The harmonicity of the Reeb vector field on contact metric 3-manifolds

摘要

A contact metric manifold whose characteristic vector field is a harmonic vector field is called an H-contact metric manifold. We introduce the notion of (kappa, mu, nu)-contact metric manifolds in terms of a specific curvature condition. Then, we prove that a contact metric 3-manifold M is an H-contact metric manifold if and only if it is a (kappa, mu, nu)-contact metric manifold on an everywhere open and dense subset of M. Also, we prove that, for dimensions greater than three, such manifolds are reduced to (kappa, mu)-contact metric manifolds whereas, in three dimensions, (kappa, mu, nu)-contact metric manifolds exist.