Good paper that extends continuous-time flows from Euclidean spaces to manifolds. The paper is an important contribution that moves the field of normalizing flows forward. Well done. One reviewer raised the concern that the determinant of the Jacobi field could be prohibitively expensive to evaluate. I would encourage the authors to clarify this point in the camera-ready, and discuss whether/when it would be the case. If indeed this is the case, this is an important issue to acknowledge in the paper, that could be useful in pointing researchers to directions for future work. I would also encourage the authors to acknowledge in the camera-ready the parallel work by Falorsi & Forré (https://arxiv.org/abs/2006.06663), and mention how it's similar and how it differs.