Stochastic characterization of harmonic sections and a Liouville theorem

TL;DR

This paper provides a stochastic and geometric characterization of harmonic sections in fiber bundles, establishes a Liouville theorem, and shows that harmonic sections are precisely the parallel ones.

Contribution

It introduces a novel stochastic characterization of harmonic sections and proves a Liouville theorem linking harmonicity to parallelism in fiber bundles.

Findings

Harmonic sections can be characterized stochastically and geometrically.

Harmonic sections are exactly the parallel sections.

A Liouville theorem for harmonic sections is established.

Abstract

Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G,P)$ be an associate fiber bundle. Our interested is to study harmonic sections of the projection $\pi_{E}$ of $E$ into $M$. Our first purpose is to give a stochastic characterization of harmonic section from $M$ into $E$ and a geometric characterization of harmonic sections with respect to its equivariant lift. The second purpose is to show a version of Liouville theorem for harmonic sections and to prove that section $M$ into $E$ is a harmonic section if and only if it is parallel.