Locally ϕϕ-symmetric normal almost contact metric manifolds of dimension 3

The object of the present work is to study a three-dimensional normal almost contact metric manifold which is locally ϕϕ-symmetric and whose Ricci tensor is ηη-parallel. First we obtain a necessary and sufficient condition for a three-dimensional normal almost contact metric manifold to be locally ϕϕ-symmetric. Next we prove that in such a manifold with α,β=constantα,β=constant if the Ricci tensor is ηη-parallel, then the manifold is locally ϕϕ-symmetric. Finally, an example of a three-dimensional locally ϕϕ-symmetric normal almost contact metric manifold is given.