*Thank you, Stephan, for correcting me.*

The generalization of Donsker's theorem from $N$-dimensional Euclidean space to general Riemannian manifolds has been worked out by Erik J&oslash;rgensen, <A HREF="http://link.springer.com/content/pdf/10.1007%2FBF00533088">The Central Limit Problem for Geodesic Random Walks</A>.


> The purpose of the present work is to
> consider the problem of defining the
> concept of a random walk in a general
> Riemannian manifold ${\cal M}$, and to
> investigate the behavior in the limit
> of a sequence of such random walks. It
> will be shown that such a sequence,
> under reasonable assumptions,
> converges to a diffusion process in
> ${\cal M}$, and in particular Brownian motion
> processes will be obtained as limits
> of sequences of random walks with
> identically distributed steps. The
> results which we arrive at in this
> paper are general versions of
> well-known classical results
> concerning the transition from random
> walks to diffusion processes, for
> instance: the central limit theorem
> and Donsker's theorem.