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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.
pdf (404 kb)
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