  We provide a general framework for constructing probability distributions on
Riemannian manifolds, taking advantage of area-preserving maps and isometries.
Control over distributions' properties, such as parameters, symmetry and
modality yield a family of flexible distributions that are straightforward to
sample from, suitable for use within Monte Carlo algorithms and latent variable
models, such as autoencoders. As an illustration, we empirically validate our
approach by utilizing our proposed distributions within a variational
autoencoder and a latent space network model. Finally, we take advantage of the
generalized description of this framework to posit questions for future work.
