• — ω1 = ω 0 .

d commutes and

id i

d

X

Homework 6: Oriented Surfaces

1. The standard symplectic form on the 2-sphere is the standard area form:

If we think of S 2 as the unit sphere in 3-space

S 2 = {u ∈ R3 such that |u| = 1} ,

then the induced area form is given by

ωu (v, w) = u, v — w

where u ∈ S 2 , v, w ∈ Tu S 2 are vectors in R3 , — is the exterior product,

and ·, · is the standard inner product. With this form, the total area of

S 2 is 4π.

Consider cylindrical polar coordinates (θ, z) on S 2 away from its poles,

where 0 ¤ θ < 2π and ’1 ¤ z ¤ 1.

Show that, in these coordinates,

ω = dθ § dz .

2. Prove the Darboux theorem in the 2-dimensional case, using the fact that

every nonvanishing 1-form on a surface can be written locally as f dg for

suitable functions f, g.

ω = df § dg is nondegenerate ⇐’ (f, g) is a local di¬eomorphism.

Hint:

3. Any oriented 2-dimensional manifold with an area form is a symplectic

manifold.

(a) Show that convex combinations of two area forms ω0 , ω1 that induce

the same orientation are symplectic.

This is wrong in dimension 4: ¬nd two symplectic forms on the vector

space R4 that induce the same orientation, yet some convex combi-

nation of which is degenerate. Find a path of symplectic forms that

connect them.

(b) Suppose that we have two area forms ω0 , ω1 on a compact 2-dimensional

manifold M representing the same de Rham cohomology class, i.e.,

2

[ω0 ] = [ω1 ] ∈ HdeRham (M ).

Prove that there is a 1-parameter family of di¬eomorphisms •t :

M ’ M such that •— ω0 = ω1 , •0 = id, and •— ω0 is symplectic for

1 t

all t ∈ [0, 1].

Exercise (a) and the Moser trick.

Hint:

Such a 1-parameter family •t is called a strong isotopy between ω0

and ω1 . In this language, this exercise shows that, up to strong

isotopy, there is a unique symplectic representative in each non-zero

2-cohomology class of M .

50

9 Weinstein Tubular Neighborhood Theorem

9.1 Observation from Linear Algebra

Let (V, „¦) be a symplectic linear space, and let U be a lagrangian subspace.

Claim. There is a canonical nondegenerate bilinear pairing „¦ : V /U — U ’ R.

Proof. De¬ne „¦ ([v], u) = „¦(v, u) where [v] is the equivalence class of v in V /U .

™¦

Exercise. Check that „¦ is well-de¬ned and nondegenerate.

Consequently, we get

=’ „¦ : V /U ’ U — de¬ned by „¦ ([v]) = „¦ ([v], ·) is an isomorphism.

=’ V /U U — are canonically identi¬ed.

In particular, if (M, ω) is a symplectic manifold, and X is a lagrangian

submanifold, then Tx X is a lagrangian subspace of (Tx M, ωx ) for each x ∈ X.

The space Nx X := Tx M/Tx X is called the normal space of X at x.

—

=’ There is a canonical identi¬cation Nx X Tx X.

=’

Theorem 9.1 The vector bundles N X and T — X are canonically identi¬ed.

9.2 Tubular Neighborhoods

Theorem 9.2 (Standard Tubular Neighborhood Theorem) Let M be

an n-dimensional manifold, X a k-dimensional submanifold, N X the normal

bundle of X in M , i0 : X ’ N X the zero section, and i : X ’ M inclusion.

Then there are neighborhoods U0 of X in N X, U of X in M and a di¬eomor-

phism ψ : U0 ’ U such that

ψ EU

U0

d

s

d

d commutes .

i0 d i

d

X

For the proof, see Lecture 6.

Theorem 9.3 (Weinstein Tubular Neighborhood Theorem) Let (M, ω)

be a symplectic manifold, X a lagrangian submanifold, ω0 canonical symplectic

form on T — X, i0 : X ’ T — X the lagrangian embedding as the zero section, and

i : X ’ M lagrangian embedding given by inclusion.

51

52 9 WEINSTEIN TUBULAR NEIGHBORHOOD THEOREM

Then there are neighborhoods U0 of X in T — X, U of X in M , and a di¬eo-

morphism • : U0 ’ U such that

• EU

U0

d

s

d

• — ω = ω0 .

d commutes and

i0 d i

d

X

Proof. This proof relies on (1) the standard tubular neighborhood theorem,

and (2) the Weinstein lagrangian neighborhood theorem.

(1) Since N X T — X, we can ¬nd a neighborhood N0 of X in T — X, a neigh-

borhood N of X in M , and a di¬eomorphism ψ : N0 ’ N such that

ψ EN

N0

d

s

d

d commutes .

i0 d i

d

X

ω0 = canonical form on T — X

symplectic forms on N0 .

Let

ω1 = ψ — ω

X is lagrangian for both ω0 and ω1 .

(2) There exist neighborhoods U0 and U1 of X in N0 and a di¬eomorphism

θ : U0 ’ U1 such that

θ E U1

U0

d

s

d

θ — ω1 = ω 0 .

d commutes and

i0 d i0

d

X

Take • = ψ —¦ θ and U = •(U0 ). Check that •— ω = θ— ψ — ω = ω0 .

ω1

Remark. Theorem 9.3 classi¬es lagrangian embeddings: up to symplectomor-

phism, the set of lagrangian embeddings is the set of embeddings of manifolds

into their cotangent bundles as zero sections.

9.3 Application 1: Tangent Space to the Group of Symplectomorphisms 53

The classi¬cation of isotropic embeddings was also carried out by Weinstein

in [103, 105]. An isotropic embedding of a manifold X into a symplectic

manifold (M, ω) is a closed embedding i : X ’ M such that i— ω = 0. Weinstein

showed that neighbourhood equivalence of isotropic embeddings is in one-to-one

correspondence with isomorphism classes of symplectic vector bundles.

The classi¬cation of coisotropic embeddings is due to Gotay [47]. A coisotropic

embedding of a manifold X carrying a closed 2-form ± of constant rank into

a symplectic manifold (M, ω) is an embedding i : X ’ M such that i— ω = ±

and i(X) is coisotropic has a submanifold of M . Let E be the characteristic

distribution of a closed form ± of constant rank on X, i.e., Ep is the kernel

of ±p at p ∈ X. Gotay showed that then E — carries a symplectic structure in a

neighbourhood of the zero section, such that X embeds coisotropically onto this

zero section, and, moreover every coisotropic embedding is equivalent to this in

™¦

some neighbourhood of the zero section.

