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 =
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
=
d(ρ∗ω|Λ)
=
ρ∗(dω|Λ)
=
ρ∗(σ|Λ)
= 0
within V . Poincar´ e’s Lemma (Theorem B.5) therefore implies α = for
some smooth function ψ : V R. Set ϕ := ψ γ =
γ∗ψ.
Then
=
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 ),
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,
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