The Infinitesimalization and Reconstruction of Locally Homogeneous Manifolds

TL;DR

This paper characterizes local homogeneity of manifolds via the existence of a special flat Cartan connection on a Lie algebroid, linking geometric structures to algebraic data such as torsion and monodromy.

Contribution

It establishes a precise equivalence between local homogeneity and the existence of a transitive Lie algebroid with a flat, geometrically closed Cartan connection.

Findings

Manifolds are locally homogeneous iff they admit a transitive Lie algebroid with a flat, geometrically closed Cartan connection.

The torsion and monodromy of the connection determine the homogeneous model G/H.

Under completeness, local homogeneity implies global homogeneity up to cover.

Abstract

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.