9.3 Application 1:

Tangent Space to the Group of Symplectomorphisms

The symplectomorphisms of a symplectic manifold (M, ω) form the group

Sympl(M, ω) = {f : M ’’ M | f — ω = ω} .

“ What is Tid (Sympl(M, ω))?

(What is the “Lie algebra” of the group of symplectomorphisms?)

“ What does a neighborhood of id in Sympl(M, ω) look like?

We use notions from the C 1 -topology:

C 1 -topology.

Let X and Y be manifolds.

De¬nition 9.4 A sequence of maps fi : X ’ Y converges in the C 0 -

topology to f : X ’ Y if and only if fi converges uniformly on compact

sets.

De¬nition 9.5 A sequence of C 1 maps fi : X ’ Y converges in the C 1 -

topology to f : X ’ Y if and only if it and the sequence of derivatives dfi :

T X ’ T Y converge uniformly on compact sets.

Let (M, ω) be a compact symplectic manifold and f ∈ Sympl(M, ω). Then

Graph f

are lagrangian subspaces of (M — M, pr— ω ’ pr— ω).

1 2

Graph id = ∆

(pri : M — M ’ M , i = 1, 2, are the projections to each factor.)

By the Weinstein tubular neighborhood theorem, there exists a neighborhood

U of ∆ ( M ) in (M — M, pr— ω ’ pr— ω) which is symplectomorphic to a neigh-

1 2

borhood U0 of M in (T — M, ω0 ). Let • : U ’ U0 be the symplectomorphism.

54 9 WEINSTEIN TUBULAR NEIGHBORHOOD THEOREM

Suppose that f is su¬ciently C 1 -close to id, i.e., f is in some su¬ciently

small neighborhood of id in the C 1 -topology. Then:

1) We can assume that Graph f ⊆ U.

j:M ’U

Let be the embedding as Graph f ,

i:M ’U be the embedding as Graph id = ∆ .

2) The map j is su¬ciently C 1 -close to i.

U0 ⊆ T — M , so the above j and i induce

3) By the Weinstein theorem, U

j0 : M ’ U 0 embedding, where j0 = • —¦ j ,

i0 : M ’ U 0 embedding as 0-section .

Hence, we have

• •

E U0 E U0

U U

d

s

d

s

d d

d d

and

id i0 jd j0

d d

M M

where i(p) = (p, p), i0 (p) = (p, 0), j(p) = (p, f (p)) and j0 (p) = •(p, f (p))

for p ∈ M .

4) The map j0 is su¬ciently C 1 -close to i0 .

“

—

The image set j0 (M ) intersects each Tp M at one point µp depending

smoothly on p.

5) The image of j0 is the image of a smooth section µ : M ’ T — M , that is,

a 1-form µ = j0 —¦ (π —¦ j0 )’1 .

—

{(p, µp ) | p ∈ M, µp ∈ Tp M }.

Therefore, Graph f

Exercise. Vice-versa: if µ is a 1-form su¬ciently C 1 -close to the zero 1-form,

then

—

{(p, µp ) | p ∈ M, µp ∈ Tp M } Graph f ,

for some di¬eomorphism f : M ’ M . By Lecture 3, we have

Graph f is lagrangian ⇐’ µ is closed. ™¦

Conclusion. A small C 1 -neighborhood of id in Sympl(M, ω) is homeomorphic

to a C 1 -neighborhood of zero in the vector space of closed 1-forms on M . So:

{µ ∈ „¦1 (M ) | dµ = 0} .

Tid (Sympl(M, ω))

In particular, Tid (Sympl(M, ω)) contains the space of exact 1-forms

{µ = dh | h ∈ C ∞ (M )} C ∞ (M )/ locally constant functions .

55

9.4 Application 2: Fixed Points of Symplectomorphisms

9.4 Application 2:

Fixed Points of Symplectomorphisms

1

Theorem 9.6 Let (M, ω) be a compact symplectic manifold with HdeRham (M ) =

0. Then any symplectomorphism of M which is su¬ciently C 1 -close to the iden-

tity has at least two ¬xed points.

Proof. Suppose that f ∈ Sympl(M, ω) is su¬ciently C 1 -close to id.

=’ Graph f closed 1-form µ on M .

dµ = 0

=’ µ = dh for some h ∈ C ∞ (M ) .

1

HdeRham (M ) = 0

M compact =’ h has at least 2 critical points.

Fixed points of f = critical points of h

Graph f © ∆ = {p : µp = dhp = 0} .

Lagrangian intersection problem:

A submanifold Y of M is C 1 -close to X when there is a di¬eomorphism

X ’ Y which is, as a map into M , C 1 -close to the inclusion X ’ M .

Theorem 9.7 Let (M, ω) be a symplectic manifold. Suppose that X is a com-

1

pact lagrangian submanifold of M with HdeRham (X) = 0. Then every lagrangian

submanifold of M which is C 1 -close to X intersects X in at least two points.

Proof. Exercise.

Arnold conjecture:

Let (M, ω) be a compact symplectic manifold, and f : M ’ M a symplecto-

morphism which is “exactly homotopic to the identity” (see below). Then

#{¬xed points of f } ≥ minimal # of critical points

a smooth function of M can have .

Together with Morse theory,8 we obtain9

#{nondegenerate ¬xed points of f } ≥ minimal # of critical points

a Morse function of M can have

2n

dim H i (M ) .

≥

i=0

8A Morse function on M is a function h : M ’ R whose critical points (i.e., points

p where dhp = 0) are all nondegenerate (i.e., the hessian at those points is nonsingular:

‚2 h

det = 0).

‚xi ‚xj p

9A ¬xed point p of f : M ’ M is nondegenerate if dfp : Tp M ’ Tp M is nonsingular.

56 9 WEINSTEIN TUBULAR NEIGHBORHOOD THEOREM

The Arnold conjecture was proved by Conley-Zehnder, Floer, Hofer-Salamon,

Ono, Futaya-Ono, Lin-Tian using Floer homology (which is an ∞-dimensional

analogue of Morse theory). There are open conjectures for sharper bounds on

the number of ¬xed points.

