Harmonic maps and sections on spheres

TL;DR

This paper explores harmonic sections and maps on spheres, demonstrating non-existence results, rigidity, classification of gradient fields, and introducing new metrics that enable harmonic Hopf vector fields.

Contribution

It introduces a family of metrics on spheres that admit harmonic Hopf vector fields, extending understanding of harmonic maps and vector fields on spherical manifolds.

Findings

Harmonic Killing vector fields do not exist on S^2.

Rigidity results for harmonic gradient vector fields on S^2.

A family of metrics making Hopf vector fields harmonic maps on S^{2n+1}.

Abstract

The absence of interesting harmonic sections for the Sasaki and Cheeger-Gromoll metrics has led to the consideration of alternatives, for example in the form of a two-parameter family of natural metrics shown to relax existence conditions for harmonicity. This article investigates harmonic Killing vector fields, proves their non-existence on S^2, obtains rigidity results for harmonic gradient vector fields on the two-sphere, classifies spherical quadratic gradient fields in all dimensions and determines the tension field, concluding with the discovery of a family of metrics making Hopf vector fields harmonic maps on S^{2n+1}.