# Contact Toric Manifolds

@article{Lerman2001ContactTM,
title={Contact Toric Manifolds},
author={Eugene Lerman},
journal={Journal of Symplectic Geometry},
year={2001},
volume={1},
pages={785-828}
}
• E. Lerman
• Published 1 July 2001
• Mathematics
• Journal of Symplectic Geometry
We complete the classification of compact connected contact toric manifolds initiated by Banyaga and Molino and by Galicki and Boyer. As an application we prove the conjectures of Toth and Zelditch on toric integrable systems on the n-torus and the 2-sphere.
163 Citations
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#### References

SHOWING 1-10 OF 20 REFERENCES
Toric integrable geodesic flows
• Mathematics
• 2000
By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics.
Contact cuts
We describe a contact analog of the symplectic cut construction [L]. As an application we show that the group of contactomorphisms for certain overtwisted contact structures on lens spaces containsExpand
A note on toric contact geometry
• Mathematics
• 1999
Abstract After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds with an effective torus action whose Reeb vector field corresponds to anExpand
A convexity theorem for torus actions on contact manifolds
We show that the cone associated with a moment map for an action of a torus on a contact compact connected manifold is a convex polyhedral cone and that the moment map has connected fibers providedExpand
Symplectic toric orbifolds
• Mathematics
• 1994
A symplectic toric orbifold is a compact connected orbifold $M$, a symplectic form $\omega$ on $M$, and an effective Hamiltonian action of a torus $T$ on $M$, where the dimension of $T$ is half theExpand
Une structure de contact, même tendue, est plus ou moins tordue
This paper proves thé existence of non isomorphic tight contact structures on T. It aiso shows that ail Lagrangian incompressible embedded tori in T x (R^^O}) are homotopic.
Symplectic Techniques in Physics
• Mathematics
• 1984
Preface 1. Introduction 2. The geometry of the moment map 3. Motion in a Yang-Mills field and the principle of general covariance 4. Complete integrability 5. Contractions of symplectic homogeneousExpand
Le thorme de rduction de Marsden-Weinstein en gomtrie cosymplectique et de contact
Abstract We show that the classical Marsden-Weinstein Reduction theorem for Hamiltonian systems with symmetries is still true for contact manifolds and cosympletic manifolds (i.e. canonical manifoldsExpand
THE GEOMETRY SURROUNDING THE ARNOLD-LIOUVILLE THEOREM
We explain how a generalized completely integrable hamil­ tonian system on a symplectic manifold (M, f2) can be viewed as a generalized Duistermaat fibration: i.e. a fibration 7r : M2n -+ wn of aExpand
Examples for obstructions to action-angle coordinates
On donne des exemples de varietes symplectiques qui sont aussi des fibres toriques principaux avec des fibres lagrangiennes. Ces fibres sont des exemples d'espaces avec une obstruction a l'existenceExpand