Meaning of “f is exactly homotopic to the identity:”

Suppose that ht : M ’ R is a smooth family of functions which is 1-

periodic, i.e., ht = ht+1 . Let ρ : M — R ’ M be the isotopy generated by the

time-dependent vector ¬eld vt de¬ned by ω(vt , ·) = dht . Then “f being exactly

homotopic to the identity” means f = ρ1 for some such ht .

In other words, f is exactly homotopic to the identity when f is the

time-1 map of an isotopy generated by some smooth time-dependent 1-periodic

hamiltonian function.

There is a one-to-one correspondence

1’1

←’ period-1 orbits of ρ : M — R ’ M

¬xed points of f

because f (p) = p if and only if {ρ(t, p) , t ∈ [0, 1]} is a closed orbit.

Proof of the Arnold conjecture in the case when h : M ’ R is independent of

t:

p is a critical point of h ⇐’ dhp = 0 ⇐’ vp = 0 =’ ρ(t, p) = p , ∀t ∈

R ⇐’ p is a ¬xed point of ρ1 .

Exercise. Compute these estimates for the number of ¬xed points on some

compact symplectic manifolds (for instance, S 2 , S 2 — S 2 and T 2 = S 1 — S 1 ). ™¦

Part IV

Contact Manifolds

Contact geometry is also known as “the odd-dimensional analogue of symplectic

geometry.” We will browse through the basics of contact manifolds and their

relation to symplectic manifolds.

10 Contact Forms

10.1 Contact Structures

De¬nition 10.1 A contact element on a manifold M is a point p ∈ M ,

called the contact point, together with a tangent hyperplane at p, H p ‚ Tp M ,

that is, a codimension-1 subspace of Tp M .

—

A hyperplane Hp ‚ Tp M determines a covector ±p ∈ Tp M \ {0}, up to

multiplication by a nonzero scalar:

(p, Hp ) is a contact element ←’ Hp = ker ±p with ±p : Tp M ’’ R linear , = 0

ker ±p = ker ±p ⇐’ ±p = »±p for some » ∈ R \ {0} .

Suppose that H is a smooth ¬eld of contact elements (i.e., of tangent hyper-

planes) on M :

H : p ’’ Hp ‚ Tp M .

Locally, H = ker ± for some 1-form ±, called a locally de¬ning 1-form for H.

(± is not unique: ker ± = ker(f ±), for any nowhere vanishing f : M ’ R.)

De¬nition 10.2 A contact structure on M is a smooth ¬eld of tangent hy-

perplanes H ‚ T M , such that, for any locally de¬ning 1-form ±, we have d±| H

nondegenerate (i.e., symplectic). The pair (M, H) is then called a contact

manifold and ± is called a local contact form.

At each p ∈ M ,

Tp M = ker ±p • ker d±p .

1’dimensional

Hp

The ker d±p summand in this splitting depends on the choice of ±.

dim Hp = 2n is even

d±p |Hp nondegenerate =’ n

(d±p ) |Hp = 0 is a volume form on Hp

±p |ker d±p nondegenerate

Therefore,

57

58 10 CONTACT FORMS

• any contact manifold (M, H) has dim M = 2n + 1 odd, and

• if ± is a (global) contact form, then ± § (d±)n is a volume form on M .

Remark. Let (M, H) be a contact manifold. A global contact form exists if and

only if the quotient line bundle T M/H is orientable. Since H is also orientable,

™¦

this implies that M is orientable.

Proposition 10.3 Let H be a ¬eld of tangent hyperplanes on M . Then

H is a contact structure ⇐’ ±§(d±)n = 0 for every locally de¬ning 1-form ± .

Proof.

=’ Done above.

⇐= Suppose that H = ker ± locally. We need to show:

d±|H nondegenerate ⇐’ ± § (d±)n = 0 .

Take a local trivialization {e1 , f1 , . . . , en , fn , r} of T M = ker ± • rest , such that

ker ± = span{e1 , f1 , . . . , en , fn } and rest = span{r}.

(± § (d±)n )(e1 , f1 , . . . , en , fn , r) = ±(r) ·(d±)n (e1 , f1 , . . . , en , fn )

=0

and hence ± § (d±)n = 0 ⇐’ (d±)n |H = 0 ⇐’ d±|H is nondegenerate .

10.2 Examples

1. On R3 with coordinates (x, y, z), consider ± = xdy + dz. Since

± § d± = (xdy + dz) § (dx § dy) = dx § dy § dz = 0 ,

± is a contact form on R3 .

The corresponding ¬eld of hyperplanes H = ker ± at (x, y, z) ∈ R3 is

‚ ‚ ‚

H(x,y,z) = {v = a | ±(v) = bx + c = 0} .

+b +c

‚x ‚y ‚z

™¦

Exercise. Picture these hyperplanes.

2. (Martinet [79], 1971) Any compact orientable 3-manifold admits a con-

tact structure.

Open Problem, 2000. The classi¬cation of compact orientable contact

3-manifolds is still not known. There is by now a huge collection of results

in contact topology related to the classi¬cation of contact manifolds. For

review of the state of the knowledge and interesting questions on contact

3-manifolds, see [33, 43, 98].

59

10.3 First Properties

3. Let X be a manifold and T — X its cotangent bundle. There are two canon-

ical contact manifolds associated to X (see Homework 7):

P(T — X) = the projectivization of T — X , and

S(T — X) = the cotangent sphere bundle .

4. On R2n+1 with coordinates (x1 , y1 , . . . , xn , yn , z), ± = xi dyi + dz is

i

contact.

10.3 First Properties

There is a local normal form theorem for contact manifolds analogous to the

Darboux theorem for symplectic manifolds.

Theorem 10.4 Let (M, H) be a contact manifold and p ∈ M . Then there

exists a coordinate system (U, x1 , y1 , . . . , xn , yn , z) centered at p such that on U

±= xi dyi + dz is a local contact form for H .

The idea behind the proof is sketched in the next lecture.

There is also a Moser-type theorem for contact forms.

Theorem 10.5 (Gray) Let M be a compact manifold. Suppose that ±t ,

t ∈ [0, 1], is a smooth family of (global) contact forms on M . Let Ht = ker ±t .

Then there exists an isotopy ρ : M — R ’’ M such that Ht = ρt— H0 , for all

0 ¤ t ¤ 1.

