  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}.
