Download Math 257A: Introduction to Symplectic Topology, Lecture 2

Math 257A: Introduction to Symplectic Topology, Lecture 2
Alex Zamorzaev
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Symplectic manifolds
Recall. (M, ω) is a symplectic manifold if the 2-form ω is closed (i.e. dω = 0) and nondegenerate (i.e.
is a volume form on M ).
1 n
n! ω
Example 1. If (M1 , ω1 ) and (M2 , ω2 ) are symplectic manifolds then (M1 × M2 , ω1 ⊕ ω2 ) is a symplectic
manifold with ω1 ⊕ ω2 := π1∗ ω1 + π2∗ ω2 where π1 , π2 are projections onto M1 and M2 , respectively.
Exercise 1. Check that ω1 ⊕ ω2 is symplectic. For example, nondegeneracy follows from:
1 ∗ n1
1 ∗ n2
1
(ω1 ⊕ ω2 )n1 +n2 =
π (ω ) ∧
π (ω ).
(n1 + n2 )!
n1 ! 1 1
n2 ! 2 2
Pn
Example 2. The torus T 2n := R2n /Z2n with ω0 := j=1 dxi ∧ dyi descending from R2n is a symplectic
manifold.
Note. Assume (M, ω) is symplectic. Then M is orientable (ω n is a volume form). If M is closed then
2
2k
[ω] ∈ HdR
(M ) is non-zero and, in fact, HdR
(M ) 6= 0 for all k with 0 ≤ k ≤ 12 dim M . Indeed, since ω is
2
closed, it defines an element [ω] of HdR (M ). And, since ω is nondegenerate [ω n ] = [ω]n defines a volume
2
form, i.e. an nonzero element of HdR
(M ) ∼
= R.
Definition 1. If ω is exact and symplectic then (M, ω) is called an exact symplectic manifold.
Note. If (M, ω) is exact then M cannot be closed (i.e. compact without boundary).
Non-example 3. The spheres S 2n for n ≥ 2 are not symplectic because H 2 (S 2n ) = 0.
Definition 2. A diffeomorphism f : (M1 , ω1 ) → (M2 , ω2 ) is called a symplectomorphism if f ∗ ω2 = ω1 .
Symplectomorphisms from (M, ω) to itself form a group Sympl(M, ω).
Example 4. Sympl(S 2 , ω0 ) = area-preserving diffeomorphisms.
Note. In contrast to the group of isometries Isom(S 2 , gstandard ) which is a finite dimensional Lie group,
Sympl(S 2 , ω0 ) is MUCH larger.
Question:. How to classify symplectic manifolds up to symplectomorphism?
Answer:. Unknown in dimension ≥ 4. In dimension 2, compact surfaces are classified by genus and total
area.
2n
Theorem 1 (Darboux’ Theorem). Any symplectic manifold (M 2n , ω)
Pnis locally symplectomorphic to (R , ω0 ),
i.e. there exist local coordinates (around each point) in which ω = j=1 dxj ∧ dyj .
Note. Contrast with the Riemannian geometry where curvature is a local invariant (and hence an obstruction
to local isometries).
Proof. Deferred for later in the course.
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Symplectic manifolds
Definition 3. Assume W is a smoothly embedded (via i : W ,→ M ) submanifold of a symplectic manifold
(M, ω). W is called symplectic/isotropic/coisotropic/Lagrangian submanifold if (Tp W, ωp |Tp M ) is a symplectic/isotropic/coisotropic/Lagrangian vector subspace of (Tp M, ωp ) for all points p ∈ W . In particular, W is
symplectic iff i∗ ω is nondegenerate and W is Lagrangian iff i∗ ω = 0 and dim W = 21 dim M .
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Symplectic vector bundles
A real vector bundle E → X is called a symplect vector bundle if each fiber (Ex , ωx ) is a symplectic vector
space and ωx varies smoothly as x varies in X.
Exercise 2. Make the definition above precise using local trivializations of E → X.
Note. We do not impose any sort of closedness condition on symplectic vector bundles.
Note. If (M, ω) is a symplectic manifold, then (T M, ω) is a symplectic vector bundle. However, if M is a
smooth manifold and (T M, ω) is a symplectic vector bundle, it does NOT necessarily follow that M is a
symplectic manifold. The correct statement is: M is a symplectic manifold iff (T M, ω) is a symplectic vector
bundle and ω is closed.
Example 5. Assume V → X is a real vector bundle. Then V ⊕ V ∗ → M is canonically a symplectic vector
bundle with
ωcan ((u, α), (v, β)) := β(u) − α(v) for all u, v ∈ Vx , and α, β ∈ Vx∗ .
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4.1
Standard examples of symplectic manifolds
Any Kähler manifold X is symplectic (Kähler form = symplectic form)
Recall. A Kähler manifold is a complex manifold M together with a hermitian form h (i.e. a hermitian form
hx : Tx M × Tx M → C for every x ∈ M varying smoothly in x) such that the imaginary part ω := 2i (h − h̄)
of h is a symplectic form (from the hermitian condition it follows immediately that ω is a nondegenerate
2-form, hence it is symplectic if and only if it is closed).