Exercise. Show that Ht = ρt— H0 ⇐’ ρ— ±t = ut · ±0 for some family

t

ut : M ’’ R, 0 ¤ t ¤ 1, of nowhere vanishing functions. ™¦

`

Proof. (A la Moser)

ρ0 = id

We need to ¬nd ρt such that For any isotopy ρ,

d d

—

dt (ρt ±t ) = dt (ut ±0 ) .

d— d±t

(ρt ±t ) = ρ— Lvt ±t + ,

t

dt dt

where vt = dρt —¦ ρ’1 is the vector ¬eld generated by ρt . By the Moser trick, it

t

dt

su¬ces to ¬nd vt and then integrate it to ρt . We will search for vt in Ht = ker ±t ;

this unnecessary assumption simpli¬es the proof.

60 10 CONTACT FORMS

We need to solve

d±t dut

ρ— ( L v t ±t + ) = ±0

t

dt dt

1

ρ— ± t

d±vt ±t +±vt d±t t

ut

d±t dut 1

ρ— ±vt d±t + · ρ — ±t

·

=’ =

t

dt ut t

dt

d±t dut 1

= (ρ— )’1

⇐’ ·

±vt d±t + ±t . ()

t

dt dt ut

Restricting to the hyperplane Ht = ker ±t , equation ( ) reads

d±t

±vt d±t |Ht = ’

dt Ht

which determines vt uniquely, since d±t |Ht is nondegenerate. After integrating

vt to ρt , the factor ut is determined by the relation ρ— ±t = ut · ±0 . Check that

t

this indeed gives a solution.

Homework 7: Manifolds of Contact Elements

Given any manifold X of dimension n, there is a canonical symplectic manifold

of dimension 2n attached to it, namely its cotangent bundle with the standard

symplectic structure. The exercises below show that there is also a canonical

contact manifold of dimension 2n ’ 1 attached to X.

The manifold of contact elements of an n-dimensional manifold X is

C = {(x, χx ) | x ∈ X and χx is a hyperplane in Tx X} .

On the other hand, the projectivization of the cotangent bundle of X is

P— X = (T — X \ zero section)/ ∼

where (x, ξ) ∼ (x, ξ ) whenever ξ = »ξ for some » ∈ R \ {0} (here x ∈ X and

ξ, ξ ∈ Tx X \ {0}). We will denote elements of P— X by (x, [ξ]), [ξ] being the ∼

—

equivalence class of ξ.

1. Show that C is naturally isomorphic to P— X as a bundle over X, i.e.,

exhibit a di¬eomorphism • : C ’ P— X such that the following diagram

commutes: •

C ’’ P— X

π“ “π

X=X

where the vertical maps are the natural projections (x, χx ) ’ x and

(x, ξ) ’ x.

—

Hint: The kernel of a non-zero ξ ∈ Tx X is a hyperplane χx ‚ Tx X.

What is the relation between ξ and ξ if ker ξ = ker ξ ?

2. There is on C a canonical ¬eld of hyperplanes H (that is, a smooth map

attaching to each point in C a hyperplane in the tangent space to C at

that point): H at the point p = (x, χx ) ∈ C is the hyperplane

Hp = (dπp )’1 χx ‚ Tp C ,

where

C Tp C

p = (x, χx )

“π “ “ dπp

X x Tx X

are the natural projections, and (dπp )’1 χx is the preimage of χx ‚ Tx X

by dπp .

P— X from exercise 1, H induces a ¬eld of

Under the isomorphism C

hyperplanes H on P— X. Describe H.

—

Hint: If ξ ∈ Tx X \ {0} has kernel χx , what is the kernel of the canonical

1-form ±(x,ξ) = (dπ(x,ξ) )— ξ?

61

62 HOMEWORK 7

3. Check that (P— X, H) is a contact manifold, and therefore (C, H) is a con-

tact manifold.

Let (x, [ξ]) ∈ P— X. For any ξ representing the class [ξ], we have

Hint:

H(x,[ξ]) = ker ((dπ(x,[ξ]) )— ξ) .

Let x1 , . . . , xn be local coordinates on X, and let x1 , . . . , xn , ξ1 , . . . , ξn be the

associated local coordinates on T — X. In these coordinates, (x, [ξ]) is given by

(x1 , . . . , xn , [ξ1 , . . . , ξn ]). Since at least one of the ξi ™s is nonzero, without loss

of generality we may assume that ξ1 = 0 so that we may divide ξ by ξ1 to obtain

a representative with coordinates (1, ξ2 , . . . , ξn ). Hence, by choosing always

the representative of [ξ] with ξ1 = 1, the set x1 , . . . , xn , ξ2 , . . . , ξn de¬nes

coordinates on some neighborhood U of (x, [ξ]) in P— X. On U , consider the

1-form

± = dx1 + ξi dxi .

i≥2

Show that ± is a contact form on U , i.e., show that ker ±(x,[ξ]) = H(x,[ξ]) , and

that d±(x,[ξ]) is nondegenerate on H(x,[ξ]) .

4. What is the symplectization of C?

What is the manifold C when X = R3 and when X = S 1 — S 1 ?

Remark. Similarly, we could have de¬ned the manifold of oriented

contact elements of X to be

χo is a hyperplane in Tx X

Co = (x, χo ) x ∈ X and x

.

x equipped with an orientation

The manifold C o is isomorphic to the cotangent sphere bundle of X

S — X := (T — X \ zero section)/ ≈

where (x, ξ) ≈ (x, ξ ) whenever ξ = »ξ for some » ∈ R+ .

A construction analogous to the above produces a canonical contact struc-

ture on C o . See [3, Appendix 4].

™¦

11 Contact Dynamics

11.1 Reeb Vector Fields

Let (M, H) be a contact manifold with a contact form ±.

±R d± = 0

Claim. There exists a unique vector ¬eld R on M such that

±R ± = 1

±R d± = 0 =’ R ∈ ker d± , which is a line bundle, and

Proof.

±R ± = 1 =’ normalizes R .

The vector ¬eld R is called the Reeb vector ¬eld determined by ±.

Claim. The ¬‚ow of R preserves the contact form, i.e., if ρt = exp tR is the

isotopy generated by R, then ρ— ± = ±, ∀t ∈ R.

t

d —

= ρ— (LR ±) = ρ— (d ±R ± + ±R d±) = 0 .

