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Transactions of the American Mathematical Society

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



Contact reduction and groupoid actions

Authors: Marco Zambon and Chenchang Zhu
Journal: Trans. Amer. Math. Soc. 358 (2006), 1365-1401
MSC (2000): Primary 53D10, 53D20, 58H05
Published electronically: June 21, 2005
MathSciNet review: 2187657
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Abstract: We introduce a new method to perform reduction of contact manifolds that extends Willett's and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map $J:M \rightarrow \Gamma_0$ from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of $\Gamma_0$ on $M$, and we show that the quotients of fibers $J^{-1}(x)$ by suitable Lie subgroups $\Gamma_x $ are either contact or locally conformal symplectic manifolds with structures induced by the one on $M$.

We show that Willett's reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett's reduction can be performed only at distinguished points. As an application we obtain Kostant's prequantizations of coadjoint orbits. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.

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Additional Information

Marco Zambon
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: Institut für Mathematik, Universität Zürich-Irchel, 8057 Zürich, Switzerland

Chenchang Zhu
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Address at time of publication: D-MATH, ETH-Zentrum, CH-8092 Zürich, Switzerland

Received by editor(s): May 25, 2004
Published electronically: June 21, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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