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 | |
+ | $[\omega]^k$ 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 $k>2.$ | ||
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 together with a differential two-form that is nondegenerate and closed. The form is called a symplectic form. The nondegeneracy means that the highest nonzero power of is a volume form on 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
The most basic example of a symplectic manifold is equipped with the form
A theorem of Darboux [McDuff-Salamon] states that locally every symplectic manifold if of this form. More precisely, if is a symplectic -manifold then for every point there exists an open neighbourhood of and a diffeomorphism such that the restriction of to is equal to the pull-back This implies that symplectic manifolds have no local invariants.
An area form on an oriented surface is symplectic.
Let be a smooth manifold and let be a one-form on the cotangent bundle defined as follows. If is a vector tangent to at a point then where is the projection. In local coordinates the form can be expressed as The differential is a symplectic form on the cotangent bundle
If is a closed, i.e. compact and without boundary, symplectic -manifold then the cohomology classes are non-zero for This follows from the fact that the cohomology class of the volume form 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 is symplectic for
3 Invariants
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4 Classification/Characterization
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5 Further discussion
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