in JuliaManifolds we have reached a relatively stable interface for manifolds. One thing I often hear about it Statistics on manifolds – basically to combine distributions and Riemannian manifolds.
We currenlty offer, what I would call a “poor-mans-version” a rand(M) that generates a random point on a manifold, but it is not always clear which one.
For me personally that is enough, I often really just need “some” point on a manifold. I am also not a strong statistician/stochastician. But if someone would like to dive into that I am very happy to help from the manifolds side as well as the existing APIs in our packages.
we have a slightly old) issue about this by @sethaxen and @mateuszbaran.
I would also be happy if we could start a bit of distributions for (just) a specific manifold 
Maybe there is someone out there with an interest in that topic.