After an extensive unsuccessful search: I need a reference (preferably a book) for the Donsker's invariance principle for Riemannian manifolds. Thanks.
The generalization of Donsker's theorem from $N$dimensional Euclidean space to general Riemannian manifolds has been worked out by Erik Jørgensen, The Central Limit Problem for Geodesic Random Walks.



D. Khoshnevisan, Lecture notes on Donsker's theorem 


You will probably interested by the paper Empirical invariance principle for ergodic torus automorphisms; Genericity. Olivier Durieu, Philippe Jouan. Stochastics and Dynamics, volume 8, 2, p. 173195, 2008. It is available here. 

