Some results on almost Kenmotsu manifolds

Abstract

First we consider almost Kenmotsu manifolds which satisfy Codazzi condition for $h$ and $\varphi h$, and we prove that in such cases the tensor $h$ vanishes. Next, we prove that an almost Kenmotsu manifold having constant $\xi$-sectional curvature $K$ which is locally symmetric is a Kenmotsu manifold of constant curvature $K=-1$. We also prove that, for a $(\kappa,\mu)'$-almost Kenmotsu manifold of $dim>3$ with $h'\neq 0$, every conformal vector field is Killing. Finally, we prove that if $M$ is a $(\kappa,\mu)'$-almost Kenmotsu manifold with $h'\neq 0$ and $\kappa \neq -2$, then the vector field $V$ which leaves the curvature tensor invariant is Killing.