Harmonic maps and Kaluza-Klein metrics on spheres

TL;DR

This paper investigates harmonic vector fields on Riemannian manifolds with Kaluza-Klein metrics, providing rigidity results, classifications on spheres, and demonstrating the existence of harmonic fields on tori.

Contribution

It introduces new classifications of harmonic vector fields on spheres and extends the class of metrics, enabling more vector fields to be harmonic.

Findings

Rigidity conditions for harmonic vector fields on surfaces.

Classification of harmonic fields on spheres.

Existence of harmonic vector fields on two-tori.

Abstract

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.