Cartan connections

A Cartan connection is a mathematical tool that provides a way to compare geometric structures locally on a space to a model geometric space, like how we compare small regions to flat planes. It generalizes concepts like curvature and allows us to describe how a space “twists” or “bends” relative to a standard reference geometry. Essentially, it assigns a meaningful way to measure how the geometry of a surface or manifold deviates from a simpler, well-understood model, enabling deeper analysis of shapes, spaces, and their intrinsic properties.