24 2. SYMPLECTIC GEOMETRY AND ANALYSIS

The meaning of (2.5.1) is that σ(u) = 0 for each point z ∈ Λ and for all

u = (u1,u2) with u1,u2 ∈ Tz(Λ), the tangent space to Λ at z.

THEOREM 2.13 (Lagrangian submanifolds). Let Λ be a Lagrangian

submanifold of

R2n.

Then each point z ∈ Λ lies in a relatively open neigh-

borhood U ⊂ Λ within which

(2.5.2) ω|Λ = dϕ

for some smooth function ϕ : U → R.

Proof. Given z ∈ Λ, we find a relatively open neighborhood U ⊂ Λ and a

smooth diffeomorphism γ : U → V , where V =

B0(0,

1) is the open unit ball

in

Rn.

Then ρ :=

γ−1

pulls back ω|Λ to the 1-form α :=

ρ∗(ω|Λ),

defined on

V .

According to (2.5.1), we have

dα =

d(ρ∗ω|Λ)

=

ρ∗(dω|Λ)

=

ρ∗(σ|Λ)

= 0

within V . Poincar´ e’s Lemma (Theorem B.5) therefore implies α = dψ for

some smooth function ψ : V → R. Set ϕ := ψ ◦ γ =

γ∗ψ.

Then

dϕ =

d(γ∗ψ)

=

γ∗dψ

=

γ∗α

= ω|Λ.

We show next that a Lagrangian submanifold is locally determined by

the graph of a generating function of appropriate coordinates:

THEOREM 2.14 (Generating functions for Lagrangian submani-

folds).

Suppose that Λ ⊂

Rn

×

Rn

is a smooth Lagrangian submanifold and that

(x0,ξ0) ∈ Λ. Then there exist a neighborhood U ⊂

Rn

×

Rn

of (x0,ξ0), a

splitting of coordinates

(2.5.3) x = (x , x ), ξ = (ξ , ξ ),

where k ∈ {0,...,n} and x , ξ ∈

Rk,x

, ξ ∈

Rn−k,

and a smooth function

(2.5.4) ϕ = ϕ(x , ξ )

such that

(2.5.5) Λ ∩ U = {(x , −∂ξ ϕ; ∂x ϕ, ξ ) | x ∈

Rk,ξ

∈

Rn−k}

∩ U.

We call ϕ = ϕ(x , ξ ) a local generating function of Λ near (x0,ξ0).

Proof. 1. Let V ⊂

Rn

be a coordinate chart for a neighborhood of (x0,ξ0)

in Λ:

ρ : V → Λ ⊂

R2n,