  We detail an approach to develop Stein's method for bounding integral metrics
on probability measures defined on a Riemannian manifold $\mathbf M$. Our
approach exploits the relationship between the generator of a diffusion on
$\mathbf M$ with target invariant measure and its characterising Stein
operator. We consider a pair of such diffusions with different starting points,
and through analysis of the distance process between the pair, derive Stein
factors, which bound the solution to the Stein equation and its derivatives.
The Stein factors contain curvature-dependent terms and reduce to those
currently available for $\mathbb R^m$, and moreover imply that the bounds for
$\mathbb R^m$ remain valid when $\mathbf M$ is a flat manifold
