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

local models for symplectic manifolds

In contrast to Riemannian geometry, symplectic manifolds are ``locally symplectomorphic.'' This means that one cannot distinguish two symplectic manifolds simply based on the symplectic geometry of a small neighborhood of a point. In this section, we sketch a proof of this fact.

theorem 0.0.1

Let \(V_0\) and \(V_1\) be closed connected \(n\) manifolds with volume forms \(\Omega_0\) and \(\Omega_1\). Suppose that they have the same total volume, that is \[\int_{V_0} \Omega_0 = \int_{V_1} \Omega_1.\] Let \(\phi_0: V_0\to V_1\) be a diffeomorphism. Then \(\phi_0\) is isotopic to a volume preserving diffeomorphism \(\phi_1\); that is \[\phi_1^*\Omega_1=\Omega_0.\]
Without loss of generality, let \(V_0=V_1\) and let \(\phi_0\) be the identity. Let \(\Omega_t=(1-t)\Omega_0+t\Omega_1\). We want to find \(\{\phi_t\}\) with \(\phi_t^*\Omega_t=\Omega_0\). We could equivalently describe such an isotopy \(\phi_t\) by the vector field \[V_t:=\left(\frac{d}{dt}\phi_t\right)\circ\phi_t.\] Taking Lie derivatives, we have the pullback condition is equivalent to \[\frac{d}{dt}(\phi_t^*\Omega_t)=\phi_t^*\left(\mathcal L_{V_t} \Omega_t+\frac{d}{dt}\Omega_t\right)\] So we are looking for \(V_t\) with \(\mathcal L_{V_t}\Omega_t+\frac{d}{dt}\Omega_t=0\). Simplifying further gives us \[ d\iota_{V_t}\Omega_t=-\frac{d}{dt}\Omega_t=\Omega_0-\Omega_1\] By Stoke's theorem, the right term is exact as \(\int_X\Omega_0=\int_X \Omega_1\). Therefore there exists some \(n-1\) form \(\eta\) so that \(d\iota_{V_t}\Omega_t=d\eta\). So now we have to solve \(\iota_{V_t}\Omega_t=\eta\). Since \(\Omega_t\) is a volume form, then there exists a \(V_t\) satisfying this equation. This tells us that symplectomorphisms of surfaces are boring because any diffeomorphism can be made into a symplectomorphism. In fact, every isotopy of symplectic forms in the same cohomology can be realized by a family of symplectomorphisms.

theorem 0.0.2

Let \(X\) be a closed \((2n)\)-manifold. Let \(\{\omega_t\}\) be a smooth family of symplectic forms in the same cohomology class. Then there exists a smooth family of diffeomorphism \(\{\phi_t\}\) with \(\phi_0=\operatorname{id}_X\) and \(\phi_t^*\omega_t=\omega_0\).
Instead of finding \(\phi_t\), we instead search for the vector \(V_t\) generating the isotopy. Because the symplectic form should be invariant under this isotopy, we obtain the following condition on \(V_t\): \begin{align*} 0=&\frac{d}{dt}(\phi_t^*\omega_t)=\phi_t^*\left(\mathcal L_{V_t}\omega_t+\frac{d}{dt}\omega_t\right)\\ =&\mathcal L_{V_t}\omega_t + \frac{d}{dt}\omega_t\\ =&d\iota_{V_t}\omega+\frac{d}{dt}\omega_t \end{align*} Since the cohomology class \([\omega_t]\) is constant, the time derivative \(\frac{d}{dt}\omega_t\) is exact. Therefore there exists 0 \(\eta_t\) with \(d\eta_t=\frac{d}{dt}\omega_t\). This reduces our previous computation to \begin{align*} =&d\iota_{V_t}\omega_t+d\eta_t \end{align*} Since \(\omega_t\) is nondegenerate, there exists unique \(V_t\) with \(\iota_{V_t}\omega_t+\eta_t=0\). We also have a relative version of the Moser theorem.

theorem 0.0.3

Let \(X\) be a manifold. Let \(Y\) be a compact submanifold of \(X\). Let \(\omega_0, \omega_1\) be symplectic forms on \(Y\) such that \(\omega_0(p)=\omega_1(p)\) for all \(p\in X\). Then there exists neighborhoods \(U_0, U_1\supset X\) and a symplectomorphism \(\phi:(U_0, \omega_0)\to (U_1, \omega_1)\) such that \(\phi(p)=p\) for all \(p\in X\).
Let \(\omega_t=(1-t)\omega_0+t\omega_1\) for \(t\in [0, 1)\). Then \(\omega_t\) is symplectic for all \(t\) in a sufficiently small tubular neighborhood \(N\) of \(X\). We want to find \(\phi_t\) with \(\phi_t^*\omega_t=\omega_0\) for all \(t\). Let \(V_t=\frac{d}{dt}\phi_t\circ \phi_t\). Recall that \begin{align*} \frac{d}{dt}(\phi_t^*\omega_t)=&\phi^*_t(\mathcal L_{V_t}\omega_t+(\omega_1-\omega_0))\\ =&\phi^*_t(di_{V_t}\omega_t+(\omega_1-\omega_0)) \end{align*} Since \(i^*\omega_0=i^*\omega_1\), we have that \([\omega_0]=[\omega_1]\in H^2(N)\) . Then there exists \(\eta\) such that \(d\eta=\omega_0-\omega_1\) and \(\eta(p)=0\) for all \(p\in X\). \begin{align*} =&\phi^*d(i_{V_t}\omega-\eta) \end{align*} Since \(\omega_t\) is nondegenerate, there exists unique vector field \(V_t\) with \(i_{V_t}\omega_t=\eta\). Since \(\eta(p)=0\) for all \(p\in X\), it follows that \(V_t(p)=0\) for all \(p\in X\). For this \(V_t\) we have that \(\frac{d}{dt}(\phi^*_t\omega_t)=0\). Therefore \(V_t\) generates a family of diffeomorphism \(\{\phi_t\}\) which are the identity on \(X\) and defined on a neighborhood of \(X\). The application of this theorem to symplectic geometry shows that there are no local invariants of symplectic manifolds. If \((X, \omega_0)\) is symplectic then for all \(p\in X\) there exists neighborhoods \(U_0\supset \{p\}\) and \(U_1\subset \RR^{2n}\) and a symplectomorphism \((U_0, \omega_0)\to (U_1, \omega_{std})\).