  This article studies the harmonicity of vector fields on Riemannian
manifolds, viewed as maps into the tangent bundle equipped with a family of
Riemannian metrics. Geometric and topological rigidity conditions are obtained,
especially for surfaces and vector fields of constant norm, and existence is
proved on two-tori. Classifications are given for conformal, quadratic and
Killing vector fields on spheres. Finally, the class of metric considered on
the tangent bundle is enlarged, permitting new vector fields to become
harmonic.
