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We consider the model space of constant curvature in dimension n andcharacterize all co-adapted couplings of Brownian motions on this space forwhich the distance between the processes is deterministic. In addition, theconstruction of the coupling is explicit for every choice of $\rho$ satisfyingthe above hypotheses.

Couplings of Brownian Motions of deterministic distance in model spaces of constant curvature