Symplectic manifolds

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(Constructions)
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===Coadjoint orbits===
===Coadjoint orbits===
===Symplectic homogeneous spaces===
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===Symplectic homogeneous spaces===[[Media:Example.ogg]]
===Surgery===
===Surgery===
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Consider two symplectic manifolds $(M_1,\omega_1),(M_2,\omega_2)$ and suppose that there are a codimension two symplectic submanifolds $V_1\subset M_1,V_2\subset M_2$ with a symplectomorphism $f:V_1\rightarrow V_2$ such that Chern classes of normal bundles satisfy $f^*c_1\nu_2=\c_1\nu_1.$
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Then one can get a new symplectic manifold by cutting neighborhods of the submanifolds and identifying the boundaries.

Revision as of 13:14, 26 November 2010


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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.

\bullet The complex projective space \mathbb C \mathbb P^n is symplectic with respect to its K\"ahler form. Its pull back to a complex projective smooth manifold X \subset \mathbb C \mathbb P^n is also symplectic. More generally, every K\"ahler manifold is symplectic.



3 Symmetries

A diffeomorphism f\colon M\to M of a symplectic manifold (M,\omega) is called symplectic if it preserves the symplectic form, f^*\omega = \omega. Sometimes such a diffeomorphism is called a symplectiomorphism. The group of all symplectic diffeomorphisms of (M,\omega) is denoted by \operatorname{Symp}(M,\omega).

It follows from the nondegeneracy of the symplectic form \omega the map X \mapsto \iota_X\omega defines an isomorphism between the vector fields and the one-forms on a symplectic manifold (M,\omega). If the flow of a vector field X preserves the symplectic form we have that 0 = 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.




4 Constructions


1 Products

The product of symplectic manifolds (M_1,\omega_1) and (M_2,\omega_2) is a symplectic manifold with respect to the form a\cdot p_1^*\omega_1 + b\cdot p_2^*\omega_2 for nonzero real numbers a,b\in \mathbb R. Here p_i\colon M_1\times M_2\to M_i is the projection.

2 Bundles

A locally trivial bundle M\to E\to B 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 T^2 \to S^3 \times S^1 \to S^2. 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 S^3 \times S^1 does not admit a symplectic form. This example shows that, in general, the total space of a symplectic bundle is not symplectic.

Let M\stackrel {i}\to E\stackrel{\pi}\to B is a compact symplectic bundle over a symplectic base. According to a theorem of Thurston, if there exists a cohomology class a\in H^2(E) 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 \alpha of the class a such that \Omega := \alpha + k\cdot \pi^*(\omega_B) is a symplectic form on E for every big enough k.

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.

3 Symplectic reduction

Let G be a Lie group acting on a symplectic manifold (M,\omega) in a hamiltonian way. Denote by \mu: M\rightarrow\frak{g}^* the moment map of this action. Since G acts on the level set \mu^{-1}(a),a\in\frak{g}^*, one can consider the orbit space \mu^{-1}(a)/G. It is an orbifold in general, but it happens to be a manifold, when G acts freely on the preimage, and a is a regular point. In this case, \tilde M=\mu^{-1}(a)/G is a symplectic manifold as well, called symplectic reduction. It is often denoted by M//G

