Cartan structures on contact manifolds
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- 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
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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