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In this paper we introduce three Markovian couplings of Brownian motions onsmooth Riemannian manifolds without boundary which sit at the crossroad of twoconcepts. The first concept is the one of shy coupling put forward in\cite{Burdzy-Benjamini} and the second concept is the lower bound on the Riccicurvature and the connection with couplings made in \cite{ReSt}.  The first construction is the shy coupling, the second one is afixed-distance coupling and the third is a coupling in which the distancebetween the processes is a deterministic exponential function of time.  The result proved here is that an arbitrary Riemannian manifold satisfyingsome technical conditions supports shy couplings. If in addition, the Riccicurvature is non-negative, there exist fixed-distance couplings. Furthermore,if the Ricci curvature is bounded below by a positive constant, then thereexists a coupling of Brownian motions for which the distance between theprocesses is a decreasing exponential function of time. The constructions usethe intrinsic geometry, and relies on an extension of the notion of frameswhich plays an important role for even dimensional manifolds.  In fact, we provide a wider class of couplings in which the distance functionis deterministic in Theorem \ref{t:100} and Corollary~\ref{Cor:9}.  As an application of the fixed-distance coupling we derive a maximumprinciple for the gradient of harmonic functions on manifolds with non-negativeRicci curvature.  As far as we are aware of, these constructions are new, though the existenceof shy couplings on manifolds is suggested by Kendall in \cite{Kendall}.

Shy and Fixed-Distance Couplings of Brownian Motions on Manifolds