An equivalence between harmonic sections and sections that are harmonic maps

TL;DR

This paper investigates the conditions under which a section of an affine submersion with a horizontal distribution is harmonic as a map and as a section, establishing an equivalence between these two notions in the Riemannian setting.

Contribution

It provides new conditions that characterize when a section is both a harmonic map and a harmonic section, linking these two concepts in the context of affine submersions.

Findings

Established an equivalence between harmonic sections and harmonic maps under certain conditions.

Derived criteria for a section to be harmonic as a map and as a section.

Contributed to the understanding of harmonicity in the setting of affine submersions.

Abstract

Let $\pi:(E,\nabla^{E}) \to (M,g)$ be an affine submersion with horizontal distribution, where $\nabla^{E}$ is a symmetric connection and $M$ is a Riemannian manifold. Let $\sigma$ be a section of $\pi$, namely, $\pi \circ \sigma = Id_{M}$. It is possible to study the harmonic property of section $\sigma$ in two ways. First, we see $\sigma$ as a harmonic map. Second, we see $\sigma$ as harmonic section. In the Riemannian context, it means that $\sigma$ is a critical point of the vertical functional energy. Our main goal is to find conditions to the assertion: $\sigma$ is a harmonic map if and only if $\sigma$ is a harmonic section.