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.

Project Euclid - mathematics and statistics online