Harmonicity of unit vector fields with respect to Riemannian g-natural metrics

Let (M,g)(M,g) be a compact Riemannian manifold and T1MT1M its unit tangent sphere bundle. Unit vector fields defining harmonic maps from (M,g)(M,g) to (T1M,g˜s), g˜s being the Sasaki metric on T1MT1M, have been extensively studied. The Sasaki metric, and other well known Riemannian metrics on T1MT1M, are particular examples of g  -natural metrics. We equip T1MT1M with an arbitrary Riemannian g  -natural metric G˜, and investigate the harmonicity of a unit vector field V of M  , thought as a map from (M,g)(M,g) to (T1M,G˜). We then apply this study to characterize unit Killing vector fields and to investigate harmonicity properties of the Reeb vector field of a contact metric manifold.