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d В
П• в€— П‰1 = П‰ 0 .
d В  commutes and
id В i
d В
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 в€€ R3 such that |u| = 1} ,
then the induced area form is given by
П‰u (v, w) = u, v Г— w
where u в€€ S 2 , v, w в€€ Tu S 2 are vectors in R3 , Г— is the exterior product,
and В·, В· is the standard inner product. With this form, the total area of
S 2 is 4ПЂ.
Consider cylindrical polar coordinates (Оё, z) on S 2 away from its poles,
where 0 в‰¤ Оё < 2ПЂ and в€’1 в‰¤ z в‰¤ 1.
Show that, in these coordinates,
П‰ = dОё в€§ 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.
П‰ = df в€§ dg is nondegenerate в‡ђв‡’ (f, g) is a local diп¬Ђeomorphism.
Hint:

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

(a) Show that convex combinations of two area forms П‰0 , П‰1 that induce
the same orientation are symplectic.
This is wrong in dimension 4: п¬Ѓnd 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 П‰0 , П‰1 on a compact 2-dimensional
manifold M representing the same de Rham cohomology class, i.e.,
2
[П‰0 ] = [П‰1 ] в€€ HdeRham (M ).
Prove that there is a 1-parameter family of diп¬Ђeomorphisms П•t :
M в†’ M such that П•в€— П‰0 = П‰1 , П•0 = id, and П•в€— П‰0 is symplectic for
1 t
all t в€€ [0, 1].
Exercise (a) and the Moser trick.
Hint:

Such a 1-parameter family П•t is called a strong isotopy between П‰0
and П‰1 . 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, в„¦) be a symplectic linear space, and let U be a lagrangian subspace.
Claim. There is a canonical nondegenerate bilinear pairing в„¦ : V /U Г— U в†’ R.

Proof. Deп¬Ѓne в„¦ ([v], u) = в„¦(v, u) where [v] is the equivalence class of v in V /U .
в™¦
Exercise. Check that в„¦ is well-deп¬Ѓned and nondegenerate.

Consequently, we get
=в‡’ в„¦ : V /U в†’ U в€— deп¬Ѓned by в„¦ ([v]) = в„¦ ([v], В·) is an isomorphism.
=в‡’ V /U U в€— are canonically identiп¬Ѓed.
In particular, if (M, П‰) is a symplectic manifold, and X is a lagrangian
submanifold, then Tx X is a lagrangian subspace of (Tx M, П‰x ) for each x в€€ X.
The space Nx X := Tx M/Tx X is called the normal space of X at x.
в€—
=в‡’ There is a canonical identiп¬Ѓcation Nx X Tx X.
=в‡’

Theorem 9.1 The vector bundles N X and T в€— X are canonically identiп¬Ѓed.

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 в†’ N X the zero section, and i : X в†’ M inclusion.
Then there are neighborhoods U0 of X in N X, U of X in M and a diп¬Ђeomor-
phism П€ : U0 в†’ U such that

П€ EU
U0
d
s 
В
d В
d В  commutes .
i0 d В i
d В
X
For the proof, see Lecture 6.

Theorem 9.3 (Weinstein Tubular Neighborhood Theorem) Let (M, П‰)
be a symplectic manifold, X a lagrangian submanifold, П‰0 canonical symplectic
form on T в€— X, i0 : X в†’ T в€— X the lagrangian embedding as the zero section, and
i : X в†’ M lagrangian embedding given by inclusion.

51
52 9 WEINSTEIN TUBULAR NEIGHBORHOOD THEOREM

Then there are neighborhoods U0 of X in T в€— X, U of X in M , and a diп¬Ђeo-
morphism П• : U0 в†’ U such that

П• EU
U0
d
s В

d В
П• в€— П‰ = П‰0 .
d В  commutes and
i0 d В i
d В
X

Proof. This proof relies on (1) the standard tubular neighborhood theorem,
and (2) the Weinstein lagrangian neighborhood theorem.

(1) Since N X T в€— X, we can п¬Ѓnd a neighborhood N0 of X in T в€— X, a neigh-
borhood N of X in M , and a diп¬Ђeomorphism П€ : N0 в†’ N such that

П€ EN
N0
d
s В

d В
d В  commutes .
i0 d В i
d В
X

П‰0 = canonical form on T в€— X
symplectic forms on N0 .
Let
П‰1 = П€ в€— П‰
X is lagrangian for both П‰0 and П‰1 .