Proof. We have dt (ρt ±) t t

1 0

Hence, ρ— ± = ρ— ± = ±, ∀t ∈ R.

0

t

De¬nition 11.1 A contactomorphism is a di¬eomorphism f of a contact

manifold (M, H) which preserves the contact structure (i.e., f— H = H).

Examples.

1. Euclidean space R2n+1 with ± = xi dyi + dz.

i

‚

dxi § dyi

±R =0

=’ R = is the Reeb vector ¬eld.

±R xi dyi + dz =1 ‚z

The contactomorphisms generated by R are translations

ρt (x1 , y1 , . . . , xn , yn , z) = (x1 , y1 , . . . , xn , yn , z + t) .

i

2. Regard the odd sphere S 2n’1 ’ R2n as the set of unit vectors

(x2 + yi ) = 1} .

2

{(x1 , y1 , . . . , xn , yn ) | i

1

Consider the 1-form on R2n , σ = (xi dyi ’ yi dxi ).

2

Claim. The form ± = i— σ is a contact form on S 2n’1 .

Proof. We need to show that ± § (d±)n’1 = 0. The 1-form on R2n

ν = d (x2 + yi ) = 2 (xi dxi + yi dyi ) satis¬es Tp S 2n’1 = ker νp , at

2

i

p ∈ S 2n’1 . Check that ν § σ § (dσ)n’1 = 0.

63

64 11 CONTACT DYNAMICS

The distribution H = ker ± is called the standard contact structure

‚ ‚

on S 2n’1 . The Reeb vector ¬eld is R = 2 xi ‚yi ’ yi ‚xi , and is also

known as the Hopf vector ¬eld on S 2n’1 , as the orbits of its ¬‚ow are

the circles of the Hopf ¬bration.

™¦

11.2 Symplectization

Example. Let M = S 2n’1 — R, with coordinate „ in the R-factor, and projec-

tion π : M ’ S 2n’1 , (p, „ ) ’ p. Under the identi¬cation M R2n \{0}, where

the R-factor represents the logarithm of the square of the radius, the projection

π becomes

R2n \{0} S 2n’1

’’

π:

X Y1 Xn Yn

(X1 , Y1 , . . . , Xn , Yn ) ’’ ( √e1„ , √e„ , . . . , √e„ , √e„ )

where e„ = (Xi2 + Yi2 ). Let ± = i— σ be the standard contact form on S 2n’1

(see the previous example). Then ω = d(e„ π — ±) is a closed 2-form on R2n \{0}.

X Yi

Since π — i— xi = √ei„ , π — i— yi = √e„ , we have

1 X Yi Y X

π — ± = π — i— σ d( √e„ ) ’

= d( √ei„ )

√i √i

2 e„ e„

1

(Xi dYi ’ Yi dXi ) .

= 2e„

Therefore, ω = dXi § dYi is the standard symplectic form on R2n \{0} ‚ R2n .

(M , ω) is called the symplectization of (S 2n’1 , ±). ™¦

Theorem 11.2 Let (M, H) be a contact manifold with a contact form ±. Let

M = M — R, and let π : M ’ M , (p, „ ) ’ p, be the projection. Then

ω = d(e„ π — ±) is a symplectic form on M , where „ is a coordinate on R.

Proof. Exercise.

Hence, M has a symplectic form ω canonically determined by a contact form

± on M and a coordinate function on R; (M , ω) is called the symplectization

of (M, ±).

Remarks.

1. The contact version of the Darboux theorem can now be derived by apply-

ing the symplectic theorem to the symplectization of the contact manifold

(with appropriate choice of coordinates); see [3, Appendix 4].

65

11.3 Conjectures of Seifert and Weinstein

2. There is a coordinate-free description of M as

—

M = {(p, ξ) | p ∈ M, ξ ∈ Tp M, such that ker ξ = Hp } .

The group R \ {0} acts on M by multiplication on the cotangent vector:

» · (p, ξ) = (p, »ξ) , » ∈ R \ {0} .

The quotient M /(R \ {0}) is di¬eomorphic to M . M has a canonical

1-form ± de¬ned at v ∈ T(p,ξ) M by

±(p,ξ) (v) = ξ((d pr)(p,ξ) v) ,

where pr : M ’ M is the bundle projection.

™¦

11.3 Conjectures of Seifert and Weinstein

Question. (Seifert, 1948) Let v be a nowhere vanishing vector ¬eld on the

3-sphere. Does the ¬‚ow of v have any periodic orbits?

Counterexamples.

• (Schweitzer, 1974) ∃ C 1 vector ¬eld without periodic orbits.

• (Kristina Kuperberg, 1994) ∃ C ∞ vector ¬eld without periodic orbits.

Question. How about volume-preserving vector ¬elds?

• (Greg Kuperberg, 1997) ∃ C 1 counterexample.

• C ∞ counterexamples are not known.

Natural generalization of this problem:

Let M = S 3 be the 3-sphere, and let γ be a volume form on M . Suppose that

v is a nowhere vanishing vector ¬eld, and suppose that v is volume-preserving,

i.e.,

Lv γ = 0 ⇐’ d±v γ = 0 ⇐’ ±v γ = d±

for some 1-form ±, since H 2 (S 3 ) = 0.

Given a 1-form ±, we would like to study vector ¬elds v such that

±v γ = d±

±v ± > 0 .

66 11 CONTACT DYNAMICS

A vector ¬eld v satisfying ±v ± > 0 is called positive. For instance, vector ¬elds

in a neighborhood of the Hopf vector ¬eld are positive relative to the standard

contact form on S 3 .

v

Renormalizing as R := ±v ± , we should study instead

±

±R d± = 0

±± = 1

± § d± is a volume form,

that is, study pairs (±, R) where

± is a contact form, and

R is its Reeb vector ¬eld.

Conjecture. (Weinstein, 1978 [104]) Suppose that M is a 3-dimensional

manifold with a (global) contact form ±. Let v be the Reeb vector ¬eld for ±.

Then v has a periodic orbit.

Theorem 11.3 (Viterbo and Hofer, 1993 [62, 63, 101]) The Weinstein

conjecture is true when

1) M = S 3 , or

2) π2 (M ) = 0, or

3) the contact structure is overtwisted.10

Open questions.

• How many periodic orbits are there?

• What do they look like?

• Is there always an unknotted one?

• What about the linking behavior?