Fact. Any smooth projective variety is a Kähler manifold, and hence a symplectic manifold.
Example 6. The complex projective space CP n has a standard (up to scaling) Kähler form (the FubiniStudy form). We can define
CP n := (Cn+1 \{0})/C∗ = S 2n+1 /S 1
where C∗ acts on Cn+1 \{0} by component wise multiplication:
λ(z0 , . . . , zn ) = (λz0 , . . . , λz0 )
(and we view S 2n+1 and S 1 as unit-norm elements of Cn+1 and C, respectively). The coordintates of Cn+1
give the homogeneous coordinates [z0 : · · · : zn ] on CP n .
The Fubini-Study form on CP n is given by
ωF S :=
i ¯
∂ ∂ log |z|2
2π
2
where z = (z0 , . . . , zn ) for homogeneous coordinates [z0 : · · · : zn ]. We can rewrite the form above as
!
Pn
Pn
i
j=0 dzj ∧ dz̄j
j=0 z̄j dzj ∧ zk dz̄k
−
ωF S =
.
2π
|z|2
|z|4
Exercise 3. The definition above is in terms of coordinates on Cn+1 \{0}. Check that it descends to CP n
to give a closed and nondegenerate form. Hint: to show it is nondegenerate use local coordinates, e.g.
set z0 := 1.
PAt [1 : 0 : ·· · : 0] (assuming we are working on the coordinate patch z0 := 1) we have
n
i
ωF S = 2π
j=1 dzj ∧ dz̄j , so ωF S is nondegenerate at [1 : 0 : · · · : 0]. The unitary group U (n + 1)
acting on Cn+1 \{0} preserves |z|2 , hence its action on CP n (which descends from Cn+1 \{0}) preserves ωF S .
Moreover, U (n + 1) transitively on S 2n+1 (and, hence, on CP n ). Therefore, ωF S is nondegenerate at every
point of CP n (since the action of U (n + 1) defines a family of diffeomorphisms from CP n to itself that send
[1 : 0 : · · · : 0] to any given point and preserve ωF S ).
Fact. Any complex submanifold M of CP n is Kähler and hence symplectic with the symplectic form ωF S |M :=
i∗ ωF S (where i is the inclusion of M into CP n ).
Proof. ωF S |M is closed because ωF S is closed and the de Rham differential d commutes with pull-backs.
Working in local complex coordinates shows that ωF S |M is nondegenerate.
Exercise 4. Show that ωF S |M is nondegenerate.
Note. There exist closed symplectic manifolds that are not Kähler. Thurston constructed the first such
example: a 4-dimensional symplectic manifold with the first Betti number b1 = 3 odd (hence, the manifold
cannot be Kähler). We will describe this example later.
4.2
Cotangent bundle T ∗ X of any smooth manifold X carries a canonical (exact)
symplectic form
Note. Cotangent bundle carries a natural structure of a symplectic manifold not of a symplectic bundle.
The idea comes from Hamilton’s formulation of mechanics where points of the manifold represent position
and cotangent vectors represent momentum.
Definition 4. Given a smooth manifold X, its cotangent bundle M := T ∗ X has a canonical 1-form λcan
(sometimes called the tautological 1-form) defined as follows: let π : T ∗ X → X be the projection map, then
for a given point (x, y) ∈ T ∗ X (x is a point in the manifold X and y is a covector at x),
λcan,(x,y) := π ∗ y
(where on the RHS we view y as a 1-form on Tx X). Informally, ”the value of the 1-form λcan at a covector
y is the covector y”.
In coordinates: given a point x ∈ X, choose local coordinates (x1 , . . . , xn ) on X (in the spirit of Hamilton
one should think of them as the position of a system of particles).
Given this choice of coordinates, Tx X has a canonical basis dx1 , . . . , dxn . Given any y ∈ Tx∗ X, write it as
y=
n
X
j=1
3
yj dxj .
This defines local coordinates on T ∗ X: point (x, y) has the coordinates (x1 , . . . , xn , y1 , . . . , yn ). Then we
define the canonical 1-form at (x, y) to be
λcan,(x,y) :=
n
X
yj dxj
j=1
(where dxj are now viewed as 1-forms on T ∗ X, not on X).
Exercise 5. (an exercise in tautology) Check that the coordinate-free definition of λcan agrees with the
definition in local coordinates.
Then we define the canonical symplectic form on T ∗ X by
ωcan := −dλcan .
Thus, T ∗ X is an exact symplectic manifold by construction. In coordinates,
ωcan =
n
X
dxj ∧ dyj .
j=1
Note. There is a natural isomorphism (can you describe it explicitly?) of vector bundles
(T (T ∗ X))|X ∼
= T X ⊕ T ∗X
(here the restriction |X to X means the pullback of a vector bundle on T ∗ X to X via the inclusion of
X as the zero section of T ∗ X). The isomorphism is in fact an isomorphism of symplectic vector bundles
where the (fiber-wise) symplectic form on the left is the one coming from the canonical symplectic manifold
structure on T ∗ X and the symplectic form on the right is the one coming from the canonical symplectic
bundle structure of V ⊕ V ∗ with V := T X (somewhat unfortunately, both forms are called ωcan ).
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