Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Cartan structures on contact manifolds
HTML articles powered by AMS MathViewer

by G. Burdet and M. Perrin PDF
Trans. Amer. Math. Soc. 266 (1981), 583-602 Request permission

Abstract:

Owing to the existence of a dilatation generator of eigenvalues $\pm 2, \pm 1,0$ the symplectic Lie algebra is considered as a $|2|$-graded Lie algebra. The corresponding decomposition of the symplectic group ${\text {Sp(2(}}n + 1{\text {),}}{\mathbf {R}}{\text {)}}$ makes the semidirect product (denoted ${L^0}$) of the $(2n + 1)$-dimensional Weyl group by the conformal symplectic group ${\text {CSp(}}2n,{\mathbf {R}}{\text {)}}$ appear as a privileged subgroup and permits one to construct a $2n + 1$-dimensional homogeneous space possessing a natural contact form. Then ${\text {Sp}}(2(n + 1),{\mathbf {R}})$-valued Cartan connections on a ${L^0}$principal fibre bundle over a $2n + 1$-dimensional manifold ${B_{2n + 1}}$ are constructed and called symplectic Cartan connections. The conditions for obtaining a unique symplectic Cartan connection are given. The existence of this unique Cartan connection is used to define the notion of contact structure over ${B_{2n + 1}}$ and it is shown that any ${L^0}$-structure of degree $2$ over ${B_{2n + 1}}$ can be considered as a contact structure on it. Moreover it is shown that a contact structure can be associated in a canonical way to any contact manifold.
References
  • G. Burdet and M. Perrin, Realizations of the central extension of the inhomogeneous symplectic algebra as time dependent invariance algebras of nonrelativistic quantum systems, J. Math. Phys. 16 (1975), 1692–1703. MR 371277, DOI 10.1063/1.522738
  • Elie Cartan, Oeuvres complètes. Partie III. Vol. 1. Divers, géométrie différentielle. Vol. 2. Géométrie différentielle (suite), Gauthier-Villars, Paris, 1955 (French). MR 0075130
  • Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège; Masson & Cie, Paris, 1951, pp. 29–55 (French). MR 0042768
  • Shoshichi Kobayashi, Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70, Springer-Verlag, New York-Heidelberg, 1972. MR 0355886
  • —, ibid., p. 28. —, op. cit., p. 127. —, op. cit., p. 13. —, op. cit., p. 15. —, op. cit., p. 130. —, op. cit., p. 137.
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C10, 53C15, 53C30
  • Retrieve articles in all journals with MSC: 53C10, 53C15, 53C30
Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 266 (1981), 583-602
  • MSC: Primary 53C10; Secondary 53C15, 53C30
  • DOI: https://doi.org/10.1090/S0002-9947-1981-0617553-3
  • MathSciNet review: 617553