Download SYMPLECTIC GEOMETRY PROBLEM SET 1 (1) (a) Prove the linear

SYMPLECTIC GEOMETRY PROBLEM SET 1
(1) (a) Prove the linear Darboux theorem (Theorem 1.3 in lecture 1).
(b) Prove the linear relative Darboux theorem (Theorem 2.6 in lecture 1).
(c) Prove the “canonical form for linear complex structure theorem” (Theorem 3.2 in lecture 1).
(2) (a) Let L : U1 → U2 be any linear isomorphism between vecter spaces.
Prove: The map
F : U1 ⊕ U1∗ → U2 ⊕ U2∗ ,
(u, α) 7→ (L(u), (L∗ )− 1(α))
is a linear symplectomorphism (with respect to the canonical symplectic
structures).
(b) Let F : (V1 , Ω1 ) → (V2 , Ω2 ) be any linear symplectomorphism. Prove:
The graph of F is a Lagrangian subspace of (V1 ⊕ V2 , Ω1 ⊕ (−Ω2 )).
(3) Prove: the image of the map
ϕ : S n−1 × S 1 → Cn = R2n ,
(4)
(5)
(6)
(7)
(8)
((x1 , · · · , xn ), eiθ ) 7→ (1 + eiθ )(x1 , · · · , xn )
is a Lagrangian submanifold of (R2n , Ω0 ).
Let X be a smooth manifold and µ any closed 2-form on X. Let π : T ∗ X →
X be the canonical projection. Prove: ωT ∗ X + π ∗ µ is a symplectic form on
T ∗ X.
(a) Prove: If ω1 and ω2 are two area forms on an oriented surface that
give the same orientation, then any convex combination of them is a
symplectic form.
(b) Given an counterexample in dimension 4: Find two symplectic forms
on R4 that induce the same orientation while a convex combination of
them is degenerate.
Let M be a compact manifold and ω a symplectic form on M . Prove: There
exists an open neighborhood U of [ω] in H 2 (M ) such that for any u ∈ U ,
there exists a symplectic structure ωu on M with [ωu ] = u.
Let U, V be two bounded open subsets of R2 having the same area and
φ : U → V a diffeomorphism that preserves the orientation. Prove: there
exists a symplectomorphism from (U, Ω0 ) to (V, Ω0 ).
Prove the linear nonsqueezing theorem: Let B2n (r) be the standard sphere
of radius r centered at 0 in R2n , and
Z2n (R) = B2 (R) × R2n−2 ⊂ R2 × R2n−2 = R2n
be the “symplectic cylinder” of of radius R. Suppose there exists a linear symplectomorphism F : (R2n , Ω0 ) → (R2n , Ω0 ) such that F (B2n (r)) ⊂
Z2n (R). Prove: r ≤ R.
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SYMPLECTIC GEOMETRY PROBLEM SET 1
[Hint:Let u = F T (e1 ), v = F T (f1 ). First prove Ω0 (u, v) = 1 and thus
kuk ≥ 1 or kvk ≥ 1. Then explain the fact F (B2n (r)) ⊂ Z2n (R) as
supkzk≤r (hu, zi2 + hv, zi2 ) ≤ R2 . ]
(9) Prove the following version of Morse lemma via Moser’s trick: Let ϕ1 and
ϕ2 be two smooth functions on R2n such that ϕ1 (0) = ϕ2 (0) = 0, ∇ϕ1 (0) =
2ϕ
2ϕ
1
2
∇ϕ2 (0) = 0 and such that the Hessian matrix [ ∂x∂ i ∂x
(0)] = [ ∂x∂ i ∂x
(0)] is
j
j
non-degenerate. Prove: There exists neighborhoods U1 and U2 of 0 in Rn
and a diffeomorphism f : U1 → U2 such that f (0) = 0 and f ∗ ϕ2 = ϕ1 .
[Hint: Lecture 5 in my other course...]