Naturally Harmonic Vector Fields

Abstract

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 g^s, the only vector fields defining harmonic maps from (M,g) to (TM,g^s) 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 ĝ^s the Sasaki metric on T₁M (the one induced on it by ĝ^s), Han and Yim characterized unit vector fields which define harmonic maps from (M,g) to (T₁M, ĝ^s). The variational problem related to the energy restricted to unit vector fields, E : X¹(M)→ ℝ, V ↦ 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.