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Transactions of the American Mathematical Society

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Cartan structures on contact manifolds

Authors: G. Burdet and M. Perrin
Journal: Trans. Amer. Math. Soc. 266 (1981), 583-602
MSC: Primary 53C10; Secondary 53C15, 53C30
MathSciNet review: 617553
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Abstract: Owing to the existence of a dilatation generator of eigenvalues $ \pm 2, \pm 1,0$ the symplectic Lie algebra is considered as a $ \vert 2\vert$-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.

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Article copyright: © Copyright 1981 American Mathematical Society