Martingales on Principal Fiber Bundles

TL;DR

This paper investigates martingales on principal fiber bundles with connection forms, providing new characterizations of harmonic maps from Riemannian manifolds to these bundles.

Contribution

It introduces methods to determine $ abla^{P}$-martingales on principal fiber bundles, advancing the understanding of harmonic maps in differential geometry.

Findings

Characterization of $ abla^{P}$-martingales on principal fiber bundles

New criteria for harmonic maps from Riemannian manifolds

Framework for analyzing stochastic processes on fiber bundles

Abstract

Let $P(M,G)$ be a principal fiber bundle, let $\omega$ be a connection form on $P(M,G)$, and consider a projectable connection $\nabla^{P}$ on $P(M,G)$. The aim of this work is to determine the $\nabla^{P}$-martingales in $P(M,G)$. Our results allow establishing new characterizations of harmonic maps from Riemannian manifolds to principal fiber bundles.