Symplectic manifolds

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If $(M,\omega)$ is a closed, i.e. compact and without boundary, symplectic $2n$-manifold then the cohomology classes
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$[\omega]^k$ are non-zero for $k=0,1.\ldots,n.$ This follows from the fact that the cohomology class of the volume
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form $\omega^n$ is nonzero on a closed manifold. This necessary condition implies that spheres of dimension greater than
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two are not symplectic. More generally, no closed manifold of the form $M \times S^k$ is symplectic for $k>2.$

Revision as of 09:59, 1 October 2010


This page has not been refereed. The information given here might be incomplete or provisional.

Contents

1 Introduction

A symplectic manifold is a smooth manifold M together with a differential two-form \omega that is nondegenerate and closed. The form \omega is called a symplectic form. The nondegeneracy means that the highest nonzero power of \omega is a volume form on M. It follows that a symplectic manifold is even dimensional.

Symplectic manifolds originated from classical mechanics. The phase space of a dynamical system is the cotangent bundle of the configuration space and it is equipped with a symplectic form. This symplectic form is preserved by the flow of the system.

2 Examples

\bullet The most basic example of a symplectic manifold is \mathbb R^{2n} equipped with the form \omega_0:=dx^1\wedge dy^1 + \ldots + dx^n\wedge dy^n.

A theorem of Darboux [McDuff-Salamon] states that locally every symplectic manifold if of this form. More precisely, if (M,\omega) is a symplectic 2n-manifold then for every point x\in M there exists an open neighbourhood U\subset M of p and a diffeomorphism f\colon U\to f(U)\subset \mathbb R^{2n} such that the restriction of \omega to U is equal to the pull-back f^*\omega_0. This implies that symplectic manifolds have no local invariants.

\bullet An area form on an oriented surface is symplectic.

\bullet Let X be a smooth manifold and let \lambda be a one-form on the cotangent bundle T^*X defined as follows. If V is a vector tangent to T^*X at a point \alpha then \lambda_{\alpha}(X) = \alpha (\pi_*(X)), where \pi\colon T^*X\to X is the projection. In local coordinates the form \lambda can be expressed as \sum y^idx^i. The differential d\lambda is a symplectic form on the cotangent bundle T^*X.

\bullet If (M,\omega) is a closed, i.e. compact and without boundary, symplectic 2n-manifold then the cohomology classes [\omega]^k are non-zero for k=0,1.\ldots,n. This follows from the fact that the cohomology class of the volume form \omega^n is nonzero on a closed manifold. This necessary condition implies that spheres of dimension greater than two are not symplectic. More generally, no closed manifold of the form M \times S^k is symplectic for k>2.





3 Invariants

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4 Classification/Characterization

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5 Further discussion

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6 References

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