HARMONIC SECTIONS OF TANGENT BUNDLES EQUIPPED WITH RIEMANNIAN g-NATURAL METRICS

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.