The Infinitesimalization and Reconstruction of Locally Homogeneous Manifolds

A linear connection on a Lie algebroid is called a Cartan connection if it is suitably compatible with the Lie algebroid structure. Here we show that a smooth connected manifold $M$ is locally homogeneous — i.e., admits an atlas of charts modeled on some homogeneous space $G/H$ — if and only if there exists a transitive Lie algebroid over $M$ admitting a flat Cartan connection that is ‘geometrically closed’. It is shown how the torsion and monodromy of the connection determine the particular form of $G/H$. Under an additional completeness hypothesis, local homogeneity becomes global homogeneity, up to cover.