The book Directional Statistics by Mardia and Jupp discusses concrete examples of distributions on:

1. Surface of the unit hypersphere
2. On Stiefel Manifolds
3. Maybe some others

I also recalled another source that might be useful to you:

Matrix Variate Distributions by Gupta and Nagar. In particular, see chapter 8.

The book Directional Statistics 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 by Gupta and Nagar. In particular, see chapter 8.

The book Directional Statistics by Mardia and Jupp discusses concrete examples of distributions on:

1. Surface of the unit hypersphere
2. On Stiefel Manifolds
3. Maybe some others

The book Directional Statistics by Mardia and Jupp discusses concrete examples of distributions on:

1. Surface of the unit hypersphere
2. On Stiefel Manifolds
3. Maybe some others

I also recalled another source that might be useful to you:

Matrix Variate Distributions by Gupta and Nagar. In particular, see chapter 8.