(2) There exist neighborhoods U0 and U1 of X in N0 and a diп¬Ђeomorphism
Оё : U0 в†’ U1 such that

Оё E U1
U0
d
s 
В
d В
Оё в€— П‰1 = П‰ 0 .
d В  commutes and
i0 d В  i0
d В
X

Take П• = П€ в—¦ Оё and U = П•(U0 ). Check that П•в€— П‰ = Оёв€— П€ в€— П‰ = П‰0 .
П‰1

Remark. Theorem 9.3 classiп¬Ѓes 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 classiп¬Ѓcation of isotropic embeddings was also carried out by Weinstein
in [103, 105]. An isotropic embedding of a manifold X into a symplectic
manifold (M, П‰) is a closed embedding i : X в†’ M such that iв€— П‰ = 0. Weinstein
showed that neighbourhood equivalence of isotropic embeddings is in one-to-one
correspondence with isomorphism classes of symplectic vector bundles.
The classiп¬Ѓcation of coisotropic embeddings is due to Gotay . A coisotropic
embedding of a manifold X carrying a closed 2-form О± of constant rank into
a symplectic manifold (M, П‰) is an embedding i : X в†’ M such that iв€— П‰ = О±
and i(X) is coisotropic has a submanifold of M . Let E be the characteristic
distribution of a closed form О± of constant rank on X, i.e., Ep is the kernel
of О±p at p в€€ X. Gotay showed that then E в€— 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
в™¦
some neighbourhood of the zero section.

9.3 Application 1:
Tangent Space to the Group of Symplectomorphisms

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

Sympl(M, П‰) = {f : M в€’в†’ M | f в€— П‰ = П‰} .

вЂ“ What is Tid (Sympl(M, П‰))?
(What is the вЂњLie algebraвЂќ of the group of symplectomorphisms?)
вЂ“ What does a neighborhood of id in Sympl(M, П‰) look like?
We use notions from the C 1 -topology:
C 1 -topology.
Let X and Y be manifolds.

Deп¬Ѓnition 9.4 A sequence of maps fi : X в†’ Y converges in the C 0 -
topology to f : X в†’ Y if and only if fi converges uniformly on compact
sets.

Deп¬Ѓnition 9.5 A sequence of C 1 maps fi : X в†’ Y converges in the C 1 -
topology to f : X в†’ Y if and only if it and the sequence of derivatives dfi :
T X в†’ T Y converge uniformly on compact sets.

Let (M, П‰) be a compact symplectic manifold and f в€€ Sympl(M, П‰). Then
Graph f
are lagrangian subspaces of (M Г— M, prв€— П‰ в€’ prв€— П‰).
1 2
Graph id = в€†
(pri : M Г— M в†’ M , i = 1, 2, are the projections to each factor.)
By the Weinstein tubular neighborhood theorem, there exists a neighborhood
U of в€† ( M ) in (M Г— M, prв€— П‰ в€’ prв€— П‰) which is symplectomorphic to a neigh-
1 2
borhood U0 of M in (T в€— M, П‰0 ). Let П• : U в†’ U0 be the symplectomorphism.
54 9 WEINSTEIN TUBULAR NEIGHBORHOOD THEOREM

Suppose that f is suп¬ѓciently C 1 -close to id, i.e., f is in some suп¬ѓciently
small neighborhood of id in the C 1 -topology. Then:

1) We can assume that Graph f вЉ† U.
j:M в†’U
Let be the embedding as Graph f ,
i:M в†’U be the embedding as Graph id = в€† .

2) The map j is suп¬ѓciently C 1 -close to i.
U0 вЉ† T в€— M , so the above j and i induce
3) By the Weinstein theorem, U
j0 : M в†’ U 0 embedding, where j0 = П• в—¦ j ,
i0 : M в†’ U 0 embedding as 0-section .
Hence, we have
П• П•
E U0 E U0
U U
d
s 
В  d
s 
В
d В  d В
d В  d В
and
id В  i0 jd В  j0
d В  d В
M M
where i(p) = (p, p), i0 (p) = (p, 0), j(p) = (p, f (p)) and j0 (p) = П•(p, f (p))
for p в€€ M .
4) The map j0 is suп¬ѓciently C 1 -close to i0 .
в‡“
в€—
The image set j0 (M ) intersects each Tp M at one point Вµp depending
smoothly on p.
5) The image of j0 is the image of a smooth section Вµ : M в†’ T в€— M , that is,
a 1-form Вµ = j0 в—¦ (ПЂ в—¦ j0 )в€’1 .
в€—
{(p, Вµp ) | p в€€ M, Вµp в€€ Tp M }.
Therefore, Graph f
Exercise. Vice-versa: if Вµ is a 1-form suп¬ѓciently C 1 -close to the zero 1-form,
then
в€—
{(p, Вµp ) | p в€€ M, Вµp в€€ Tp M } Graph f ,
for some diп¬Ђeomorphism f : M в†’ M . By Lecture 3, we have
Graph f is lagrangian в‡ђв‡’ Вµ is closed. в™¦
Conclusion. A small C 1 -neighborhood of id in Sympl(M, П‰) is homeomorphic
to a C 1 -neighborhood of zero in the vector space of closed 1-forms on M . So:

