\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
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 \((U_0, \omega_0)\to (U_1, \omega_{std})\).