The book [*Directional Statistics*][1] by Mardia and Jupp discusses concrete examples of distributions on: 1. Surface of the unit hypersphere 2. On Stiefel Manifolds 3. Maybe some others **EDIT** In particular, have a look at §13.4.2 that discusses distributions on more general manifolds (e.g., on compact Riemannian manifolds). That section also provides several useful references. I also recalled another source that might be useful to you: [*Matrix Variate Distributions*][2] by Gupta and Nagar. In particular, see chapter 8. [1]: http://books.google.com/books?id=PTNiCm4Q-M0C&dq=directional+statistics&source=gbs_navlinks_s [2]: http://www.amazon.com/Variate-Distributions-Monographs-Surveys-Mathematics/dp/1584880465