4 Coadjoint orbits

===Symplectic homogeneous spaces===Media:Example.ogg

5 Surgery

Consider two symplectic manifolds (M_1,\omega_1),(M_2,\omega_2) and suppose that there are a codimension two symplectic submanifolds V_1\subset M_1,V_2\subset M_2 with a symplectomorphism f:V_1\rightarrow V_2 such that Chern classes of normal bundles satisfy
Tex 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== ===Products=== The product of symplectic manifolds $(M_1,\omega_1)$ and $(M_2,\omega_2)$ is a symplectic manifold with respect to the form $a\cdot p_1^*\omega_1 + b\cdot p_2^*\omega_2$ for nonzero real numbers $a,b\in \mathbb R.$ Here $p_i\colon M_1\times M_2\to M_i$ is the projection. ===Bundles=== A locally trivial bundle $M\to E\to B$ 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 $T^2 \to S^3 \times S^1 \to S^2.$ 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 $S^3 \times S^1$ does not admit a symplectic form. This example shows that, in general, the total space of a symplectic bundle is not symplectic. Let $M\stackrel {i}\to E\stackrel{\pi}\to B$ is a compact symplectic bundle over a symplectic base. According to a theorem of Thurston, if there exists a cohomology class $a\in H^2(E)$ 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 $\alpha $ of the class $a$ such that $\Omega := \alpha + k\cdot \pi^*(\omega_B)$ is a symplectic form on $E$ for every big enough $k.$ 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'''. ===Symplectic reduction=== Let $G$ be a Lie group acting on a symplectic manifold $(M,\omega)$ in a hamiltonian way. Denote by $\mu: M\rightarrow\frak{g}^*$ the moment map of this action. Since $G$ acts on the level set $\mu^{-1}(a),a\in\frak{g}^*$, one can consider the orbit space $\mu^{-1}(a)/G$. It is an orbifold in general, but it happens to be a manifold, when $G$ acts freely on the preimage, and $a$ is a regular point. In this case, $\tilde M=\mu^{-1}(a)/G$ is a symplectic manifold as well, called '''symplectic reduction'''. It is often denoted by $M//G$ ===Coadjoint orbits=== ===Symplectic homogeneous spaces=== ===Surgery=== == Invariants == ; ... == Classification/Characterization == ; ... == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]]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.

\bullet The complex projective space \mathbb C \mathbb P^n is symplectic with respect to its K\"ahler form. Its pull back to a complex projective smooth manifold X \subset \mathbb C \mathbb P^n is also symplectic. More generally, every K\"ahler manifold is symplectic.



3 Symmetries

A diffeomorphism f\colon M\to M of a symplectic manifold (M,\omega) is called symplectic if it preserves the symplectic form, f^*\omega = \omega. Sometimes such a diffeomorphism is called a symplectiomorphism. The group of all symplectic diffeomorphisms of (M,\omega) is denoted by \operatorname{Symp}(M,\omega).

It follows from the nondegeneracy of the symplectic form \omega the map X \mapsto \iota_X\omega defines an isomorphism between the vector fields and the one-forms on a symplectic manifold (M,\omega). If the flow of a vector field X preserves the symplectic form we have that 0 = 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.




4 Constructions


1 Products

The product of symplectic manifolds (M_1,\omega_1) and (M_2,\omega_2) is a symplectic manifold with respect to the form a\cdot p_1^*\omega_1 + b\cdot p_2^*\omega_2 for nonzero real numbers a,b\in \mathbb R. Here p_i\colon M_1\times M_2\to M_i is the projection.

2 Bundles

A locally trivial bundle M\to E\to B 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 T^2 \to S^3 \times S^1 \to S^2. 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 S^3 \times S^1 does not admit a symplectic form. This example shows that, in general, the total space of a symplectic bundle is not symplectic.

Let M\stackrel {i}\to E\stackrel{\pi}\to B is a compact symplectic bundle over a symplectic base. According to a theorem of Thurston, if there exists a cohomology class a\in H^2(E) 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 \alpha of the class a such that \Omega := \alpha + k\cdot \pi^*(\omega_B) is a symplectic form on E for every big enough k.

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.

3 Symplectic reduction

Let G be a Lie group acting on a symplectic manifold (M,\omega) in a hamiltonian way. Denote by \mu: M\rightarrow\frak{g}^* the moment map of this action. Since G acts on the level set \mu^{-1}(a),a\in\frak{g}^*, one can consider the orbit space \mu^{-1}(a)/G. It is an orbifold in general, but it happens to be a manifold, when G acts freely on the preimage, and a is a regular point. In this case, \tilde M=\mu^{-1}(a)/G is a symplectic manifold as well, called symplectic reduction. It is often denoted by M//G

4 Coadjoint orbits

===Symplectic homogeneous spaces===Media:Example.ogg

5 Surgery

Consider two symplectic manifolds (M_1,\omega_1),(M_2,\omega_2) and suppose that there are a codimension two symplectic submanifolds V_1\subset M_1,V_2\subset M_2 with a symplectomorphism f:V_1\rightarrow V_2 such that Chern classes of normal bundles satisfy
Tex 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

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