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Stein's method has been widely used to achieve distributional approximationsfor probability distributions defined in Euclidean spaces. Recently, techniquesto extend Stein's method to manifold-valued random variables with distributionsdefined on the respective manifolds have been reported. However, several ofthese methods impose strong regularity conditions on the distributions as wellas the manifolds and/or consider very special cases. In this paper, we presenta novel framework for Stein's method on Riemannian manifolds using theFriedrichs extension technique applied to self-adjoint unbounded operators.This framework is applicable to a variety of conventional and unconventionalsituations, including but not limited to, intrinsically defined non-smoothdistributions, truncated distributions on Riemannian manifolds, distributionson incomplete Riemannian manifolds, etc. Moreover, the stronger the regularityconditions imposed on the manifolds or target distributions, the stronger willbe the characterization ability of our novel Stein pair, which facilitates theapplication of Stein's method to problem domains hitherto uncharted. We presentseveral (non-numeric) examples illustrating the applicability of the presentedtheory.

A Framework for Improving the Characterization Scope of Stein's Method on Riemannian Manifolds