Harmonic maps and Kaluza-Klein metrics on spheres

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.