Deterministic couplings of Brownian motions on Riemannian manifolds

We will introduce special types of couplings of Brownian motions on Riemannian manifolds.  We discuss the shy  couplings in which the motions do not meet in finite time and then the ones in which the distance between the motions is a given deterministic function.  On the model manifolds of constant curvature we show a complete characterization of such distance functions.  This is joint work with Mihai Pascu.