  Markov chain Monte Carlo methods explicitly defined on the manifold of
probability distributions have recently been established. These methods are
constructed from diffusions across the manifold and the solution of the
equations describing geodesic flows in the Hamilton--Jacobi representation.
This paper takes the differential geometric basis of Markov chain Monte Carlo
further by considering methods to simulate from probability distributions that
themselves are defined on a manifold, with common examples being classes of
distributions describing directional statistics. Proposal mechanisms are
developed based on the geodesic flows over the manifolds of support for the
distributions and illustrative examples are provided for the hypersphere and
Stiefel manifold of orthonormal matrices.
