Abstract
The harmonic map equations for unit vector fields are discussed for the particular case of Reeb fields in contact metric geometry. It is always a harmonic vector field and it is a harmonic map under appropriate conditions on the coefficients determining the fixed g-natural metric, allowing one to exhibit large families of harmonic maps defined on a compact Riemannian manifold and having a target space with a highly nontrivial geometry. Vertical harmonicity with respect to Riemannian g-natural metrics appears to be worth further investigation. In the theory of harmonic maps, a fundamental question concerns the existence of harmonic maps between two given Riemannian manifolds. It is therefore important to find examples of harmonic maps into Riemannian manifolds whose sectional curvature is not necessarily non-positive.
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