10 A surface S inside a contact 3-manifold determines a singular foliation on S, called the

characteristic foliation of S, by the intersection of the contact planes with the tangent

spaces to S. A contact structure on a 3-manifold M is called overtwisted if there exists

an embedded 2-disk whose characteristic foliation contains one closed leaf C and exactly

one singular point inside C; otherwise, the contact structure is called tight. Eliashberg [32]

showed that the isotopy classi¬cation of overtwisted contact structures on closed 3-manifolds

coincides with their homotopy classi¬cation as tangent plane ¬elds. The classi¬cation of tight

contact structures is still open.

Part V

Compatible Almost Complex

Structures

The fact that any symplectic manifold possesses almost complex structures, and

even so in a compatible sense, establishes a link from symplectic geometry to

complex geometry, and is the point of departure for the modern technique of

counting pseudo-holomorphic curves, as ¬rst proposed by Gromov [49].

12 Almost Complex Structures

12.1 Three Geometries

1. Symplectic geometry:

geometry of a closed nondegenerate skew-symmetric bilinear form.

2. Riemannian geometry:

geometry of a positive-de¬nite symmetric bilinear map.

3. Complex geometry:

geometry of a linear map with square -1.

Example. The euclidean space R2n with the standard linear coordinates

(x1 , . . . , xn , y1 , . . . , yn ) has standard structures:

dxj § dyj , standard symplectic structure;

ω0 =

·, · ,

g0 = standard inner product; and

√

if we identify R2n with Cn with coordinates zj = xj + ’1 yj , then multi-

√

plication by ’1 induces a constant linear map J0 on the tangent spaces of

R2n :

‚ ‚ ‚ ‚

)=’

J0 ( )= , J0 ( ,

‚xj ‚yj ‚yj ‚xj

‚ ‚ ‚ ‚

2

with J0 = ’Id. Relative to the basis ‚x1 , . . . , ‚xn , ‚y1 , . . . , ‚yn , the maps J0 ,

ω0 and g0 are represented by

0 ’Id

J0 (u) = u

Id 0

0 ’Id

= vt

ω0 (u, v) u

Id 0

= vt u

g0 (u, v)

67

68 12 ALMOST COMPLEX STRUCTURES

where u, v ∈ R2n and v t is the transpose of v. The following compatibility

relation holds:

ω0 (u, v) = g0 (J0 (u), v) .

™¦

12.2 Complex Structures on Vector Spaces

De¬nition 12.1 Let V be a vector space. A complex structure on V is a

linear map:

J 2 = ’Id .

J :V ’V with

The pair (V, J) is called a complex vector space.

A complex structure J is equivalent to a structure of vector space over C if

√

we identify the map J with multiplication by ’1.

De¬nition 12.2 Let (V, „¦) be a symplectic vector space. A complex structure

J on V is said to be compatible (with „¦, or „¦-compatible) if

∀u, v ∈ V , is a positive inner product on V .

GJ (u, v) := „¦(u, Jv) ,

That is,

„¦(Ju, Jv) = „¦(u, v) [symplectomorphism]

J is „¦-compatible ⇐’

„¦(u, Ju) > 0, ∀u = 0 [taming condition]

Compatible complex structures always exist on symplectic vector spaces:

Proposition 12.3 Let (V, „¦) be a symplectic vector space. Then there is a

compatible complex structure J on V .

Proof. Choose a positive inner product G on V . Since „¦ and G are nondegen-

erate,

’’ „¦(u, ·) ∈ V —

u∈V

are isomorphisms between V and V — .

’’ G(w, ·) ∈ V —

w∈V

Hence, „¦(u, v) = G(Au, v) for some linear map A : V ’ V . This map A is

skew-symmetric because

G(A— u, v) = G(u, Av) = G(Av, u)

= „¦(v, u) = ’„¦(u, v) = G(’Au, v) .

Also:

• AA— is symmetric: (AA— )— = AA— .

• AA— is positive: G(AA— u, u) = G(A— u, A— u) > 0, for u = 0.

69

12.2 Complex Structures on Vector Spaces

These properties imply that AA— diagonalizes with positive eigenvalues »i ,

AA— = B diag {»1 , . . . , »2n } B ’1 .

We may hence de¬ne an arbitrary real power of AA— by rescaling the eigenspaces,

in particular, √

AA— := B diag { »1 , . . . , »2n } B ’1 .

√

Then AA— is symmetric and positive-de¬nite. Let

√

J = ( AA— )’1 A .

√

The factorization A = AA— J is called the √ polar decomposition of A. Since

√

— , J commutes with AA— . Check that J is orthogonal,

A commutes with AA

JJ — = Id, as well as skew-adjoint, J — = ’J, and hence it is a complex structure

on V :

J 2 = ’JJ — = ’Id .

Compatibility:

„¦(Ju, Jv) = G(AJu, Jv) = G(JAu, Jv) = G(Au, v)

= „¦(u, v)

„¦(u, Ju) = G(Au, Ju) = G(’JAu, u)

√

= G( AA— u, u) > 0 , for u = 0 .

Therefore, J is a compatible complex structure on V .

As indicated in the proof, in general, the positive inner product de¬ned by

√

„¦(u, Jv) = G( AA— u, v) is di¬erent from G(u, v) .

Remarks.

1. This construction is canonical after an initial choice of G. To see this,

√

notice that AA— does not depend on the choice of B nor of the ordering

√ √

of the eigenvalues in diag { »1 , . . . , »2n }. The linear transformation

√

AA— is completely determined by its e¬ect on each eigenspace√ AA— :

of

on the eigenspace corresponding to the eigenvalue »k , the map AA— is

√

de¬ned to be multiplication by »k .

2. If (Vt , „¦t ) is a family of symplectic vector spaces with a family Gt of

positive inner products, all depending smoothly on a real parameter t,

then, adapting the proof of the previous proposition, we can show that

there is a smooth family Jt of compatible complex structures on Vt .

3. To check just the existence of compatible complex structures on a sym-

plectic vector space (V, „¦), we could also proceed as follows. Given a

symplectic basis e1 , . . . , en , f1 , . . . , fn (i.e., „¦(ei , ej ) = „¦(fi , fj ) = 0 and

70 12 ALMOST COMPLEX STRUCTURES

„¦(ei , fj ) = δij ), one can de¬ne Jej = fj and Jfj = ’ej . This is a com-

patible complex structure on (V, „¦). Moreover, given „¦ and J compatible

on V , there exists a symplectic basis of V of the form:

e1 , . . . , en , f1 = Je1 , . . . , fn = Jen .