{Вµ в€€ в„¦1 (M ) | dВµ = 0} .
Tid (Sympl(M, П‰))

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

{Вµ = dh | h в€€ C в€ћ (M )} C в€ћ (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, П‰) be a compact symplectic manifold with HdeRham (M ) =
0. Then any symplectomorphism of M which is suп¬ѓciently C 1 -close to the iden-
tity has at least two п¬Ѓxed points.

Proof. Suppose that f в€€ Sympl(M, П‰) is suп¬ѓciently C 1 -close to id.
=в‡’ Graph f closed 1-form Вµ on M .
dВµ = 0
=в‡’ Вµ = dh for some h в€€ C в€ћ (M ) .
1
HdeRham (M ) = 0
M compact =в‡’ h has at least 2 critical points.
Fixed points of f = critical points of h

Graph f в€© в€† = {p : Вµp = dhp = 0} .

Lagrangian intersection problem:
A submanifold Y of M is C 1 -close to X when there is a diп¬Ђeomorphism
X в†’ Y which is, as a map into M , C 1 -close to the inclusion X в†’ M .

Theorem 9.7 Let (M, П‰) 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, П‰) be a compact symplectic manifold, and f : M в†’ M a symplecto-
morphism which is вЂњexactly homotopic to the identityвЂќ (see below). Then
#{п¬Ѓxed points of f } в‰Ґ minimal # of critical points
a smooth function of M can have .

Together with Morse theory,8 we obtain9
#{nondegenerate п¬Ѓxed points of f } в‰Ґ minimal # of critical points
a Morse function of M can have
2n
dim H i (M ) .
в‰Ґ
i=0
8A Morse function on M is a function h : M в†’ R whose critical points (i.e., points
p where dhp = 0) are all nondegenerate (i.e., the hessian at those points is nonsingular:
в€‚2 h
det = 0).
в€‚xi в€‚xj p
9A п¬Ѓxed point p of f : M в†’ M is nondegenerate if dfp : Tp M в†’ 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 в€ћ-dimensional
analogue of Morse theory). There are open conjectures for sharper bounds on
the number of п¬Ѓxed points.
Meaning of вЂњf is exactly homotopic to the identity:вЂќ
Suppose that ht : M в†’ R is a smooth family of functions which is 1-
periodic, i.e., ht = ht+1 . Let ПЃ : M Г— R в†’ M be the isotopy generated by the
time-dependent vector п¬Ѓeld vt deп¬Ѓned by П‰(vt , В·) = dht . Then вЂњf being exactly
homotopic to the identityвЂќ means f = ПЃ1 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
1в€’1
в†ђв†’ period-1 orbits of ПЃ : M Г— R в†’ M
п¬Ѓxed points of f

because f (p) = p if and only if {ПЃ(t, p) , t в€€ [0, 1]} is a closed orbit.
Proof of the Arnold conjecture in the case when h : M в†’ R is independent of
t:
p is a critical point of h в‡ђв‡’ dhp = 0 в‡ђв‡’ vp = 0 =в‡’ ПЃ(t, p) = p , в€Ђt в€€
R в‡ђв‡’ p is a п¬Ѓxed point of ПЃ1 .

Exercise. Compute these estimates for the number of п¬Ѓxed points on some
compact symplectic manifolds (for instance, S 2 , S 2 Г— S 2 and T 2 = S 1 Г— S 1 ). в™¦
Part IV
Contact Manifolds
Contact geometry is also known as вЂњthe odd-dimensional analogue of symplectic
geometry.вЂќ We will browse through the basics of contact manifolds and their
relation to symplectic manifolds.

