2.5. LAGRANGIAN SUBMANIFOLDS 23 have η(u, v) = n i,j=1 xiyjη(ei,ej) + ξiηjη(fi,fj) + xiηjσ(ei,fj) + ξiyjσ(fi,ej) = ξ, y− x, η = σ((x, ξ), (y, η)). We leave finding L as a linear algebra exercise. 2. Next, define ηt := tη + (1 − t)σ for 0 ≤ t ≤ 1. Our intention is to find κt so that κ∗ηt t = σ near (0, 0) then κ := κ1 solves our problem. We will construct κt by solving the flow (2.4.12) ˙ = Vt(w) (0 ≤ t ≤ 1) w(0) = z and setting κt := ϕt. For this to work, we must design the vector fields Vt in (2.4.12) so that ∂t(κt ∗ ηt) = 0. Let us therefore calculate ∂t(κt ∗ ηt) = κ∗ t (∂tηt) + κ∗L t Vt ηt = κt ∗ [(η − σ) + d(Vt ηt) + Vt dηt] , where we used Cartan’s formula, Theorem B.3. Now dηt = tdη + (1 − t)dσ, and hence (d/dt)(κ∗ηt) t = 0 provided (2.4.13) (η − σ) + d(Vt ηt) = 0. 3. According to Poincar´ e’s Lemma (Theorem B.4), we can write η − σ = dα near (0, 0). So (2.4.13) will hold if (2.4.14) Vt ηt = −α (0 ≤ t ≤ 1). Since η = σ at (0, 0), ηt = σ at (0, 0). In particular, ηt is nondegenerate for 0 ≤ t ≤ 1 in a neighborhood of (0, 0), and hence we can solve (2.4.13) for the vector field Vt. 2.5. LAGRANGIAN SUBMANIFOLDS This section provides some further geometric interpretations of generating functions, introduced earlier in Example 4 in Section 2.3. DEFINITION. A Lagrangian submanifold Λ in R2n is an n-dimensional submanifold for which (2.5.1) σ|Λ = 0.

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