The harmonicity of the Reeb vector field with respect to Riemannian g-natural metrics

In this paper we show that a 3-dimensional non-Sasakian contact metric manifold [M,(η,ξ,ϕ,g)][M,(η,ξ,ϕ,g)] is a (κ,μ,ν)(κ,μ,ν)-contact metric manifold with ν=const.ν=const., if and only if there exists a Riemannian g  -natural metric G˜ on T1MT1M for which ξ:(M,g)↦(T1M,G˜) is a harmonic map. Furthermore, we give examples of 3-dimensional non-Sasakian contact metric manifolds [M,(η,ξ,ϕ,g)][M,(η,ξ,ϕ,g)] such that the corresponding Reeb vector fields ξ:(M,g)↦(T1M,G˜) are harmonic maps, for suitable Riemannian g  -natural metrics G˜ on T1MT1M which are not of Kaluza–Klein type. Finally, we prove that if (M,g)(M,g) is an Einstein manifold and (η˜,ξ˜,ϕ˜,G˜) a g  -natural contact metric structure on T1MT1M, then the contact metric manifold [T1M,(η˜,ξ˜,ϕ˜,G˜)] is H  -contact if and only if (M,g)(M,g) is 2-stein.