10 Contact Forms

10.1 Contact Structures

Deп¬Ѓnition 10.1 A contact element on a manifold M is a point p в€€ M ,
called the contact point, together with a tangent hyperplane at p, H p вЉ‚ Tp M ,
that is, a codimension-1 subspace of Tp M .
в€—
A hyperplane Hp вЉ‚ Tp M determines a covector О±p в€€ Tp M \ {0}, up to
multiplication by a nonzero scalar:

(p, Hp ) is a contact element в†ђв†’ Hp = ker О±p with О±p : Tp M в€’в†’ R linear , = 0

ker О±p = ker О±p в‡ђв‡’ О±p = О»О±p for some О» в€€ R \ {0} .
Suppose that H is a smooth п¬Ѓeld of contact elements (i.e., of tangent hyper-
planes) on M :
H : p в€’в†’ Hp вЉ‚ Tp M .
Locally, H = ker О± for some 1-form О±, called a locally deп¬Ѓning 1-form for H.
(О± is not unique: ker О± = ker(f О±), for any nowhere vanishing f : M в†’ R.)

Deп¬Ѓnition 10.2 A contact structure on M is a smooth п¬Ѓeld of tangent hy-
perplanes H вЉ‚ T M , such that, for any locally deп¬Ѓning 1-form О±, we have dО±| H
nondegenerate (i.e., symplectic). The pair (M, H) is then called a contact
manifold and О± is called a local contact form.

At each p в€€ M ,

Tp M = ker О±p вЉ• ker dО±p .
1в€’dimensional
Hp

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

dim Hp = 2n is even
dО±p |Hp nondegenerate =в‡’ n
(dО±p ) |Hp = 0 is a volume form on Hp
О±p |ker dО±p nondegenerate

Therefore,

57
58 10 CONTACT FORMS

вЂў any contact manifold (M, H) has dim M = 2n + 1 odd, and
вЂў if О± is a (global) contact form, then О± в€§ (dО±)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,
в™¦
this implies that M is orientable.

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

H is a contact structure в‡ђв‡’ О±в€§(dО±)n = 0 for every locally deп¬Ѓning 1-form О± .

Proof.
=в‡’ Done above.
в‡ђ= Suppose that H = ker О± locally. We need to show:

dО±|H nondegenerate в‡ђв‡’ О± в€§ (dО±)n = 0 .

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

(О± в€§ (dО±)n )(e1 , f1 , . . . , en , fn , r) = О±(r) В·(dО±)n (e1 , f1 , . . . , en , fn )
=0

and hence О± в€§ (dО±)n = 0 в‡ђв‡’ (dО±)n |H = 0 в‡ђв‡’ dО±|H is nondegenerate .

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

О± в€§ dО± = (xdy + dz) в€§ (dx в€§ dy) = dx в€§ dy в€§ dz = 0 ,

О± is a contact form on R3 .
The corresponding п¬Ѓeld of hyperplanes H = ker О± at (x, y, z) в€€ R3 is
в€‚ в€‚ в€‚
H(x,y,z) = {v = a | О±(v) = bx + c = 0} .
+b +c
в€‚x в€‚y в€‚z
в™¦
Exercise. Picture these hyperplanes.

2. (Martinet , 1971) Any compact orientable 3-manifold admits a con-
tact structure.
Open Problem, 2000. The classiп¬Ѓcation 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 classiп¬Ѓcation 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 в€— X its cotangent bundle. There are two canon-
ical contact manifolds associated to X (see Homework 7):

P(T в€— X) = the projectivization of T в€— X , and
S(T в€— X) = the cotangent sphere bundle .

4. On R2n+1 with coordinates (x1 , y1 , . . . , xn , yn , z), О± = 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 в€€ M . Then there
exists a coordinate system (U, x1 , y1 , . . . , xn , yn , z) centered at p such that on U

О±= 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 О±t ,
t в€€ [0, 1], is a smooth family of (global) contact forms on M . Let Ht = ker О±t .
Then there exists an isotopy ПЃ : M Г— R в€’в†’ M such that Ht = ПЃtв€— H0 , for all
0 в‰¤ t в‰¤ 1.

