This paper extends/generalizes the results from Vempala and Wibisono (2019) on Langevin diffusions to Riemannian manifolds. The problem is quite interesting, has many potential applications and the derivations are sound. The result relating the diameter of a compact manifold to the Langevin step-size is neat. There is overall concern between reviewers that accessibility of the current draft to the broader Neurips community is rather limited.