Symplectic manifolds

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1 Introduction

A symplectic manifold is a smooth manifold $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ together with a differential two-form $\omega$$\omega$ that is nondegenerate and closed. The form $\omega$$\omega$ is called a symplectic form. The nondegeneracy means that the highest nonzero power of $\omega$$\omega$ is a volume form on $M.$$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$$\bullet$ The most basic example of a symplectic manifold is $\mathbb R^{2n}$$\mathbb R^{2n}$ equipped with the form $\omega_0:=dx^1\wedge dy^1 + \ldots + dx^n\wedge dy^n.$$\omega_0:=dx^1\wedge dy^1 + \ldots + dx^n\wedge dy^n.$

A theorem of Darboux [McDuff&Salamon1998] states that locally every symplectic manifold if of this form. More precisely, if $(M,\omega)$$(M,\omega)$ is a symplectic $2n$$2n$-manifold then for every point $x\in M$$x\in M$ there exists an open neighbourhood $U\subset M$$U\subset M$ of $p$$p$ and a diffeomorphism $f\colon U\to f(U)\subset \mathbb R^{2n}$$f\colon U\to f(U)\subset \mathbb R^{2n}$ such that the restriction of $\omega$$\omega$ to $U$$U$ is equal to the pull-back $f^*\omega_0.$$f^*\omega_0.$ This implies that symplectic manifolds have no local invariants.

$\bullet$$\bullet$ An area form on an oriented surface is symplectic.

$\bullet$$\bullet$ Let $X$$X$ be a smooth manifold and let $\lambda$$\lambda$ be a one-form on the cotangent bundle $T^*X$$T^*X$ defined as follows. If $V$$V$ is a vector tangent to $T^*X$$T^*X$ at a point $\alpha$$\alpha$ then $\lambda_{\alpha}(X) = \alpha (\pi_*(X)),$$\lambda_{\alpha}(X) = \alpha (\pi_*(X)),$ where $\pi\colon T^*X\to X$$\pi\colon T^*X\to X$ is the projection. In local coordinates the form $\lambda$$\lambda$ can be expressed as $\sum y^idx^i.$$\sum y^idx^i.$ The differential $d\lambda$$d\lambda$ is a symplectic form on the cotangent bundle $T^*X.$$T^*X.$

$\bullet$$\bullet$ If $(M,\omega)$$(M,\omega)$ is a closed, i.e. compact and without boundary, symplectic $2n$$2n$-manifold then the cohomology classes $[\omega]^k$$[\omega]^k$ are non-zero for $k=0,1.\ldots,n.$$k=0,1.\ldots,n.$ This follows from the fact that the cohomology class of the volume form $\omega^n$$\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$$M \times S^k$ is symplectic for $k>2.$$k>2.$

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

3 Symmetries

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

It follows from the nondegeneracy of the symplectic form $\omega$$\omega$ the map $X \mapsto \iota_X\omega$$X \mapsto \iota_X\omega$ defines an isomorphism between the vector fields and the one-forms on a symplectic manifold $(M,\omega).$$(M,\omega).$ If the flow of a vector field $X$$X$ preserves the symplectic form we have that $0 = L_X\omega = d\iota_X \omega + \iota _X d\omega.$$0 = L_X\omega = d\iota_X \omega + \iota _X d\omega.$ Then the closedness of the symplectic form implies that the one-form $\iota_X\omega$$\iota_X\omega$ is closed. It follows that the Lie algebra of the group of symplectic diffeomorhism consists of the vector fields $X$$X$ for which the one-form $\iota _X \omega$$\iota _X \omega$ is closed. Hence it can be identified with the space of closed one-forms.

If the one-form $\iota _X \omega$$\iota _X \omega$ is exact, i.e. $\iota _X \omega = dH$$\iota _X \omega = dH$ for some function $H\colon M\to \mathbb R$$H\colon M\to \mathbb R$ then the vector field $X$$X$ is called Hamiltonian. Symplectic diffeomorphism generated by Hamiltonian flows form a group $\operatorname{Ham}(M,\omega)$$\operatorname{Ham}(M,\omega)$ called the group of Hamiltonian diffeomorphism. Its Lie algebra can be identified with the quotient of the space of smooth functions on $M$$M$ by the constants.