Exercise. Show that Ht = ПЃtв€— H0 в‡ђв‡’ ПЃв€— О±t = ut В· О±0 for some family
t
ut : M в€’в†’ R, 0 в‰¤ t в‰¤ 1, of nowhere vanishing functions. в™¦

`
Proof. (A la Moser)
ПЃ0 = id
We need to п¬Ѓnd ПЃt such that For any isotopy ПЃ,
d d
в€—
dt (ПЃt О±t ) = dt (ut О±0 ) .

dв€— dО±t
(ПЃt О±t ) = ПЃв€— Lvt О±t + ,
t
dt dt

where vt = dПЃt в—¦ ПЃв€’1 is the vector п¬Ѓeld generated by ПЃt . By the Moser trick, it
t
dt
suп¬ѓces to п¬Ѓnd vt and then integrate it to ПЃt . We will search for vt in Ht = ker О±t ;
this unnecessary assumption simpliп¬Ѓes the proof.
60 10 CONTACT FORMS

We need to solve

dО±t dut
ПЃв€— ( L v t О±t + ) = О±0
t
dt dt
1
ПЃв€— О± t
dД±vt О±t +Д±vt dО±t t
ut

dО±t dut 1
ПЃв€— Д±vt dО±t + В· ПЃ в€— О±t
В·
=в‡’ =
t
dt ut t
dt

dО±t dut 1
= (ПЃв€— )в€’1
в‡ђв‡’ В·
Д±vt dО±t + О±t . ()
t
dt dt ut

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

dО±t
Д±vt dО±t |Ht = в€’
dt Ht

which determines vt uniquely, since dО±t |Ht is nondegenerate. After integrating
vt to ПЃt , the factor ut is determined by the relation ПЃв€— О±t = ut В· О±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 в€’ 1 attached to X.

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

C = {(x, П‡x ) | x в€€ X and П‡x is a hyperplane in Tx X} .

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

Pв€— X = (T в€— X \ zero section)/ в€ј

where (x, Оѕ) в€ј (x, Оѕ ) whenever Оѕ = О»Оѕ for some О» в€€ R \ {0} (here x в€€ X and
Оѕ, Оѕ в€€ Tx X \ {0}). We will denote elements of Pв€— X by (x, [Оѕ]), [Оѕ] being the в€ј
в€—

equivalence class of Оѕ.

1. Show that C is naturally isomorphic to Pв€— X as a bundle over X, i.e.,
exhibit a diп¬Ђeomorphism П• : C в†’ Pв€— X such that the following diagram
commutes: П•
C в€’в†’ Pв€— X
ПЂв†“ в†“ПЂ
X=X
where the vertical maps are the natural projections (x, П‡x ) в†’ x and
(x, Оѕ) в†’ x.
в€—
Hint: The kernel of a non-zero Оѕ в€€ Tx X is a hyperplane П‡x вЉ‚ Tx X.
What is the relation between Оѕ and Оѕ if ker Оѕ = ker Оѕ ?

2. There is on C a canonical п¬Ѓeld 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, П‡x ) в€€ C is the hyperplane

Hp = (dПЂp )в€’1 П‡x вЉ‚ Tp C ,

where
C Tp C
p = (x, П‡x )
в†“ПЂ в†“ в†“ dПЂp
X x Tx X
are the natural projections, and (dПЂp )в€’1 П‡x is the preimage of П‡x вЉ‚ Tx X
by dПЂp .
Pв€— X from exercise 1, H induces a п¬Ѓeld of
Under the isomorphism C
hyperplanes H on Pв€— X. Describe H.
в€—
Hint: If Оѕ в€€ Tx X \ {0} has kernel П‡x , what is the kernel of the canonical
1-form О±(x,Оѕ) = (dПЂ(x,Оѕ) )в€— Оѕ?

61
62 HOMEWORK 7

3. Check that (Pв€— X, H) is a contact manifold, and therefore (C, H) is a con-
tact manifold.
Let (x, [Оѕ]) в€€ Pв€— X. For any Оѕ representing the class [Оѕ], we have
Hint:
H(x,[Оѕ]) = ker ((dПЂ(x,[Оѕ]) )в€— Оѕ) .
Let x1 , . . . , xn be local coordinates on X, and let x1 , . . . , xn , Оѕ1 , . . . , Оѕn be the
associated local coordinates on T в€— X. In these coordinates, (x, [Оѕ]) is given by
(x1 , . . . , xn , [Оѕ1 , . . . , Оѕn ]). Since at least one of the Оѕi вЂ™s is nonzero, without loss
of generality we may assume that Оѕ1 = 0 so that we may divide Оѕ by Оѕ1 to obtain
a representative with coordinates (1, Оѕ2 , . . . , Оѕn ). Hence, by choosing always
the representative of [Оѕ] with Оѕ1 = 1, the set x1 , . . . , xn , Оѕ2 , . . . , Оѕn deп¬Ѓnes
coordinates on some neighborhood U of (x, [Оѕ]) in Pв€— X. On U , consider the
1-form
О± = dx1 + Оѕi dxi .
iв‰Ґ2
Show that О± is a contact form on U , i.e., show that ker О±(x,[Оѕ]) = H(x,[Оѕ]) , and
that dО±(x,[Оѕ]) is nondegenerate on H(x,[Оѕ]) .

