locally isometric to $mathbb{H}^{n+1}(-4)$ $times$ $mathbb{R}^n$ or $V$ is an eigenvector of $h'$.