Almost Kenmotsu manifolds with a condition of eta-parallelism

We consider almost Kenmotsu manifolds (M^{2n+1}, phi, xi, eta, g) with eta-parallel tensor h' = h o phi, 2h being the Lie derivative of the structure tensor phi with respect to the Reeb vector field xi. We describe the Riemannian geometry of an integral submanifold of the distribution orthogonal to xi, characterizing the CR-integrability of the structure. Under the additional condition nabla_xi h'=0, the almost Kenmotsu manifold is locally a warped product. Finally, some lightlike structures on M^{2n+1} are introduced and studied.