Riemannian Geometry

A Riemannian manifold is a differentiable-manifold equipped with a metric . This is a smoothly varying inner product on each tangent space . The metric measures lengths of tangent vectors and angles between them. Integrating the length of gives the length of a curve , hence a distance on .

Property. The Levi-Civita connection and geodesics

The Levi-Civita connection is the unique connection that is compatible with and torsion-free. Its parallel transport moves tangent vectors along curves without turning or stretching them. A geodesic is a curve that parallel-transports its own velocity, . It is the manifold’s straight line, and locally the shortest path. The failure of transport around a loop to be the identity is the curvature.

Example. Geometry the models here run on

Flow matching transports along geodesics of a chosen ; the space of probability distributions carries the Fisher metric, making information-geometry a Riemannian geometry; and the grassmannian is the Riemannian manifold of subspaces. Whenever “smoothly interpolate while respecting a metric” appears, the metric is a Riemannian one.