  Stein's method has been widely used to achieve distributional approximations
for probability distributions defined in Euclidean spaces. Recently, techniques
to extend Stein's method to manifold-valued random variables with distributions
defined on the respective manifolds have been reported. However, several of
these methods impose strong regularity conditions on the distributions as well
as the manifolds and/or consider very special cases. In this paper, we present
a novel framework for Stein's method on Riemannian manifolds using the
Friedrichs extension technique applied to self-adjoint unbounded operators.
This framework is applicable to a variety of conventional and unconventional
situations, including but not limited to, intrinsically defined non-smooth
distributions, truncated distributions on Riemannian manifolds, distributions
on incomplete Riemannian manifolds, etc. Moreover, the stronger the regularity
conditions imposed on the manifolds or target distributions, the stronger will
be the characterization ability of our novel Stein pair, which facilitates the
application of Stein's method to problem domains hitherto uncharted. We present
several (non-numeric) examples illustrating the applicability of the presented
theory.