4 Constructions

4.1 Products

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

4.2 Bundles

A locally trivial bundle $M\to E\to B$$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 symplectic bundle $T^2 \to S^3 \times S^1 \to S^2.$$T^2 \to S^3 \times S^1 \to S^2.$ Indeed, the structure group is a group of rotations of the torus and hence it preserves the area. As we have seen above the product $S^3 \times S^1$$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$$M\stackrel {i}\to E\stackrel{\pi}\to B$ is a compact symplectic bundle over a symplectic base. According to a theorem of Thurston, if there exists a cohomology class $a\in H^2(E)$$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$$\alpha$ of the class $a$$a$ such that $\Omega := \alpha + k\cdot \pi^*(\omega_B)$$\Omega := \alpha + k\cdot \pi^*(\omega_B)$ is a symplectic form on $E$$E$ for every big enough $k.$$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. Let there be given a principal fiber bundle

$\displaystyle G\rightarrow P\rightarrow M.$ $$\displaystyle G\rightarrow P\rightarrow M.$$

Let $\theta$$\theta$ be a connection form, $\Omega$$\Omega$ the curvature form of this connection, and $\mathcal{H}$$\mathcal{H}$ be the horizontal distribution. A vector $v\in\frak{g}^*$$v\in\frak{g}^*$ is called fat (with respect to the given connection), if the 2-form

$\displaystyle (X,Y)\rightarrow \langle \Omega(X,Y),v\rangle$ $$\displaystyle (X,Y)\rightarrow \langle \Omega(X,Y),v\rangle$$

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 $(F,\omega)$$(F,\omega)$ endowed with a hamiltonian action of a Lie group $G$$G$. Let $\mu: F\rightarrow\frak{g}^*$$\mu: F\rightarrow\frak{g}^*$ be the moment map of the $G$$G$-action. If $\mu(F)\subset\frak{g}^*$$\mu(F)\subset\frak{g}^*$ consists of fat vectors, then the associated bundle

$\displaystyle F\rightarrow P\times_GF\rightarrow M$ $$\displaystyle F\rightarrow P\times_GF\rightarrow M$$

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 $2n$$2n$-dimensional manifold $M$$M$:

$\displaystyle SO(2n)\rightarrow P\rightarrow M.$ $$\displaystyle SO(2n)\rightarrow P\rightarrow M.$$

Let $F=SO(2n)/U(n)$$F=SO(2n)/U(n)$. The associated bundle with fiber $SO(2n)/U(n)$$SO(2n)/U(n)$ is called the twistor bundle. It is easy to see that $SO(2n)/U(n)$$SO(2n)/U(n)$ can be identified with a coadjoint orbit $F_{\xi}$$F_{\xi}$ of some $\xi\in\frak{g}^*$$\xi\in\frak{g}^*$, where $\frak{g}^*$$\frak{g}^*$ denotes the Lie algebra of $SO(2n)$$SO(2n)$. Moreover, if $M$$M$ admits Riemannian metric of pinched curvature with sufficiently small pinching constant then $\xi$$\xi$ is fat with respect to the Levi-Civitta connection in the frame bundle. As a result, the whole coadjoint orbit (which is the image of the moment map of the $SO(2n)$$SO(2n)$-action) consists of fat vectors. Thus, we obtain a fiberwise symplectic structure on the total space of any twistor bundle

$\displaystyle F_{\xi}\rightarrow P\times_{SO(2n)}F_{\xi}\rightarrow M$ $$\displaystyle F_{\xi}\rightarrow P\times_{SO(2n)}F_{\xi}\rightarrow M$$

over even-dimensional manifolds of pinched curvature. In particular, twistor bundles over spheres $S^{2n}$$S^{2n}$ or hyperbolic manifolds, admit fiberwise symplectic structures. The simplest example of this construction is the fibering of $\mathbb{C}P^3$$\mathbb{C}P^3$ over $S^4$$S^4$ with fiber $\mathbb{C}P^1$$\mathbb{C}P^1$, since it is known that the total space of the twistor bundle over $S^4$$S^4$ is $\mathbb{C}P^3$$\mathbb{C}P^3$.

