Martingales on Frame Bundles

Abstract

Let M be a smooth manifold endowed with a symmetric connection \(\nabla\). There are two important ways of lift the connection \(\nabla\) of M to the frame bundle BM, the canonical lift \(\nabla^{c}\) and the horizontal lift \(\nabla^{h}\). The aim of this work is determine the \(\nabla^{c}\)-martingales and the \(\nabla^{h}\)-martingales on BM. Our results allow to establish new characterizations of harmonic maps from Riemannian manifolds to frame bundles.