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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
Published electronically: March 24, 2006
MathSciNet review: 2218993
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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.

References [Enhancements On Off] (What's this?)

  • 1. D. V. Alekseevsky and P. W. Michor.
    Differential geometry of Cartan connections.
    Publ. Math. Debrecen, 47(3-4):349-375, 1995. MR 1362298 (96h:53029)
  • 2. A. D. Blaom.
    Lie algebroids and Cartan's method of equivalence.
    In preparation, 2005.
  • 3. A. Cannas da Silva and A. Weinstein.
    Geometric Models for Noncommutative Algebras, volume 10 of Berkeley Mathematics Lecture Notes.
    American Mathematical Society, Providence, RI, 1999. MR 1747916 (2001m:58013)
  • 4. A. Cap and A. R. Gover.
    Tractor calculi for parabolic geometries.
    Trans. Amer. Math. Soc., 354(4):1511-1548 (electronic), 2002. MR 1873017 (2003j:53033)
  • 5. M. Crainic and R. L. Fernandes.
    Integrability of Lie brackets.
    Ann. of Math. (2), 157(2):575-620, 2003. MR 1973056 (2004h:58027)
  • 6. M. Crainic and R. L. Fernandes.
    Secondary characteristic classes of Lie algebroids. Quantum Field Theory and Noncommutative Geometry, volume 662 of Lecture Notes in Physics. Springer, Berlin, 2005, 157-176. MR 2179182
  • 7. P. Dazord.
    Groupoïde d'holonomie et géométrie globale.
    C. R. Acad. Sci. Paris Sér. I Math., 324(1):77-80, 1997. MR 1435591 (97m:58214)
  • 8. R. L. Fernandes.
    Lie algebroids, holonomy and characteristic classes.
    Adv. Math., 170(1):119-179, 2002. MR 1929305 (2004b:58023)
  • 9. R. B. Gardner.
    The Method of Equivalence and its Applications.
    SIAM, Philadelphia, 1989. MR 1062197 (91j:58007)
  • 10. P. Griffiths.
    On the theory of variation of structures defined by transitive, continuous pseudogroups.
    Osaka J. Math., 1:175-199, 1964. MR 0180993 (31:5223)
  • 11. P. J. Higgins and K. C. H. Mackenzie.
    Algebraic constructions in the category of Lie algebroids.
    J. Algebra, 129:194-230, 1990. MR 1037400 (92e:58241)
  • 12. S. Kobayashi.
    Transformation Groups in Differential Geometry.
    Springer-Verlag, New York, 1972. MR 0355886 (50:8360)
  • 13. A. Kumpera and D. Spencer.
    Lie Equations. Vol. I: General Theory.
    Princeton University Press, Princeton, N.J., 1972. MR 0380908 (52:1805)
  • 14. K. C. H. Mackenzie.
    General Theory of Lie Groupoids and Lie Algebroids, volume 213 of London Mathematical Society Lecture Note Series.
    Cambridge University Press, Cambridge, 2003. MR 2157566
  • 15. J. E. Marsden and T. S. Ratiu.
    Introduction to Mechanics and Symmetry, volume 17 of Texts in Applied Mathematics.
    Springer, 1994. MR 1304682 (95i:58073)
  • 16. R. W. Sharpe.
    Differential Geometry: Cartan's generalization of Klein's Erlangen program, volume 166 of Graduate Texts in Mathematics.
    Springer, 1997. MR 1453120 (98m:53033)
  • 17. J. Slovák.
    Parabolic Geometries.
    Research Lecture Notes, Masaryk University, 1997.

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Additional Information

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

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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