Abstract
This article studies the harmonicity of vector fields on Rie- mannian manifolds, viewed as maps into the tangent bundle equipped with a family of Riemannian metrics. Geometric and topological rigid- ity 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. Fi- nally, the class of metric considered on the tangent bundle is enlarged, permitting new vector fields to become harmonic.
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