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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