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 $t\in [0,1]$.

Lagrangian Submanifold

A Lagrangian submanifold of a symplectic manifold $(M,\omega )$ is a submanifold $L\subset M$ such that $\dim L=\frac{1}{2}\dim M$ and $\omega {|}_L=0$. In other words, the restriction of the symplectic form to $L$ vanishes.

Moser’s Theorem

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 $X=\{p\}$.