Connections and martingales

The main point in this chapter is the possibility, on some manifolds, of distinguishing among semimartingales a smaller class, that of martingales. The name ‘local martingale’ would fit these processes better, since when the manifold is the flat space ℝ or ℝn they are exactly the usual (continuous) local martingales; but we won’t consider any subclass corresponding in that case to usual martingales, so no ambiguity is created by calling them just martingales (except precisely in that flat case!).