### 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 T_{1}M. Being T_{1}M 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 T_{1}M (the one induced on it by ĝ^{s}), Han and Yim characterized unit vector fields which define harmonic maps from (M,g) to (T_{1}M, ĝ^{s}). The variational problem related to the energy restricted to unit vector fields, E : X^{1}(M)→ ℝ, V ↦ E(V), has been studied by Wood in [18]. We equipped T_{1}M 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) → (T_{1}M, Ĝ) corresponding to a unit vector field.DOI Code:
10.1285/i15900932v28n1supplp107

Keywords:
tangent bundle; harmonic vector fields;

*g*-natural metrics; Reeb vector fieldClassification:
53C43; 53C07; 53C15

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