2.5. LAGRANGIAN SUBMANIFOLDS 25

with ρ(0) = (x0,ξ0). The Jacobian ∂ρ(0) has full rank and hence has n inde-

pendent rows. We choose n such rows and call the corresponding coordinates

x ∈

Rk

and ξ ∈

Rn−k.

2. Define p :

R2n

→

Rk

×

Rn−k

by

p(x, ξ) := (x , ξ ).

Our choice of the coordinates (x , ξ ) means that

p ◦ ρ : V →

Rk

×

Rn−k

has an invertible Jacobian at 0 ∈

Rn.

Hence the Implicit Function Theorem

implies p ◦ γ is invertible in a neighborhood of 0. This means that we can

use (x , ξ ) as local coordinates on Λ, and so there exists a neighborhood

U ⊂

R2n

of (x0,ξ0) and smooth functions

f :

Rk

×

Rn−k

→

Rn−k,

g :

Rk

×

Rn−k

→

Rk

such that

Λ ∩ U = {(x , f(x , ξ ),g(x , ξ ),ξ ) | (x , ξ ) ∈

Rk

×

Rn−k}

∩ U.

3. Recalling that ω = ξdx, we use Theorem 2.13 to see that for some

function ψ = ψ(x , ξ ),

ω|Λ = g, dx + ξ , ∂x fdx + ∂ξ fdξ

= g + (∂x

f)T

ξ , dx + (∂ξ

f)T

ξ , dξ

= ∂x ψ, dx + ∂ξ ψ, dξ .

That is,

ψx = g + (∂x

f)T

ξ = g + ∂x f, ξ , ψξ = (∂ξ

f)T

ξ .

If we put

ϕ(x , ξ ) := ψ(x , ξ ) − f(x , ξ ),ξ ,

then

f = −∂ξ ϕ, g = ∂x ϕ.

EXAMPLES.

(i) The simplest case is k = n, for which

Λ ∩ U = {(x, ∂ϕ(x)) : x ∈

Rn}.

Then (2.5.2) reads ω|Λ = dϕ = ∂ϕdx.

(ii) Theorem 2.14 generalizes Theorem 2.7. To see this, consider the

twisted graph of κ:

(2.5.6) Λκ = (x, y, ξ, −η) | (x, ξ) = κ(y, η), (y, η) ∈

R2n

.