\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip: examples of symplectic manifolds

examples of symplectic manifolds

example 0.0.1

The simplest example comes from \(\RR^{2n}\), which we give the coordinates \((q_i, p_i)\). In these local coordinates, we can define a symplectic form by \[\omega_{std}=\sum_{i=1}^n d p_i\wedge d q_i.\] Note that when \(n=1\), this gives the standard area form on \(\RR^2\). In these coordinates, it is easy to check that \(\frac{\omega_{std}^n}{n!}=\text{vol}_{\RR^{2n}}\), the standard volume form.

example 0.0.2

Let \((X, g)\) be an oriented surface. Then \(g\) prescribes a volume form \(\omega:=\text{vol}_g\in \Omega^2(X;\RR)\), which is an example of a non-degenerate 2-form. Because \(\Omega^3(X;\RR)=0\), it trivially follows that \(\omega\) is closed. This example raises the possibility of the same space having many different symplectic forms, as an oriented surface can be equipped with several different metrics.

example 0.0.3

An example that will be especially relevant later is \((\CC^*)^n\). We will equip this with a different symplectic form than the one inherited as a subset of \(\CC^n=\RR^{2n}\). Since \((\CC^*)^n\) is a group, it is natural to ask for a symplectic form on \((\CC^*)^n\) which is invariant under the group action. The symplectic form \[ \omega=\frac{1}{2\pi} d(\log |z|)\wedge d\theta \] gives an example of such a symplectic form. When \(n=1\), then this is the area form on \((\CC^*)\) which embeds into three dimensional space as an infinitely long cylinder, as drawn in figure 0.0.4.
figure 0.0.4:The symplectic structure that we choose for \(\CC^*\) makes it an infinitely long cylinder.
The most important example of symplectic manifold comes from physics, which is the historical origin of symplectic geometry.

example 0.0.5

Let \(Q\) be a smooth \(n\)-dimensional manifold. We now describe a canonical symplectic form on the cotangent bundle, \(T^*Q\). At every point \(q\in Q\), there exists chart \(q\in U\subset Q\) which we can parameterize with coordinates \((q_1, \ldots, q_n)\). The cotangent bundle \(T^*U\) inherits coordinates \((q_1, p_1, q_2, p_2, \ldots, q_n, p_n)\), where the \(p_i\) linearly parameterize the fibers of the cotangent bundle in the direction of the basis element \(dq_i\). 0 In these coordinates, the canonical symplectic form on this chart is: \[\omega=\sum_{i=1}^n dq_i \wedge dp_i=-d(p dq).\]
In the physical literature, \(T^*Q\) is called the ``phase-space'' which encodes both the position and momentum of a particle. The canonical symplectic form describes the natural pairing of momentum and velocity. There is also a coordinate-free description of \(\omega\). To do so, we first define a canonical 1-form \(\eta\in \Omega^1(T^*Q)\). Let \((q,p)\in T^*Q\) be a point, where the coordinates take values \(q\in Q\) and \(p\in T^*_qQ\). The map \(\pi: T^*Q\to Q\) induces a map \(\pi_*: T(T^*Q)\to TQ\). The canonical 1-form is defined by its value on tangent vectors \(v\in T_{(q,p)}(T^*M)\) \[\eta_{(q,p)}(v):=p(\pi_*(v)).\] This describes the pairing between the \(Q\)-component (velocity in the base) of \(v\) and the momentum coordinate \(p\). From the canonical form, we obtain a coordinate-free definition of canonical symplectic form as \[\omega=-d\eta.\] Note that \(\omega\) is not simply a closed 2-form, but rather an exact symplectic form. One can also prove that the cotangent bundle serves as a general kind of local model for symplectic manifolds. We've already seen the cotangent bundle appear in the examples of symplectic structures on \(\RR^{2n}\) and \((\CC^*)^n\), which can be interpreted as the symplectic structures on the cotangent bundles \(T^*\RR^n\) and \(T^*T^n\) respectively. This example also gives an example of how an almost complex structure and symplectic structure can interact. Let \(Q\) be a manifold equipped with a connection. The tangent bundle of \(Q\) comes with an almost complex structure. With a choice of connection we obtain a splitting \[T_{(q,v)}TQ= T_v (T_q Q)\oplus T_q Q\] with an isomorphism \(A: T_v(T_q Q)\to T_qQ.\) One can then construct an almost complex structure by taking the matrix \[J:=\begin{pmatrix} 0 & A\\ -A^{-1} & 0\end{pmatrix}: T_{(q,v)}TQ\to T_{(q,v)}TQ.\] One can similarly (but not canonically) construct an almost complex structure for the cotangent bundle. Pick \(g\) a metric for \(Q\), which induces a bundle isomorphism between the tangent and cotangent bundle. Let \((q_1, \ldots, q_n, p_1, \ldots p_n)\) be local coordinates for \(T^*Q\) chosen so that \(\partial q_1, \ldots \partial q_n\) form an orthonormal basis at the origin and the coordinates \(p_1, \ldots, p_n\) parameterize the linear coordinates determined by the basis \(\{\iota_{\partial q_i}g\}\). Then an almost complex structure is specified by \begin{align*} J \partial q_i=\partial p_i && J\partial p_i = -\partial q_i. \end{align*} Note that the resulting almost complex structure depends on the choice of metric, which modifies the ``size'' of the cotangent fiber relative to the base. The metric \(\omega(J, -)\) is then the standard induced metric on the cotangent bundle.