Almost Kenmotsu manifolds with conformal Reeb foliation

Abstract

We consider almost Kenmotsu manifolds with conformal Reeb foliation. We prove that such a foliation produces harmonic morphisms, we study the $k$-nullity distributions and we discuss the isometrical immersion of such a manifold $M$ as hypersurface in a real space form $\widetilde{M}(c)$ of constant curvature $c$ proving that $c \leq -1$ and, if $c<-1$, $M$ is totally umbilical, Kenmotsu and locally isometric to the hyperbolic space of constant curvature $-1$. Finally, the Einstein and $\eta$-Einstein conditions are discussed.