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