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Transactions of the American Mathematical Society

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Lie algebroids and Cartan's method of equivalence


Author: Anthony D. Blaom
Journal: Trans. Amer. Math. Soc. 364 (2012), 3071-3135
MSC (2010): Primary 53C15, 58H15; Secondary 53B15, 53C07, 53C05, 58H05, 53A55, 53A30, 58A15
DOI: https://doi.org/10.1090/S0002-9947-2012-05441-9
Published electronically: February 3, 2012
MathSciNet review: 2888239
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Abstract: Élie Cartan's general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan's method of equivalence via reduction and prolongation. We show how to construct certain normal forms (Cartan algebroids) for objects of finite-type, and are able to interpret these directly as `infinitesimal symmetries deformed by curvature'.

Details are developed for transitive structures, but rudiments of the theory include intransitive structures (intransitive symmetry deformations). Detailed illustrations include subriemannian contact structures and conformal geometry.


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Additional Information

Anthony D. Blaom
Affiliation: 22 Ridge Road, Waiheke Island, New Zealand
Email: anthony.blaom@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2012-05441-9
Keywords: Lie algebroid, Cartan algebroid, equivalence, geometric structure, Cartan geometry, Cartan connection, deformation, differential invariant, pseudogroup, connection theory, G-structure, conformal, prolongation, reduction, subriemannian
Received by editor(s): November 27, 2008
Received by editor(s) in revised form: August 9, 2010
Published electronically: February 3, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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