4. What is the symplectization of C?
What is the manifold C when X = R3 and when X = S 1 Г— S 1 ?

Remark. Similarly, we could have deп¬Ѓned the manifold of oriented
contact elements of X to be
П‡o is a hyperplane in Tx X
Co = (x, П‡o ) x в€€ X and x
.
x equipped with an orientation

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

S в€— X := (T в€— X \ zero section)/ в‰€

where (x, Оѕ) в‰€ (x, Оѕ ) whenever Оѕ = О»Оѕ for some О» в€€ R+ .
A construction analogous to the above produces a canonical contact struc-
ture on C o . See [3, Appendix 4].
в™¦
11 Contact Dynamics

11.1 Reeb Vector Fields

Let (M, H) be a contact manifold with a contact form О±.
Д±R dО± = 0
Claim. There exists a unique vector п¬Ѓeld R on M such that
Д±R О± = 1

Д±R dО± = 0 =в‡’ R в€€ ker dО± , which is a line bundle, and
Proof.
Д±R О± = 1 =в‡’ normalizes R .

The vector п¬Ѓeld R is called the Reeb vector п¬Ѓeld determined by О±.

Claim. The п¬‚ow of R preserves the contact form, i.e., if ПЃt = exp tR is the
isotopy generated by R, then ПЃв€— О± = О±, в€Ђt в€€ R.
t

d в€—
= ПЃв€— (LR О±) = ПЃв€— (d Д±R О± + Д±R dО±) = 0 .
Proof. We have dt (ПЃt О±) t t
1 0
Hence, ПЃв€— О± = ПЃв€— О± = О±, в€Ђt в€€ R.
0
t

Deп¬Ѓnition 11.1 A contactomorphism is a diп¬Ђeomorphism f of a contact
manifold (M, H) which preserves the contact structure (i.e., fв€— H = H).

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

в€‚
dxi в€§ dyi
Д±R =0
=в‡’ R = is the Reeb vector п¬Ѓeld.
Д±R xi dyi + dz =1 в€‚z

The contactomorphisms generated by R are translations

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

i
2. Regard the odd sphere S 2nв€’1 в†’ R2n as the set of unit vectors

(x2 + yi ) = 1} .
2
{(x1 , y1 , . . . , xn , yn ) | i

1
Consider the 1-form on R2n , Пѓ = (xi dyi в€’ yi dxi ).
2

Claim. The form О± = iв€— Пѓ is a contact form on S 2nв€’1 .

Proof. We need to show that О± в€§ (dО±)nв€’1 = 0. The 1-form on R2n
ОЅ = d (x2 + yi ) = 2 (xi dxi + yi dyi ) satisп¬Ѓes Tp S 2nв€’1 = ker ОЅp , at
2
i
p в€€ S 2nв€’1 . Check that ОЅ в€§ Пѓ в€§ (dПѓ)nв€’1 = 0.

63
64 11 CONTACT DYNAMICS

The distribution H = ker О± is called the standard contact structure
в€‚ в€‚
on S 2nв€’1 . The Reeb vector п¬Ѓeld is R = 2 xi в€‚yi в€’ yi в€‚xi , and is also
known as the Hopf vector п¬Ѓeld on S 2nв€’1 , as the orbits of its п¬‚ow are
the circles of the Hopf п¬Ѓbration.

