Symplectic manifolds
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&Salamon1998] 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 Kähler form. Its pull back to a complex projective smooth manifold is also symplectic. More generally, every Kähler 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
4.1 Products
The product of symplectic manifolds and is a symplectic manifold with respect to the form for nonzero real numbers Here is the projection.
4.2 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 symplectic 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 particular connections.
Theorem 3.1. 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.
using this theorem, one can construct examples of symplectic fiber bundles with fiberwise symplectic form on the total space (see examples below).
Example (twistor bundles) Consider the principal bundle of the orthogonal frame bundles over -dimensional manifold :
Let . The associated bundle with fiber is called the twistor bundle. It is easy to see that can be identified with a coadjoint orbit of some , where denotes the Lie algebra of . Moreover, if admits Riemannian metric of pinched curvature with sufficiently small pinching constant then is fat with respect to the Levi-Civitta connection in the frame bundle. As a result, the whole coadjoint orbit (which is the image of the moment map of the -action) consists of fat vectors. Thus, we obtain a fiberwise symplectic structure on the total space of any twistor bundle
over even-dimensional manifolds of pinched curvature. In particular, twistor bundles over spheres or hyperbolic manifolds, admit fiberwise symplectic structures. The simplest example of this construction is the fibering of over with fiber , since it is known that the total space of the twistor bundle over is .
Example (locally homogeneous complex manifolds) Let be a Lie group of non-compact type, which is a real form of a complex Lie group . Choose a parabolic subgroup and a maximal compact subgroup in . Assume that is compact. Then one can show that can be identified with a coadjoint orbit of some vector in , which is fat with respect to a -invariant connection in the principal bundle
It follows that the associated bundle
is a symplectic fiber bundle with fiberwise symplectic structure. This construction can be compactified by taking lattices in which intersect trivially with . A particular example is given by the fiber bundle
and its compactification by lattices.
4.3 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 .
4.4 Symplectic cut
Let be a symplectic manifold with a hamiltonian action of the circle If is the moment map, is a regular level, then the action restricted to has no fixed points, hence is the boundary of the associated disk bundle W. This is a manifold if the action is free and an orbifold if a non-trivial isotropy occurs.
4.5 Coadjoint orbits
4.6 Symplectic homogeneous spaces
Nilmanifolds, solvmanifolds, homogeneous spaces of semisimple Lie groups
4.7 Donaldson's theorem on submanifolds
4.8 Surgery in codimension 2
Consider two symplectic manifolds of equal dimension and suppose that there are codimension two symplectic submanifolds and a symplectomorphism such that Chern classes of normal bundles satisfy Then by removing tubular neighborhods of and we get manifolds with boundaries. The map induces a diffeomorphism of the boundaries, one can form a new manifold identifying the boundaries by this diffeomorphism and define on it a symplectic form which coincides with and outside of a tubular neigborhood of the trace of glueing [Gompf1995]. The same works if are symplectic submanifolds of a connected symplectic manifold.
4.9 Symplectic blow-up
5 Invariants
...
6 Classification/Characterization
...
7 Further discussion
...
8 References
- [Gompf1995] R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), no.3, 527–595. MR1356781 (96j:57025) Zbl 0849.53027
- [McDuff&Salamon1998] D. McDuff and D. Salamon, Introduction to symplectic topology, The Clarendon Press Oxford University Press, 1998. MR1698616 (2000g:53098) Zbl 0978.53120
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&Salamon1998] 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 Kähler form. Its pull back to a complex projective smooth manifold is also symplectic. More generally, every Kähler 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
4.1 Products
The product of symplectic manifolds and is a symplectic manifold with respect to the form for nonzero real numbers Here is the projection.
4.2 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 symplectic 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 particular connections.
Theorem 3.1. 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.
using this theorem, one can construct examples of symplectic fiber bundles with fiberwise symplectic form on the total space (see examples below).
Example (twistor bundles) Consider the principal bundle of the orthogonal frame bundles over -dimensional manifold :
Let . The associated bundle with fiber is called the twistor bundle. It is easy to see that can be identified with a coadjoint orbit of some , where denotes the Lie algebra of . Moreover, if admits Riemannian metric of pinched curvature with sufficiently small pinching constant then is fat with respect to the Levi-Civitta connection in the frame bundle. As a result, the whole coadjoint orbit (which is the image of the moment map of the -action) consists of fat vectors. Thus, we obtain a fiberwise symplectic structure on the total space of any twistor bundle
over even-dimensional manifolds of pinched curvature. In particular, twistor bundles over spheres or hyperbolic manifolds, admit fiberwise symplectic structures. The simplest example of this construction is the fibering of over with fiber , since it is known that the total space of the twistor bundle over is .
Example (locally homogeneous complex manifolds) Let be a Lie group of non-compact type, which is a real form of a complex Lie group . Choose a parabolic subgroup and a maximal compact subgroup in . Assume that is compact. Then one can show that can be identified with a coadjoint orbit of some vector in , which is fat with respect to a -invariant connection in the principal bundle
It follows that the associated bundle
is a symplectic fiber bundle with fiberwise symplectic structure. This construction can be compactified by taking lattices in which intersect trivially with . A particular example is given by the fiber bundle
and its compactification by lattices.
4.3 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 .
4.4 Symplectic cut
Let be a symplectic manifold with a hamiltonian action of the circle If is the moment map, is a regular level, then the action restricted to has no fixed points, hence is the boundary of the associated disk bundle W. This is a manifold if the action is free and an orbifold if a non-trivial isotropy occurs.
4.5 Coadjoint orbits
4.6 Symplectic homogeneous spaces
Nilmanifolds, solvmanifolds, homogeneous spaces of semisimple Lie groups
4.7 Donaldson's theorem on submanifolds
4.8 Surgery in codimension 2
Consider two symplectic manifolds of equal dimension and suppose that there are codimension two symplectic submanifolds and a symplectomorphism such that Chern classes of normal bundles satisfy Then by removing tubular neighborhods of and we get manifolds with boundaries. The map induces a diffeomorphism of the boundaries, one can form a new manifold identifying the boundaries by this diffeomorphism and define on it a symplectic form which coincides with and outside of a tubular neigborhood of the trace of glueing [Gompf1995]. The same works if are symplectic submanifolds of a connected symplectic manifold.
4.9 Symplectic blow-up
5 Invariants
...
6 Classification/Characterization
...
7 Further discussion
...
8 References
- [Gompf1995] R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), no.3, 527–595. MR1356781 (96j:57025) Zbl 0849.53027
- [McDuff&Salamon1998] D. McDuff and D. Salamon, Introduction to symplectic topology, The Clarendon Press Oxford University Press, 1998. MR1698616 (2000g:53098) Zbl 0978.53120