Harmonic Vector Fields on Space Forms

TL;DR

This paper studies harmonic vector fields on space forms, classifying harmonic conformal and quadratic gradient fields, and providing explicit examples on hyperbolic planes and spheres.

Contribution

It introduces a new framework for harmonic vector fields via generalized Cheeger-Gromoll metrics and classifies harmonic conformal and quadratic gradient fields on space forms.

Findings

Examples of harmonic conformal vector fields on hyperbolic plane

Classification of harmonic Killing and conformal gradient fields

Complete classification of harmonic quadratic gradient fields on spheres

Abstract

A vector field s on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised Cheeger-Gromoll metrics on TM with respect to which s is a harmonic section. If M is a simply-connected non-flat space form other than the 2-sphere, examples are obtained of conformal vector fields that are harmonic. In particular, the harmonic Killing fields and conformal gradient fields are classified, a loop of non-congruent harmonic conformal fields on the hyperbolic plane constructed, and the 2-dimensional classification achieved for conformal fields. A classification is then given of all harmonic quadratic gradient fields on spheres.