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Transactions of the American Mathematical Society

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Geometric structures as deformed infinitesimal symmetries


Author: Anthony D. Blaom
Journal: Trans. Amer. Math. Soc. 358 (2006), 3651-3671
MSC (2000): Primary 53C15, 58H15; Secondary 53B15, 53C07, 53C05, 58H05
DOI: https://doi.org/10.1090/S0002-9947-06-04057-8
Published electronically: March 24, 2006
MathSciNet review: 2218993
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Abstract | References | Similar Articles | Additional Information

Abstract: A general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries. Specifically, this model consists of a Lie algebroid, equipped with an affine connection compatible with the Lie algebroid structure. The curvature of this connection vanishes precisely when the structure is locally symmetric.

This model generalizes Cartan geometries, a substantial class, to the intransitive case. Simple examples are surveyed and corresponding local obstructions to symmetry are identified. These examples include foliations, Riemannian structures, infinitesimal $ G$-structures, symplectic and Poisson structures.


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Additional Information

Anthony D. Blaom
Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand

DOI: https://doi.org/10.1090/S0002-9947-06-04057-8
Keywords: Lie algebroid, geometric structure, Cartan geometry, Cartan connection, action Lie algebroid, deformation, connection theory
Received by editor(s): April 28, 2004
Published electronically: March 24, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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