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 ,
= ∂x ψ, dx + ∂ξ ψ, .
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 = = ∂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
.
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