An equivalence between harmonic sections and sections that are harmonic maps

Abstract

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