Advanced Riemannian Structures for Geometric World Models: Parallel Transport, Geodesics, and Information Geometry: Volume II — Extending the Intrinsic Calculus of Latent Manifolds

Abstract

This second volume extends the differential-geometric foundations of geometric world models beyond the Riemannian metric, gradient, Levi-Civita connection, and Riemann curvature tensor. It develops the theory of parallel transport and holonomy, the geodesic equation and the exponential and logarithmic maps, the relationship between curvature and the Hessian, and the interpretation of latent geometries through the lens of information geometry. Particular attention is given to the computational and algorithmic consequences of these structures for Riemannian optimisation, natural gradients, and geometry-aware planning. The exposition remains coordinate-independent where possible and is oriented toward the design of latent spaces that respect physical and statistical constraints.