Symplectic manifolds
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is nondegenerate on the horizontal distribution of the connection. The following theorem yields a construction of a symplectic form on the total space of fiber bundles associated with principal bundles equipped with particlar connections. | is nondegenerate on the horizontal distribution of the connection. The following theorem yields a construction of a symplectic form on the total space of fiber bundles associated with principal bundles equipped with particlar connections. | ||
− | {{beginthm|Theorem|}} Let there be given a symplectic manifold $(F,\omega)$ endowed with a hamiltonian action of a Lie group $G$. Let $\mu: F\rightarrow\frak{g}^*$ be the moment map of the $G$-action. If $\mu(F)\subset\frak{ | + | {{beginthm|Theorem|}} Let there be given a symplectic manifold $(F,\omega)$ endowed with a hamiltonian action of a Lie group $G$. Let $\mu: F\rightarrow\frak{g}^*$ be the moment map of the $G$-action. If $\mu(F)\subset\frak{g}^*$ consists of fat vectors, then the associated bundle |
$$F\rightarrow P\times_GF\rightarrow M$$ | $$F\rightarrow P\times_GF\rightarrow M$$ | ||
admits a fiberwise symplectic form on the total space. | admits a fiberwise symplectic form on the total space. |
Revision as of 16:27, 26 November 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
The complex projective space is symplectic with respect to its K\"ahler form. Its pull back to a complex projective smooth manifold is also symplectic. More generally, every K\"ahler manifold is symplectic.
3 Symmetries
A diffeomorphism of a symplectic manifold is called symplectic if it preserves the symplectic form, Sometimes such a diffeomorphism is called a symplectiomorphism. The group of all symplectic diffeomorphisms of is denoted by
It follows from the nondegeneracy of the symplectic form the map defines an isomorphism between the vector fields and the one-forms on a symplectic manifold If the flow of a vector field preserves the symplectic form we have that Then the closedness of the symplectic form implies that the one-form is closed. It follows that the Lie algebra of the group of symplectic diffeomorhism consists of the vector fields for which the one-form is closed. Hence it can be identified with the space of closed one-forms.
If the one-form is exact, i.e. for some function then the vector field is called Hamiltonian. Symplectic diffeomorphism generated by Hamiltonian flows form a group called the group of Hamiltonian diffeomorphism. Its Lie algebra can be identified with the quotient of the space of smooth functions on by the constants.
4 Constructions
1 Products
The product of symplectic manifolds and is a symplectic manifold with respect to the form for nonzero real numbers Here is the projection.
2 Bundles
A locally trivial bundle is called symplectic (resp. Hamiltonian) if its structure group is a subgroup of the group of symplectic (resp. Hamiltonian) diffeomorphisms.
Example. The product of the Hopf bundle with the circle is a symplecti bundle Indeed, the structure group is a group of rotations of the torus and hence it preserves the area. As we have seen above the product does not admit a symplectic form. This example shows that, in general, the total space of a symplectic bundle is not symplectic.
Let is a compact symplectic bundle over a symplectic base. According to a theorem of Thurston, if there exists a cohomology class such that its pull back to every fibre is equal to the class of the symplectic form of the fibre then there exists a representative of the class such that is a symplectic form on for every big enough
A symplectic fiber bundle may have a symplectic form on the total space which restricts symplectically to the fibers, even if the base is not symplectic. Such bundles are constructed using fat connections. Let there be given a principal fiber bundle
Let be a connection form, the curvature form of this connection, and be the horizontal distribution. A vector is called fat (with respect to the given connection), if the 2-form
is nondegenerate on the horizontal distribution of the connection. The following theorem yields a construction of a symplectic form on the total space of fiber bundles associated with principal bundles equipped with particlar connections.
Theorem 4.1. Let there be given a symplectic manifold endowed with a hamiltonian action of a Lie group . Let be the moment map of the -action. If consists of fat vectors, then the associated bundle
admits a fiberwise symplectic form on the total space.
3 Symplectic reduction
Let be a Lie group acting on a symplectic manifold in a hamiltonian way. Denote by the moment map of this action. Since acts on the level set , one can consider the orbit space . It is an orbifold in general, but it happens to be a manifold, when acts freely on the preimage, and is a regular point. In this case, is a symplectic manifold as well, called symplectic reduction. It is often denoted by
4 Coadjoint orbits
===Symplectic homogeneous spaces===Media:Example.ogg
5 Surgery
Consider two symplectic manifolds and suppose that there are a codimension two symplectic submanifolds with a symplectomorphism such that Chern classes of normal bundles satisfyTex syntax error
Then one can get a new symplectic manifold by cutting neighborhods of the submanifolds and identifying the boundaries.
