Martingales on Frame Bundles

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. Finally, we consider on the associated vector bundles to $BM$ the canonical lift $\nabla^+$ and give a characterization of the $\nabla^+$-martingales.