Example (locally homogeneous complex manifolds) Let $G$$G$ be a Lie group of non-compact type, which is a real form of a complex Lie group $G^c$$G^c$. Choose a parabolic subgroup $B\subset G^c$$B\subset G^c$ and a maximal compact subgroup $K$$K$ in $G$$G$. Assume that $V=B\subset G$$V=B\subset G$ is compact. Then one can show that $K/V$$K/V$ can be identified with a coadjoint orbit of some vector in $\frak{k}^*$$\frak{k}^*$, which is fat with respect to a $K$$K$-invariant connection in the principal bundle

$\displaystyle K\rightarrow G/V\rightarrow G/K.$ $$\displaystyle K\rightarrow G/V\rightarrow G/K.$$

It follows that the associated bundle

$\displaystyle K/V\rightarrow G/V\rightarrow G/K$ $$\displaystyle K/V\rightarrow G/V\rightarrow G/K$$

is a symplectic fiber bundle with fiberwise symplectic structure. This construction can be compactified by taking lattices in $G$$G$ which intersect trivially with $K$$K$. A particular example is given by the fiber bundle

$\displaystyle SO(2n)/U(n)\rightarrow SO(2n,p)/U(n)\times SO(p)\rightarrow SO(2n,p)/SO(2n)\times SO(p),$ $$\displaystyle SO(2n)/U(n)\rightarrow SO(2n,p)/U(n)\times SO(p)\rightarrow SO(2n,p)/SO(2n)\times SO(p),$$

and its compactification by lattices.

4.3 Symplectic reduction

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

4.4 Symplectic cut

Let $(M,\omega)$$(M,\omega)$ be a symplectic manifold with a hamiltonian action of the circle $S^1.$$S^1.$ If $\mu:\rightarrow \mathbb R$$\mu:\rightarrow \mathbb R$ is the moment map, $M_a=\mu^{-1}(a)$$M_a=\mu^{-1}(a)$ is a regular level, then the action restricted to $M_a$$M_a$ has no fixed points, hence $M_a$$M_a$ 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.

4.6 Symplectic homogeneous spaces

Nilmanifolds, solvmanifolds, homogeneous spaces of semisimple Lie groups

4.8 Surgery in codimension 2

Consider two symplectic manifolds $(M_1,\omega_1),(M_2,\omega_2)$$(M_1,\omega_1),(M_2,\omega_2)$ of equal dimension and suppose that there are codimension two symplectic submanifolds $V_1\subset M_1,V_2\subset M_2$$V_1\subset M_1,V_2\subset M_2$ and a symplectomorphism $f:V_1\rightarrow V_2$$f:V_1\rightarrow V_2$ such that Chern classes of normal bundles satisfy $f^*c_1(\nu_2)=-c_1(\nu_1).$$f^*c_1(\nu_2)=-c_1(\nu_1).$ Then by removing tubular neighborhods of $V_1$$V_1$ and $V_2$$V_2$ we get manifolds with boundaries. The map $f$$f$ induces a diffeomorphism of the boundaries, one can form a new manifold identifying the boundaries by this diffeomorphism and define on it a symplectic form which coincides with $\omega_1$$\omega_1$ and $\omega_2$$\omega_2$ outside of a tubular neigborhood of the trace of glueing [Gompf1995]. The same works if $V_1,V_2$$V_1,V_2$ are symplectic submanifolds of a connected symplectic manifold.

5 Invariants

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6 Classification/Characterization

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7 Further discussion

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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