Cartan connection

Summary

A Cartan connection describes the geometry of a manifold as a curved version of a model homogeneous space $G/H$. At each point the manifold is identified to first order with the model, and the connection says how these identifications fit together. Its curvature measures how far the manifold is from being locally the model, and it vanishes exactly when the manifold is locally isomorphic to $G/H$. Riemannian, conformal and projective geometry all fit in this frame, with models Euclidean space, the conformal sphere and projective space.