  The standard theory of stochastic approximation (SA) is extended to the case
when the constraint set is a Riemannian manifold. Specifically, the standard
ODE method for analyzing SA schemes is extended to iterations constrained to
stay on a manifold using a retraction mapping. In addition, for submanifolds of
a Euclidean space, a framework is developed for a projected SA scheme with
approximate retractions. The framework is also extended to non-differentiable
constraint sets.
