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

翻译：我们详细阐述了一种在黎曼流形$\mathbf M$上发展Stein方法以界定概率测度积分度量的途径。该方法利用$\mathbf M$上具有目标不变测度的扩散生成元与其特征性Stein算子之间的关系。我们考虑一对起始点不同的此类扩散过程，通过分析两者间的距离过程，推导出Stein因子，该因子对Stein方程的解及其导数进行界定。这些Stein因子包含依赖于曲率的项，并简化为$\mathbb R^m$中现有的形式，且进一步表明当$\mathbf M$为平坦流形时，$\mathbb R^m$的界仍然成立。