Appendix for Riemannian Continuous Normalizing Flows A Constant curvature manifolds

In the following, we provide a brief overview of Riemannian geometry and constant curvature manifolds, specifically the Poincaré ball and the hypersphere models. In the following, we provide a brief overview of key concepts related to hyperbolic geometry. In the following, we discuss key concepts related to positively curved spaces known as elliptic spaces, and in particular to the hypersphere model. It is endowed with the pull-back metric of the ambient Euclidean space. Unfortunately, conventional probabilistic models implicitly assume a flat geometry.