Let (M, g) be a Riemannian manifold. When M is compact and the tangent bundle TM is equipped with the Sasaki metric g^s, the only vector fields that define harmonic maps from (M, g) to (TM, g^s) are the parallel ones. The Sasaki metric and other well-known Riemannian metrics on TM are particular examples of g-natural metrics. We equip TM with an arbitrary Riemannian g-natural metric G and investigate the harmonicity of a vector field V of M, thought as a map from (M, g) to (TM, G). We then apply this study to the Reeb vector field of a contact metric manifold and, in particular, to Hopf vector fields on odd-dimensional spheres.

We prove that all g-natural contact metric structures on a two-point homogeneous space are homogeneous contact. The converse is also proved for metrics of Kaluza–Klein type. We also show that if (M,g) is an Einstein manifold and is a Riemannian g-natural metric on T1M of Kaluza–Klein type, then is H-contact if and only if (M,g) is 2-stein, so proving that the main result of Chun et al. [‘H-contact unit tangent sphere bundles of Einstein manifolds’, Q. J. Math., to appear. DOI: 10.1093/qmath/hap025] is invariant under a two-parameter deformation of the standard contact metric structure on T1M. Moreover, we completely characterize Riemannian manifolds admitting two distinct H-contact g-natural contact metric structures, with associated metric of Kaluza–Klein type.