Harmonicity of sections of sphere bundles

TL;DR

This paper studies the energy functional on sections of sphere bundles over Riemannian manifolds, providing methods to identify harmonic sections and constructing new examples of harmonic forms and maps.

Contribution

It introduces a method for computing the tension field of sections, enabling the construction of many new harmonic sections and maps on manifolds with G-structures.

Findings

Developed a technique for tension field computation

Constructed numerous new harmonic sections and maps

Extended harmonicity concepts to tensor fields with G-structures

Abstract

We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold (M, <,>) equipped with the Sasaki metric and we discuss the characterising condition for critical points. Likewise, we provide a useful method for computing the tension field in some particular situations. Such a method is shown to be adequate for many tensor fields defined on manifolds M equipped with a G-structure compatible with <,>. This leads to the construction of a lot of new examples of differential forms which are harmonic sections or determine a harmonic map from (M,<,>) into its sphere bundle.