в™¦

11.2 Symplectization

Example. Let M = S 2nв€’1 Г— R, with coordinate П„ in the R-factor, and projec-
tion ПЂ : M в†’ S 2nв€’1 , (p, П„ ) в†’ p. Under the identiп¬Ѓcation M R2n \{0}, where
the R-factor represents the logarithm of the square of the radius, the projection
ПЂ becomes

R2n \{0} S 2nв€’1
в€’в†’
ПЂ:
X Y1 Xn Yn
(X1 , Y1 , . . . , Xn , Yn ) в€’в†’ ( в€љe1П„ , в€љeП„ , . . . , в€љeП„ , в€љeП„ )

where eП„ = (Xi2 + Yi2 ). Let О± = iв€— Пѓ be the standard contact form on S 2nв€’1
(see the previous example). Then П‰ = d(eП„ ПЂ в€— О±) is a closed 2-form on R2n \{0}.
X Yi
Since ПЂ в€— iв€— xi = в€љeiП„ , ПЂ в€— iв€— yi = в€љeП„ , we have

1 X Yi Y X
ПЂ в€— О± = ПЂ в€— iв€— Пѓ d( в€љeП„ ) в€’
= d( в€љeiП„ )
в€љi в€љi
2 eП„ eП„
1
(Xi dYi в€’ Yi dXi ) .
= 2eП„

Therefore, П‰ = dXi в€§ dYi is the standard symplectic form on R2n \{0} вЉ‚ R2n .
(M , П‰) is called the symplectization of (S 2nв€’1 , О±). в™¦

Theorem 11.2 Let (M, H) be a contact manifold with a contact form О±. Let
M = M Г— R, and let ПЂ : M в†’ M , (p, П„ ) в†’ p, be the projection. Then
П‰ = d(eП„ ПЂ в€— О±) is a symplectic form on M , where П„ is a coordinate on R.

Proof. Exercise.

Hence, M has a symplectic form П‰ canonically determined by a contact form
О± on M and a coordinate function on R; (M , П‰) is called the symplectization
of (M, О±).

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
в€—
M = {(p, Оѕ) | p в€€ M, Оѕ в€€ Tp M, such that ker Оѕ = Hp } .

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

О» В· (p, Оѕ) = (p, О»Оѕ) , О» в€€ R \ {0} .

The quotient M /(R \ {0}) is diп¬Ђeomorphic to M . M has a canonical
1-form О± deп¬Ѓned at v в€€ T(p,Оѕ) M by

О±(p,Оѕ) (v) = Оѕ((d pr)(p,Оѕ) v) ,

where pr : M в†’ M is the bundle projection.

в™¦

11.3 Conjectures of Seifert and Weinstein

Question. (Seifert, 1948) Let v be a nowhere vanishing vector п¬Ѓeld on the
3-sphere. Does the п¬‚ow of v have any periodic orbits?

Counterexamples.

вЂў (Schweitzer, 1974) в€ѓ C 1 vector п¬Ѓeld without periodic orbits.

вЂў (Kristina Kuperberg, 1994) в€ѓ C в€ћ vector п¬Ѓeld without periodic orbits.

Question. How about volume-preserving vector п¬Ѓelds?

вЂў (Greg Kuperberg, 1997) в€ѓ C 1 counterexample.

вЂў C в€ћ counterexamples are not known.

Natural generalization of this problem:
Let M = S 3 be the 3-sphere, and let Оі be a volume form on M . Suppose that
v is a nowhere vanishing vector п¬Ѓeld, and suppose that v is volume-preserving,
i.e.,
Lv Оі = 0 в‡ђв‡’ dД±v Оі = 0 в‡ђв‡’ Д±v Оі = dО±
for some 1-form О±, since H 2 (S 3 ) = 0.
Given a 1-form О±, we would like to study vector п¬Ѓelds v such that

Д±v Оі = dО±
Д±v О± > 0 .
66 11 CONTACT DYNAMICS

A vector п¬Ѓeld v satisfying Д±v О± > 0 is called positive. For instance, vector п¬Ѓelds
in a neighborhood of the Hopf vector п¬Ѓeld are positive relative to the standard
contact form on S 3 .
v
Renormalizing as R := Д±v О± , we should study instead
пЈ±
пЈІ Д±R dО± = 0
Д±О± = 1
пЈі
О± в€§ dО± is a volume form,

that is, study pairs (О±, R) where

О± is a contact form, and
R is its Reeb vector п¬Ѓeld.

Conjecture. (Weinstein, 1978 ) Suppose that M is a 3-dimensional
manifold with a (global) contact form О±. Let v be the Reeb vector п¬Ѓeld for О±.
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) ПЂ2 (M ) = 0, or
3) the contact structure is overtwisted.10

Open questions.

вЂў How many periodic orbits are there?
вЂў What do they look like?
вЂў Is there always an unknotted one?

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 
showed that the isotopy classiп¬Ѓcation of overtwisted contact structures on closed 3-manifolds
coincides with their homotopy classiп¬Ѓcation as tangent plane п¬Ѓelds. The classiп¬Ѓcation 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 п¬Ѓrst proposed by Gromov .