The proof is part of Homework 8.

4. Conversely, given (V, J), there is always a symplectic structure „¦ such that

J is „¦-compatible: pick any positive inner product G such that J — = ’J

and take „¦(u, v) = G(Ju, v).

™¦

12.3 Compatible Structures

De¬nition 12.4 An almost complex structure on a manifold M is a smooth

¬eld of complex structures on the tangent spaces:

2

x ’’ Jx : Tx M ’ Tx M Jx = ’Id .

linear, and

The pair (M, J) is then called an almost complex manifold.

De¬nition 12.5 Let (M, ω) be a symplectic manifold. An almost complex

structure J on M is called compatible (with ω or ω-compatible) if the as-

signment

x ’’ gx : Tx M — Tx M ’ R

gx (u, v) := ωx (u, Jx v)

is a riemannian metric on M .

For a manifold M ,

=’ x ’’ ωx : Tx M — Tx M ’ R is bilinear,

ω is a symplectic form

nondegenerate, skew-symmetric;

=’ x ’’ gx : Tx M — Tx M ’ R

g is a riemannian metric

is a positive inner product;

=’ x ’’ Jx : Tx M ’ Tx M

J almost complex structure

is linear and J 2 = ’Id .

The triple (ω, g, J) is called a compatible triple when g(·, ·) = ω(·, J·).

Proposition 12.6 Let (M, ω) be a symplectic manifold, and g a riemannian

metric on M . Then there exists a canonical almost complex structure J on M

which is compatible.

71

12.3 Compatible Structures

Proof. The polar decomposition is canonical (after a choice of metric), hence

this construction of J on M is smooth; cf. Remark 2 of the previous section.

Remark. In general, gJ (·, ·) := ω(·, J·) = g(·, ·). ™¦

Since riemannian metrics always exist, we conclude:

Corollary 12.7 Any symplectic manifold has compatible almost complex struc-

tures.

“ How di¬erent can compatible almost complex structures be?

Proposition 12.8 Let (M, ω) be a symplectic manifold, and J0 , J1 two almost

complex structures compatible with ω. Then there is a smooth family J t , 0 ¤ t ¤

1, of compatible almost complex structures joining J0 to J1 .

Proof. By compatibility, we get

g0 (·, ·) = ω(·, J0 ·)

ω, J0

two riemannian metrics on M .

g1 (·, ·) = ω(·, J1 ·)

ω, J1

Their convex combinations

gt (·, ·) = (1 ’ t)g0 (·, ·) + tg1 (·, ·) , 0¤t¤1,

form a smooth family of riemannian metrics. Apply the polar decomposition to

(ω, gt ) to obtain a smooth family of Jt ™s joining J0 to J1 .

Corollary 12.9 The set of all compatible almost complex structures on a sym-

plectic manifold is path-connected.

Homework 8: Compatible Linear Structures

1. Let „¦(V ) and J(V ) be the spaces of symplectic forms and complex struc-

tures on the vector space V , respectively. Take „¦ ∈ „¦(V ) and J ∈ J(V ).

Let GL(V ) be the group of all isomorphisms of V , let Sp(V, „¦) be the

group of symplectomorphisms of (V, „¦), and let GL(V, J) be the group of

complex isomorphisms of (V, J).

Show that

„¦(V ) GL(V )/Sp(V, „¦) and J(V ) GL(V )/GL(V, J) .

Hint: The group GL(V ) acts on „¦(V ) by pullback. What is the stabilizer of

a given „¦?

2. Let (R2n , „¦0 ) be the standard 2n-dimensional symplectic euclidean space.

The symplectic linear group is the group of all linear transformations

of R2n which preserve the symplectic structure:

Sp(2n) := {A ∈ GL(2n; R) | „¦0 (Au, Av) = „¦0 (u, v) for all u, v ∈ R2n } .

Identifying the complex n — n matrix X + iY with the real 2n — 2n matrix

X ’Y

, consider the following subgroups of GL(2n; R):

Y X

Sp(2n) , O(2n) , GL(n; C) and U(n) .

Show that the intersection of any two of them is U(n). (From [82, p.41].)

d

d

d

Sp(2n)

d

d

d

d U(n)

d

d

d

d

GL(n; C) O(2n)

d

d

d

d

d

d d

d d

72

73

HOMEWORK 8

3. Let (V, „¦) be a symplectic vector space of dimension 2n, and let J : V ’

V , J 2 = ’Id, be a complex structure on V .

(a) Prove that, if J is „¦-compatible and L is a lagrangian subspace of

(V, „¦), then JL is also lagrangian and JL = L⊥ , where ⊥ denotes

orthogonality with respect to the positive inner product GJ (u, v) =

„¦(u, Jv).

(b) Deduce that J is „¦-compatible if and only if there exists a symplectic

basis for V of the form

e1 , e2 , . . . , en , f1 = Je1 , f2 = Je2 , . . . , fn = Jen

where „¦(ei , ej ) = „¦(fi , fj ) = 0 and „¦(ei , fj ) = δij .

13 Compatible Triples

13.1 Compatibility

Let (M, ω) be a symplectic manifold. As shown in the previous lecture, compat-

ible almost complex structures always exist on (M, ω). We also showed that the

set of all compatible almost complex structures on (M, ω) is path-connected. In

fact, the set of all compatible almost complex structures is even contractible.

(This is important for de¬ning invariants.) Let J (Tx M, ωx ) be the set of all

compatible complex structures on (Tx M, ωx ) for x ∈ M .

Proposition 13.1 The set J (Tx M, ωx ) is contractible, i.e., there exists a ho-

motopy

ht : J (Tx M, ωx ) ’’ J (Tx M, ωx ) , 0 ¤ t ¤ 1 ,

starting at the identity h0 = Id,

¬nishing at a trivial map h1 : J (Tx M, ωx ) ’ {J0 },

and ¬xing J0 (i.e., ht (J0 ) = J0 , ∀t) for some J0 ∈ J (Tx M, ωx ).

Proof. Homework 9.

Consider the ¬ber bundle J ’ M with ¬ber

Jx := J (Tx M, ωx ) over x ∈ M .

A compatible almost complex structure J on (M, ω) is a section of J . The

space of sections of J is contractible because the ¬bers are contractible.

