The Symplectic Slice Theorem

2010

Memoirs of the AMS

We classify symplectic actions of 2-tori on compact, connected symplectic 4-manifolds, up to equivariant symplectomorphisms. This extends results of Atiyah, Guillemin-Sternberg, Delzant and Benoist. The classification is in terms of a collection of invariants, which are invariants of the topology of the manifold, of the torus action and of the symplectic form. We construct explicit models of such symplectic manifolds with torus actions, defined in terms of these invariants.We also classify, up to equivariant symplectomorphisms, symplectic actions of (2n − 2)dimensional tori on 2n-dimensional symplectic manifolds, when at least one orbit is a (2n−2)--dimensional symplectic submanifold. Then we show that a 2n-dimensional symplectic manifold (M, σ) equipped with a free symplectic action of a (2n − 2)-dimensional torus with at least one (2n − 2)-dimensional symplectic orbit is equivariantly diffeomorphic to M/T × T equipped with the translational action of T . Thus two such symplectic manifolds are equivariantly diffeomorphic if and only if their orbit spaces are surfaces of the same genus. * Partly funded by a Rackham Predoctoral Fellowship (2006-2007) and a Rackham Dissertation Fellowship (2005-2006. He is grateful to professor J.J. Duistermaat for discussions, specifically on sections 2.4.4, 2.5, 2.6, for hospitality on three visits to Utrecht, and for comments on a preliminary version.He thanks professor A. Uribe for conversations on symplectic normal forms, and professor P. Scott for discussions on orbifold theory, and helpful feedback and remarks on Section 4.2. He also has benefited from conversations with professors sat through several talks of the author on the paper and offered feedback. He thanks professor M. The orbit space M/TWe describe the structure of the orbit space M/T , c.f. Definition 2.13. Symplectic form on the T -orbitsWe prove that the symplectic form on every T -orbit of M is given by the same non-degenerate antisymmetric bilinear form.Let X be an element of the Lie algebra t of T , and denote by X M the smooth vector field on M obtained as the infinitesimal action of X on M. Let ω be a smooth differential form, let L v denote the Lie derivative with respect to a vector field v, and let i v ω denote the usual inner product of ω with v. Since the symplectic form σ is T -invariant, we have that d(i X M σ) = L X M σ = 0, where the first equality follows by combining d σ = 0 and the homotopy identityThe following result follows from [12, Lem. 2.1].Lemma 2.1. Let (M, σ) be a compact, connected, symplectic manifold equipped with an effective symplectic action of a torus T for which there is at least one T -orbit which is a dim T -dimensional symplectic submanifold of (M, σ). Then there exists a unique non-degenerate antisymmetric bilinear form σ t : t × t → R on the Lie algebra t of T such thatfor every X, Y ∈ t, and every x ∈ M.Proof. In [12, Lem. 2.1] it was shown that there is a unique antisymmetric bilinear form σ t : t×t → R on the Lie algebra t of T such that expression (2.1) holds for every ...

“…Then by the symplectic tube theorem of Benoist [3, Prop. 1.9], Ortega and Ratiu [42] and Weinstein, which we use as it was formulated in [12,Sec. 11], there exists an open neighborhood…”

Section: Orbifold Structure Of M/tmentioning

confidence: 99%

Symplectic actions of 2-tori on 4-manifolds

2010

Memoirs of the AMS

“…Benoist [3] proved a symplectic tube theorem for symplectic actions with coisotropic orbits and convexity result in the spirit of the of the Atiyah-Guillemin-Sternberg theorem [3]; Ortega-Ratiu [30] proved a local normal form theorem for symplectic torus actions with coisotropic orbits. These appear to be the most general results prior to the classification of symplectic torus actions with coisotropic principal orbits in Duistermaat-Pelayo [11] and Pelayo [33].…”

Section: Theorem 11 If M Is Compact Connected Symplectic Manifold Omentioning

confidence: 99%

“…Lemma 3.1 leads to the local models of the symplectic T -space described in the paragraph after Theorem 3.5. These local models can also be obtained by applying results of Benoist [3, Proposition 1.9] or Ortega and Ratiu [30,] to the case of a symplectic torus action with symplectic orbits. The proof of Lemma 3.1 in [33] uses these local models.…”

Section: Lemma 31 the Distribution Is T -Invariant And Integrablementioning

confidence: 99%

Topology of symplectic torus actions with symplectic orbits

Duistermaat

2010

Rev Mat Complut

We give a concise overview of the classification theory of symplectic manifolds equipped with torus actions for which the orbits are symplectic (this is equivalent to the existence of a symplectic principal orbit), and apply this theory to study the structure of the leaf space induced by the action. In particular we show that if M is a symplectic manifold on which a torus T acts effectively with symplectic orbits, then the leaf space M/T is a very good orbifold with first Betti number b 1 (M/T ) = b 1 (M) − dim T .

“…In this paper we have covered symplectic Hamiltonian actions as contained in the works of Audin, Ahara, Hattori, Delzant, Duistermaat, Heckman Kostant, Atiyah, Guillemin, Karshon, Sternberg, Tolman, Weitsman [3,12,13,35,87,97,10,73,31] among others, and more general symplectic actions as in the works of Benoist, Duistermaat, Frankel, McDuff, Ortega, Ratiu, and the author [15,16,38,117,125] among others.…”

Section: Final Remarksmentioning

confidence: 99%

Hamiltonian and symplectic symmetries: An introduction

2017

Bull. Amer. Math. Soc.

Abstract. Classical mechanical systems are modeled by a symplectic manifold (M, ω), and their symmetries are encoded in the action of a Lie group G on M by diffeomorphisms which preserve ω. These actions, which are called symplectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact Abelian Lie groups that are, in addition, of Hamiltonian type, i.e., they also satisfy Hamilton's equations. Since then a number of connections with combinatorics, finitedimensional integrable Hamiltonian systems, more general symplectic actions, and topology have flourished. In this paper we review classical and recent results on Hamiltonian and non-Hamiltonian symplectic group actions roughly starting from the results of these authors. This paper also serves as a quick introduction to the basics of symplectic geometry.