Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 074, 19 pages      arXiv:1304.7838      https://doi.org/10.3842/SIGMA.2013.074

The Infinitesimalization and Reconstruction of Locally Homogeneous Manifolds

Anthony D. Blaom
22 Ridge Road, Waiheke Island, New Zealand

Received May 08, 2013, in final form November 19, 2013; Published online November 26, 2013

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.

Key words: locally homogeneous; Lie algebroid; Cartan connection; completeness.

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