  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
which 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 and, in particular, to Hopf vector fields on
odd-dimensional spheres.
