Symplectic manifolds

Revision as of 15:05, 1 October 2010

1 Introduction

A symplectic manifold is a smooth manifold $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; 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. </wikitex> == Examples == <wikitex>; $\bullet$ 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 {{cite|McDuff-Salamon}} states that locally every symplectic manifold if of this form. More precisely, if $(M,\omega)$ is a symplectic n$-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 n$-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 $k>2.$ </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[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 $k.$

3 Surgery

5 Invariants

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6 Classification/Characterization

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7 Further discussion

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8 References

• [McDuff-Salamon] Template:McDuff-Salamon