Contact metric manifolds whose characteristic vector field is a harmonic vector field

In this paper, contact metric manifolds whose characteristic vector field ξ is a harmonic vector field are called H-contact manifolds. We show that a (2n + 1)-dimensional contact metric manifold is an H-contact manifold if and only if ξ is an eigenvector of the Ricci operator. Consequently, the class of H-contact manifolds is very large: η-Einstein contact metric manifolds, K-contact manifolds (which we characterize in terms of the rough Laplacian), (k,μ)-spaces and strongly locally φ-symmetric spaces are H-contact manifolds. Then, we give some results on the topology of a compact H-contact manifold. In particular, using a Geiges’ result, we obtain that a compact three-manifold admits an H-contact structure with critical metric for the Chern–Hamilton energy functional if and only if it is diffeomorphic to a left quotient of the Lie group G under a discrete subgroup, where G is one of SU(2), H^3 (the Heisenberg group), or \tilde SL(2,R).