Abstract: The inhomogeneity of medium, time-varying and other factors determine that the underwater acoustic propagation channel is a curved Riemannian manifold. Based on Riemannian geometry theory, a generalized form of underwater acoustic Eikonal equation on Riemannian manifold is given in this paper, which is still applicable to Euclidean space. By comparing two ideal scenes of a plane and a curved sphere with homogeneous medium, the difference of underwater acoustic ray propagation between curved Riemannian manifold and flat Euclid space is revealed. The ray simulation results for Munk sound speed profile based on geodesic eikonal equation and Bellhop-3D model are basically consistent, which verifies the correctness of the theory presented in this paper. The results of this paper lay a conceptual foundation for further modeling and calculation on the curved underwater acoustic channels.