Naturally Harmonic Vector Fields

This paper is a survey on recent results obtained in collaboration with M.T.K. Abbassi and D. Perrone. Let (M,g) be a compact Riemannian manifold. If we equip the tangent bundle TM with the Sasaki metric gs, the only vector fields defining harmonic maps from (M,g) to (TM,gs) are the parallel ones, as Nouhaud and Ishihara proved independently. The Sasaki metric is just a particular example of Riemannian g-natural metric. Equipping TM with an arbitrary Riemannian g-natural metric G and investigating the harmonicity of a vector field V of M, thought as a map from (M,g) to (TM,G), several interesting behaviours are found. If V is a unit vector field, then it also defines a smooth map from M to the unit tangent sphere bundle T1M. Being T1M an hypersurface of TM, any Riemannian metric on TM induces one on the unit tangent sphere bundle. Denoted by ĝs the Sasaki metric on T1M (the one induced on it by ĝs), Han and Yim characterized unit vector fields which define harmonic maps from (M,g) to (T1M, ĝs). The variational problem related to the energy restricted to unit vector fields, E : X1(M)→ &#x211D, V &#x21A6 E(V), has been studied by Wood in [18]. We equipped T1M with an arbitrary Riemannian metric Ĝ induced by a Riemannian g-natural metric G on TM, and we studied harmonicity properties of the map V : (M,g) → (T1M, Ĝ) corresponding to a unit vector field