The harmonicity of the Reeb vector field with respect to Riemannian \(g\)-natural metrics (Q428921)

It is proved that a 3-dimensional non-Sasakian contact metric manifold \(M(\varphi,\xi,\eta,g)\) is \((\kappa,\mu,\nu)\)-contact with \(\nu= {\text{constant}}\) if and only if there exists a Riemannian \(g\)-natural metric \(\tilde G\) on the unit tangent sphere bundle \(T_1M\) for which \((\ast)\;\xi:(M,g)\to(T_1M,\tilde G)\) is a harmonic map. If \((M,g)\) is Einstein and \((\tilde\varphi,\tilde\xi,\tilde\eta,\tilde G)\) a \(g\)-natural contact metric structure on \(T_1M\), then the contact metric manifold \(T_1M(\tilde\varphi,\tilde\xi,\tilde\eta,\tilde G)\) is \(H\)-contact (i.e., \((\ast)\) holds) if and only if \((M,g)\) is 2-stein. Examples are constructed.