We present a comprehensive framework for probabilistic modeling on Riemannian manifolds,encompassing diffusion processes, continuous normalizing flows, energy-based models,and information-theoretic measures adapted to curved geometries. Our unified approachextends classical probabilistic methods from Euclidean spaces to arbitrary Riemannianmanifolds, providing principled tools for modeling data with inherent geometric structure.We develop complete mathematical foundations including forward and reverse stochasticdifferential equations, probability-flow ordinary differential equations, intrinsic Langevindynamics, and manifold-aware information measures. The framework is demonstrated oncanonical manifolds including spheres, rotation groups SO(3), symmetric positive definitematrices, and hyperbolic spaces, with applications spanning computer vision, robotics,neuroscience, and network analysis.