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 := + (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) σ) + d(Vt ηt) = 0.
3. According to Poincar´ e’s Lemma (Theorem B.4), we can write
η σ = 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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