5 Invariants
...
6 Classification/Characterization
...
7 Further discussion
...
8 References
= L_X\omega = d\iota_X \omega + \iota _X d\omega.$ Then the closedness of the symplectic form implies that the one-form $\iota_X\omega$ is closed. It follows that the Lie algebra of the group of symplectic diffeomorhism consists of the vector fields $X$ for which the one-form $\iota _X \omega$ is closed. Hence it can be identified with the space of closed one-forms. If the one-form $\iota _X \omega$ is exact, i.e. $\iota _X \omega = dH$ for some function $H\colon M\to \mathbb R$ then the vector field $X$ is called Hamiltonian. Symplectic diffeomorphism generated by Hamiltonian flows form a group $\operatorname{Ham}(M,\omega)$ called the group of Hamiltonian diffeomorphism. Its Lie algebra can be identified with the quotient of the space of smooth functions on $M$ by the constants. ==Constructions==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
The complex projective space is symplectic with respect to its K\"ahler form. Its pull back to a complex projective smooth manifold is also symplectic. More generally, every K\"ahler manifold is symplectic.
3 Symmetries
A diffeomorphism of a symplectic manifold is called symplectic if it preserves the symplectic form, Sometimes such a diffeomorphism is called a symplectiomorphism. The group of all symplectic diffeomorphisms of is denoted by
It follows from the nondegeneracy of the symplectic form the map defines an isomorphism between the vector fields and the one-forms on a symplectic manifold If the flow of a vector field preserves the symplectic form we have that Then the closedness of the symplectic form implies that the one-form is closed. It follows that the Lie algebra of the group of symplectic diffeomorhism consists of the vector fields for which the one-form is closed. Hence it can be identified with the space of closed one-forms.
If the one-form is exact, i.e. for some function then the vector field is called Hamiltonian. Symplectic diffeomorphism generated by Hamiltonian flows form a group called the group of Hamiltonian diffeomorphism. Its Lie algebra can be identified with the quotient of the space of smooth functions on by the constants.
4 Constructions
1 Products
The product of symplectic manifolds and is a symplectic manifold with respect to the form for nonzero real numbers Here is the projection.
2 Bundles
A locally trivial bundle is called symplectic (resp. Hamiltonian) if its structure group is a subgroup of the group of symplectic (resp. Hamiltonian) diffeomorphisms.
Example. The product of the Hopf bundle with the circle is a symplecti bundle Indeed, the structure group is a group of rotations of the torus and hence it preserves the area. As we have seen above the product does not admit a symplectic form. This example shows that, in general, the total space of a symplectic bundle is not symplectic.
Let is a compact symplectic bundle over a symplectic base. According to a theorem of Thurston, if there exists a cohomology class such that its pull back to every fibre is equal to the class of the symplectic form of the fibre then there exists a representative of the class such that is a symplectic form on for every big enough
A symplectic fiber bundle may have a symplectic form on the total space which restricts symplectically to the fibers, even if the base is not symplectic. Such bundles are constructed using fat connections. Let there be given a principal fiber bundle
Let be a connection form, the curvature form of this connection, and be the horizontal distribution. A vector is called fat (with respect to the given connection), if the 2-form
is nondegenerate on the horizontal distribution of the connection. The following theorem yields a construction of a symplectic form on the total space of fiber bundles associated with principal bundles equipped with particlar connections.
Theorem 4.1. Let there be given a symplectic manifold endowed with a hamiltonian action of a Lie group . Let be the moment map of the -action. If consists of fat vectors, then the associated bundle
admits a fiberwise symplectic form on the total space.
3 Symplectic reduction
Let be a Lie group acting on a symplectic manifold in a hamiltonian way. Denote by the moment map of this action. Since acts on the level set , one can consider the orbit space . It is an orbifold in general, but it happens to be a manifold, when acts freely on the preimage, and is a regular point. In this case, is a symplectic manifold as well, called symplectic reduction. It is often denoted by
4 Coadjoint orbits
===Symplectic homogeneous spaces===Media:Example.ogg
5 Surgery
Consider two symplectic manifolds and suppose that there are a codimension two symplectic submanifolds with a symplectomorphism such that Chern classes of normal bundles satisfyTex syntax error
Then one can get a new symplectic manifold by cutting neighborhods of the submanifolds and identifying the boundaries.
5 Invariants
...
6 Classification/Characterization
...
7 Further discussion
...