Almost Kenmotsu manifolds with a condition of η-parallelism

Abstract

We consider almost Kenmotsu manifolds ( M 2 n + 1 , φ , ξ , η , g ) with η-parallel tensor h ′ = h ○ φ , 2 h 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 ∇ ξ h ′ = 0 , the almost Kenmotsu manifold is locally a warped product. Finally, some lightlike structures on M 2 n + 1 are introduced and studied.