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 κt

∗ηt

= σ near (0, 0); then κ := κ1 solves our problem. We will

construct κt by solving the flow

(2.4.12)

˙ w = 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)

= κt

∗

(∂tηt) + κt

∗LVt

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)(κt

∗ηt)

= 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.