An equivalence between harmonic sections and sections that are harmonic maps

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, $\pi \circ \sigma =I{d}_M$. 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.

arXiv comment14 pages