Darboux Theorem
傅杰, 范佳轩
2022 年 11 月 24 日
Definition (M, ω) is a symplectic manifold if m is a manifold and ω is a
symplectic form. Namely, ω is closed and for each p ∈ M , the map ωp is
symplectic.
Theorem(Darboux) Let (M, ω) be a 2n-dimensional symplectic manifold. Then ∀p ∈ M , ∃ coordinate chart (U , x1 , . . . , xn , y1 , . . . , yn ) centered
at p such that
ω=
n
∑
dxi ∧ dyi
i=1
on U
Theorem(Moser, Version 1)
Suppose M is compact, [ω0 ] = [ω1 ] and ωt = (1 − t)ω0 + tω1 is symplectic
for t ∈ [0, 1]. Then ∃ isotopy ρ : M × R such that ρ∗t ωt = ω0 , t ∈ [0, 1]
Proof
For desired ρ let
vt =
Then
0=
dρt
◦ ρ−1
t
dt
dωt
d ∗
(ρt ωt ) = ρ∗t (Lvt ωt +
)
dt
dt
is equivalent to
dωt
=0
dt
If the equation is satisfied, we can integrate vt to obtain desired ρt
L vt ω t +
First,
dωt
= ω1 − ω0
dt
1
2
Second, [ω0 ] = [ω1 ] implies ∃ 1-form µ such that
ω1 − ω0 = dµ
Third, by Cartan magic formula,
Lvt ωt = dιvt ωt + ιvt dωt
Altogether we just have to find vt such that
dιvt ωt + dµ = 0
It is sufficient to solve ιvt ωt + µ=0. By the nondegeneracy of ωt , we can
solve the equation pointwise to obtain solution vt .
Theorem(Moser, Relative Version) Let M be a manifold, X a compact submanifold of M with i : X ,→ M the inclusion map. ω0 , ω1 are
symplectic forms on M such that
ω0 |p = ω1 |p , ∀p ∈ X
Then ∃ neighborhood U0 , U1 of X in M , diffeomorphism ϕ : U0 → U1 such
that
φ
U0
i
U1
i
X
commutes and ϕ∗ ω1 = ω0
Proof
Pick a tubular neighborhood U0 of X. ω1 − ω0 is closed on U0 and (ω1 −
ω0 )|p = 0, ∀p ∈ X. The homotopy formula on the tubular neighborhood
implies ∃ 1-form µ on U0 such that ω1 − ω0 = dµ and µp = 0, ∀p ∈ X
Let ωt = (1 − t)ω0 + tω1 . Under proper shrinking of U0 we can suppose ωt
is symplectic for 0 ≤ t ≤ 1
We can determine vt from Moser equation
ιvt ωt = −µ
3
Notice that vt = 0 on X
Integrate vt , we obtain isotopy ρ : U × [0, 1] → M with ρ∗t ωt = ω0 , t ∈ [0, 1].
vt = 0 on X gives ρt |X = idX
Just let ϕ = ρ1 , U1 = ρ1 (U0 ).
Proof of Darboux Theorem
Consider any symplectic basis (x′1 , . . . , x′n , y1′ , . . . , yn′ ) of Tp M .
∑n
Let ω0 = ω, ω1 = i=1 dx′i ∧ dyi′ .
Applying Moser theorem to X = p, we obtain there are neighborhoods U0
and U1 of p, and diffeomorphism ϕ : U0 → U1 such that
∗
ϕ(p) = p, ϕ (
n
∑
dx′i ∧ dyi′ ) = ω
i=1
Thus
ω=
n
∑
d(x′i ◦ ϕ)′ ∧ d(yi′ ◦ ϕ) =
i=1
where xi = x′i ◦ ϕ, yi = yi′ ◦ ϕ
n
∑
i=1
dxi ∧ dyi