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,
