Almost Kenmotsu manifolds with a condition of η-parallelism

We consider almost Kenmotsu manifolds (M2n+1,φ,ξ,η,g)(M2n+1,φ,ξ,η,g) with η  -parallel tensor h′=h○φh′=h○φ, 2h being the Lie derivative of the structure tensor φ with respect to the Reeb vector field ξ. We describe the Riemannian geometry of an integral submanifold of the distribution orthogonal to ξ, characterizing the CR  -integrability of the structure. Under the additional condition ∇ξh′=0∇ξh′=0, the almost Kenmotsu manifold is locally a warped product. Finally, some lightlike structures on M2n+1M2n+1 are introduced and studied.