The Infinitesimalization and Reconstruction of Locally Homogeneous Manifolds

Symmetry Integrability and Geometry-Methods and Applications

article

Anthony D. Blaom

关键词: locally homogeneous;    Lie algebroid;    Cartan connection;    completeness;   

DOI  :  10.3842/SIGMA.2013.074

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown