Local Forms

Homotopy between Manifold

A homotopy from a smooth function $f:M\to N$ to a smooth function $g:M\to N$ is a smooth function $\rho :M\times [0,1]\to N$ such that $\rho (x,0)=f(x)$ and $H(x,1)=g(x)$ for all $x\in M$. A homotopy is an isotopy if ${\rho }_t$ is an embedding for all $t\in [0,1]$, i.e. each ${\rho }_t:M\to M$ is a diffeomorphism, and ${\rho }_0={\text{id}}_M$.

Proposition

Given an isotopy $\rho$ we get a time-dependent vector field that is, a family of vector fields $v_t$, $t\in \mathbb{R}$, which at $p\in M$ satisfy

i.e.,\frac{d\rho_t}{dt} = v_t \circ \rho_t ;.$$

Conversely, given a time-dependent vector field $v_t$, if $M$ is compact or if the $v_t$‘s are compactly supported there exists an isotopy $\rho$ satisfying $\frac{d{\rho }_t}{dt}=v_t\circ {\rho }_t$. So for compact manifold, we get a one-to-one correspondence between isotopies of $M$ and time-dependent vector fields of $M$.

Tubular Neighborhood Theorem

Let M be an n-dimensional manifold, and let X be a k-dimensional submanifold where $k<n$ and with inclusion map $i:X\to M$. Consider normal space $N_{x}X:=T_{x}M/T_{x}X$, a (n-k) dimensional vector space. The normal bundle of $X$ is $NX=\{(x,v)|x\in X,v\in N_{x}X\}$ The zero section of $NX$, $i_0:X\to NX,x\to (x,0)$ , embeds $X$ as a closed submanifold of $NX$. A neighborhood ${\mathcal{U}}_0$ of the zero section $X$ in $NX$ is called convex if the intersection $U_0\cap N_{x}X$ with each fiber is convex. There exist a convex neighbourhood ${\mathcal{U}}_0$ of $X$ in $NX$, a neighbourhood $\mathcal{U}$ of $X$ in M, and a diffeomorphism $\phi :{\mathcal{U}}_0\to \mathcal{U}$ such that

\mathcal{U}{0}\subset NX & \xrightarrow{\quad \varphi \quad} & \mathcal{U} \subset M \ & \nwarrow i{0} \quad \nearrow i & \ & X & \end{matrix}$$

commutes.

Proposition

If a closed $\ell$-form $\omega$ on $\mathcal{U}$ has restriction $i^{\ast }\omega =0$, then $\omega$ is exact, i.e., $\omega =d\mu$ for some $\mu \in {\Omega }^{d-1}(\mathcal{U})$. Moreover, we can choose $\mu$ such that ${\mu }_x=0$ at all $x\in X$.

Proof Sketch Via $\phi :{\mathcal{U}}_0\stackrel{\cong }{\to }\mathcal{U}$, it is equivalent to work over ${\mathcal{U}}_0$. Define for every $0\le t\le 1$ a map ${\rho }_t:\begin{array}{ccc}{\mathcal{U}}_0& ⟶& {\mathcal{U}}_0\\ (x,v)& ⟼& (x,tv)\end{array}.$This is well-defined since ${\mathcal{U}}_0$ is convex. The map ${\rho }_1$ is the identity, ${\rho }_0=i_0\circ {\pi }_0$, and each ${\rho }_t$ fixes $X$, that is, ${\rho }_t\circ i_0=i_0$. We hence say that the family $\{{\rho }_t\mid 0\le t\le 1\}$ is a homotopy from $i_0\circ {\pi }_0$ to the identity fixing $X$. The map ${\pi }_0:{\mathcal{U}}_0\to X$ is called a retraction because ${\pi }_0\circ i_0$ is the identity. The submanifold $X$ is then called a deformation retract of $\mathcal{U}$. Consider concrete operator $Q$ is given by the formula: $Q\omega ={\int }_0^1{\rho }_t^{\ast }({ı}_{v_t}\omega )dt,$ we get to check $Q$ satisfies homotopy formula, that is $\text{Id}-(i_0\circ {\pi }_0{)}^{\ast }=dQ+Qd.$ When $d\omega =0$ and $i_0^{\ast }\omega =0$, the operator $Q$ gives $\omega =dQ\omega$, so for $\mu =Q\omega$, we get $\omega =d\mu$.

Notions of Equivalence for Symplectic Structures

Definition

$(M,{\omega }_0)$ and $(M,{\omega }_1)$ are symplectomorphic if there is a diffeomorphism $\phi :M\to M$ with ${\phi }^{\ast }{\omega }_1={\omega }_0$; $(M,{\omega }_0)$ and $(M,{\omega }_1)$ are strongly isotopic if there is an isotopy ${\rho }_t:M\to M$ such that ${\rho }_1^{\ast }{\omega }_1={\omega }_0$; $(M,{\omega }_0)$ and $(M,{\omega }_1)$ are deformation-equivalent if there is a smooth family ${\omega }_t$ of symplectic forms joining ${\omega }_0$ to ${\omega }_1$; $(M,{\omega }_0)$ and $(M,{\omega }_1)$ are isotopic if they are deformation-equivalent with $[{\omega }_t]$ independent of $t$.

Proposition

Clearly, we have $\text{strongly isotopic}⟹\text{symplectomorphic}$, $\text{isotopic}⟹\text{deformation-equivalent}$, $\text{strongly isotopic}⟹\text{isotopic}$, and Moser’s Theorem states on compact manifold, $\text{isotopic}⟹\text{strongly isotopic}$

Moser’s Theorem

Remark

Given a $2n$-dimensional manifold $M$, a $k$-dimensional submanifold $X$, neighborhoods $U_0,U_1$ of $X$, and symplectic forms ${\omega }_0,{\omega }_1$ on $U_0,U_1$, does there exist a symplectomorphism preserving $X$? More precisely, does there exist a diffeomorphism $\varphi :U_0\to U_1$ with ${\varphi }^{\ast }{\omega }_1={\omega }_0$ and $\varphi (X)=X$?
At the two extremes, we have: Case $X=\text{point}$: Darboux theorem, Case $X=M$: Moser theorem

Moser's Theorem (first version)

Suppose $[{\omega }_0]=[{\omega }_1]$ on a compact manifold and the 2-form ${\omega }_t=(1-t){\omega }_0+t{\omega }_1$ is symplectic for each $t\in [0,1]$. Then there exists an isotopy $\rho :M\times \mathbb{R}\to M$ such that ${\rho }_t^{\ast }{\omega }_t={\omega }_0$ for all $t\in [0,1]$.

Moser's Theorem (second version)

Let M be a compact manifold with symplectic forms ${\omega }_0$ and ${\omega }_1$. Suppose that ${\omega }_t$, $0\le t\le 1$, is a smooth family of closed 2-forms joining ${\omega }_0$ to ${\omega }_1$ and satisfying: (1) cohomology assumption: $[{\omega }_t]$ is independent of t, i.e., $\frac{d}{dt}[{\omega }_t]=[\frac{d}{dt}{\omega }_t]=0$, (2) nondegeneracy assumption: ${\omega }_t$ is nondegenerate for $0\le t\le 1$. Then there exists an isotopy ${\rho }_t^{\ast }{\omega }_t={\omega }_0$ for all $t\in [0,1]$.

Moser Relative Theorem

Let $M$ be a manifold, $X$ a compact submanifold of $M$, $i:X\hookrightarrow M$ the inclusion map, ${\omega }_0$ and ${\omega }_1$ symplectic forms on $M$ such that ${\omega }_0{|}_X={\omega }_1{|}_X$. Then there exists neighbourhoods $U_0$, $U_1$ of $X$ and a diffeomorphism $\phi :U_0\to U_1$ such that the following diagram commutes: and ${\phi }^{\ast }{\omega }_1={\omega }_0$.

Darboux Theorem

Let $(M,\omega )$ be a symplectic manifold, and let $p$ be any point in $M$. Then we can find a coordinate $(x_1,\dots ,x_n,y_1,\dots ,y_n)$ on $U$ containing $p$ such that $\omega ={\sum }_{j=1}^n\text{d}{x}_j\wedge \text{d}{y}_j.$

Proof Apply the Moser relative theorem to $X=\{p\}$.