Riemannian Geometry of Latent Spaces: From Vector Fields to Curvature in Geometric World Models: A Synthesis of Differential-Geometric Foundations for Geometry-Aware Latent Dynamics

This article synthesises the differential-geometric foundations underlying geometric world models, beginning from the role of vector fields on latent Riemannian manifolds and proceeding through the Riemannian metric and its inverse, the transformation properties of vectors and covectors, the Levi-Civita connection, and the Riemann curvature tensor. Drawing on recent expositions of vector fields for geometric world models, we develop a coherent account of why the latent space of a world model is properly treated as a Riemannian manifold, how the Riemannian gradient is obtained by raising the index of the differential with the inverse metric, and how the Levi-Civita connection and Riemann tensor furnish the intrinsic calculus of parallel transport, geodesics, and curvature required for geometry-aware planning and optimisation. The treatment is self-contained and emphasises the coordinate-independent character of the constructions.