Almost Kenmotsu manifolds and local symmetry

We consider locally symmetric almost Kenmotsu manifolds showing that such a manifold is a Kenmotsu manifold if and only if the Lie derivative of the structure, with respect to the Reeb vector field ξ, vanishes. Furthermore, assuming that for a (2n + 1)-dimensional locally symmetric almost Kenmotsu manifold such Lie derivative does not vanish and the curvature satisfies RXY ξ = 0 for any X,Y orthogonal to ξ, we prove that the manifold is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant curvature −4 and a flat n-dimensional manifold. We give an example of such a manifold. Introduction An almost contact structure on a differentiable manifold M is given by a tensor field φ of type (1, 1), a vector field ξ and a 1-form η satisfying φ = − I + η⊗ ξ and η(ξ) = 1, which imply that φ(ξ) = 0 and η ◦ φ = 0. Furthermore, on the product manifold M × R one can define an almost complex structure J by J (