Remarks.

• We never used the closedness of ω to construct compatible almost complex

structures. The construction holds for an almost symplectic manifold

(M, ω), that is, a pair of a manifold M and a nondegenerate 2-form ω, not

necessarily closed.

• Similarly, we could de¬ne a symplectic vector bundle to be a vector

bundle E ’ M over a manifold M equipped with a smooth ¬eld ω of

¬berwise nondegenerate skew-symmetric bilinear maps

ωx : Ex — Ex ’’ R .

The existence of such a ¬eld ω is equivalent to being able to reduce the

structure group of the bundle from the general linear group to the linear

symplectic group. As a consequence of our discussion, a symplectic vector

bundle is always a complex vector bundles, and vice-versa.

™¦

74

75

13.2 Triple of Structures

13.2 Triple of Structures

If (ω, J, g) is a compatible triple, then any one of ω, J or g can be written in

terms of the other two:

g(u, v) = ω(u, Jv)

ω(u, v) = g(Ju, v)

J(u) = g ’1 (ω(u))

where

’’ T — M u ’’ ω(u, ·)

ω : TM

’’ T — M u ’’ g(u, ·)

g : TM

are the linear isomorphisms induced by the bilinear forms ω and g.

The relations among ω, J and g can be summarized in the following table.

The last column lists di¬erential equations these structures are usually asked to

satisfy.

Data Condition/Technique Consequence Question

ω(Ju, Jv) = ω(u, v) g(u, v) := ω(u, Jv)

ω, J (g ¬‚at?)

ω(u, Ju) > 0, u = 0 is positive inner product

g(Ju, Jv) = g(u, v) ω(u, v) := g(Ju, v)

g, J ω closed?

(i.e., J is orthogonal) is nondeg., skew-symm.

ω, g polar decomposition J almost complex str. J integrable?

An almost complex structure J on a manifold M is called integrable if and

only if J is induced by a structure of complex manifold on M . In Lecture 15

we will discuss tests to check whether a given J is integrable.

13.3 First Consequences

Proposition 13.2 Let (M, J) be an almost complex manifold. Suppose that J

is compatible with two symplectic structures ω0 , ω1 Then ω0 , ω1 are deformation-

equivalent, that is, there exists a smooth family ωt , 0 ¤ t ¤ 1, of symplectic

forms joining ω0 to ω1 .

Proof. Take ωt = (1 ’ t)ω0 + tω1 , 0 ¤ t ¤ 1. Then:

• ωt is closed.

• ωt is nondegenerate, since

gt (·, ·) := ωt (·, J·) = (1 ’ t)g0 (·, ·) + tg1 (·, ·)

is positive, hence nondegenerate.

76 13 COMPATIBLE TRIPLES

Remark. The converse of this proposition is not true. A counterexample is

provided by the following family in R4 :

ωt = cos πt dx1 dy1 + sin πt dx1 dy2 + sin πt dy1 dx2 + cos πt dx2 dy2 , 0 ¤ t ¤ 1 .

There is no J in R4 compatible with both ω0 and ω1 . ™¦

De¬nition 13.3 A submanifold X of an almost complex manifold (M, J) is an

almost complex submanifold when J(T X) ⊆ T X, i.e., for all x ∈ X, v ∈

Tx X, we have Jx v ∈ Tx X.

Proposition 13.4 Let (M, ω) be a symplectic manifold equipped with a com-

patible almost complex structure J. Then any almost complex submanifold X of

(M, J) is a symplectic submanifold of (M, ω).

Proof. Let i : X ’ M be the inclusion. Then i— ω is a closed 2-form on X.

Nondegeneracy:

∀x ∈ X , ∀u, v ∈ Tx X .

ωx (u, v) = gx (Jx u, v) ,

Since gx |Tx X is nondegenerate, so is ωx |Tx X . Hence, i— ω is symplectic.

“ When is an almost complex manifold a complex manifold? See Lecture 15.

Examples.

S 2 is an almost complex manifold and it is a complex manifold.

S 4 is not an almost complex manifold (proved by Ehresmann and Hopf).

S 6 is almost complex and it is not yet known whether it is complex.

S 8 and higher spheres are not almost complex manifolds.

™¦

Homework 9: Contractibility

The following proof illustrates in a geometric way the relation between la-

grangian subspaces, complex structures and inner products; from [11, p.45].

Let (V, „¦) be a symplectic vector space, and let J (V, „¦) be the space of

all complex structures on (V, „¦) which are „¦-compatible; i.e., given a complex

structure J on V we have

J ∈ J (V, „¦) ⇐’ GJ (·, ·) := „¦(·, J·) is a positive inner product on V .

Fix a lagrangian subspace L0 of (V, „¦). Let L(V, „¦, L0 ) be the space of all

lagrangian subspaces of (V, „¦) which intersect L0 transversally. Let G(L0 ) be

the space of all positive inner products on L0 .

Consider the map

Ψ : J (V, „¦) ’ L(V, „¦, L0 ) — G(L0 )

’ (JL0 , GJ |L0 )

J

Show that:

1. Ψ is well-de¬ned.

2. Ψ is a bijection.

Hint: Given (L, G) ∈ L(V, „¦, L0 ) — G(L0 ), de¬ne J in the following manner:

For v ∈ L0 , v ⊥ = {u ∈ L0 | G(u, v) = 0} is a (n ’ 1)-dimensional space of L0 ;

its symplectic orthogonal (v ⊥ )„¦ is (n+1)-dimensional. Check that (v ⊥ )„¦ ©L is

1-dimensional. Let Jv be the unique vector in this line such that „¦(v, Jv) = 1.

Check that, if we take v™s in some G-orthonormal basis of L0 , this de¬nes the

required element of J (V, „¦).

3. L(V, „¦, L0 ) is contractible.

Hint: Prove that L(V, „¦, L0 ) can be identi¬ed with the vector space of all

symmetric n — n matrices. Notice that any n-dimensional subspace L of V

which is transversal to L0 is the graph of a linear map S : JL0 ’ L0 , i.e.,

L = span of {Je1 + SJe1 , . . . , Jen + SJen }

when L0 = span of {e1 , . . . , en } .

4. G(L0 ) is contractible.

G(L0 ) is even convex.

Hint:

Conclude that J (V, „¦) is contractible.

77

14 Dolbeault Theory

14.1 Splittings