Computational and Equivariant Extensions of Riemannian Geometry for Geometric World Models: Volume III — Discrete Approximations, Stochastic Dynamics, and Geometric Priors

Abstract

This third volume continues the series by addressing the computational realisation of Riemannian structures on latent manifolds and their integration with equivariant architectures and stochastic dynamics. It treats discrete differential geometry on graphs and meshes, numerical approximations of the Levi-Civita connection and curvature, Riemannian Langevin dynamics and stochastic gradient flows, and the systematic incorporation of geometric priors through equivariant neural networks and geometric deep learning. The discussion emphasises practical consequences for the design, training, and analysis of geometric world models, including JEPA-style predictors and spatial intelligence systems.