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-deп¬Ѓnite 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 в€§ dyj , standard symplectic structure;
П‰0 =
В·, В· ,
g0 = standard inner product; and
в€љ
if we identify R2n with Cn with coordinates zj = xj + в€’1 yj , then multi-
в€љ
plication by в€’1 induces a constant linear map J0 on the tangent spaces of
R2n :
в€‚ в€‚ в€‚ в€‚
)=в€’
J0 ( )= , J0 ( ,
в€‚xj в€‚yj в€‚yj в€‚xj
в€‚ в€‚ в€‚ в€‚
2
with J0 = в€’Id. Relative to the basis в€‚x1 , . . . , в€‚xn , в€‚y1 , . . . , в€‚yn , the maps J0 ,
П‰0 and g0 are represented by

0 в€’Id
J0 (u) = u
Id 0
0 в€’Id
= vt
П‰0 (u, v) u
Id 0
= vt u
g0 (u, v)

67
68 12 ALMOST COMPLEX STRUCTURES

where u, v в€€ R2n and v t is the transpose of v. The following compatibility
relation holds:
П‰0 (u, v) = g0 (J0 (u), v) .
в™¦

12.2 Complex Structures on Vector Spaces

Deп¬Ѓnition 12.1 Let V be a vector space. A complex structure on V is a
linear map:
J 2 = в€’Id .
J :V в†’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
в€љ
we identify the map J with multiplication by в€’1.

Deп¬Ѓnition 12.2 Let (V, в„¦) be a symplectic vector space. A complex structure
J on V is said to be compatible (with в„¦, or в„¦-compatible) if

в€Ђu, v в€€ V , is a positive inner product on V .
GJ (u, v) := в„¦(u, Jv) ,

That is,

в„¦(Ju, Jv) = в„¦(u, v) [symplectomorphism]
J is в„¦-compatible в‡ђв‡’
в„¦(u, Ju) > 0, в€Ђu = 0 [taming condition]

Compatible complex structures always exist on symplectic vector spaces:

Proposition 12.3 Let (V, в„¦) 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 в„¦ and G are nondegen-
erate,

в€’в†’ в„¦(u, В·) в€€ V в€—
uв€€V
are isomorphisms between V and V в€— .
в€’в†’ G(w, В·) в€€ V в€—
wв€€V

Hence, в„¦(u, v) = G(Au, v) for some linear map A : V в†’ V . This map A is
skew-symmetric because

G(Aв€— u, v) = G(u, Av) = G(Av, u)
= в„¦(v, u) = в€’в„¦(u, v) = G(в€’Au, v) .

Also:
вЂў AAв€— is symmetric: (AAв€— )в€— = AAв€— .
вЂў AAв€— is positive: G(AAв€— u, u) = G(Aв€— u, Aв€— u) > 0, for u = 0.
69
12.2 Complex Structures on Vector Spaces

These properties imply that AAв€— diagonalizes with positive eigenvalues О»i ,

AAв€— = B diag {О»1 , . . . , О»2n } B в€’1 .

We may hence deп¬Ѓne an arbitrary real power of AAв€— by rescaling the eigenspaces,
in particular, в€љ
AAв€— := B diag { О»1 , . . . , О»2n } B в€’1 .
в€љ
Then AAв€— is symmetric and positive-deп¬Ѓnite. Let
в€љ
J = ( AAв€— )в€’1 A .
в€љ
The factorization A = AAв€— J is called the в€љ polar decomposition of A. Since
в€љ
в€— , J commutes with AAв€— . Check that J is orthogonal,
A commutes with AA
JJ в€— = Id, as well as skew-adjoint, J в€— = в€’J, and hence it is a complex structure
on V :
J 2 = в€’JJ в€— = в€’Id .
Compatibility:

в„¦(Ju, Jv) = G(AJu, Jv) = G(JAu, Jv) = G(Au, v)
= в„¦(u, v)
в„¦(u, Ju) = G(Au, Ju) = G(в€’JAu, u)
в€љ
= G( AAв€— 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 deп¬Ѓned by
в€љ
в„¦(u, Jv) = G( AAв€— u, v) is diп¬Ђerent from G(u, v) .

Remarks.

1. This construction is canonical after an initial choice of G. To see this,
в€љ
notice that AAв€— does not depend on the choice of B nor of the ordering
в€љ в€љ
of the eigenvalues in diag { О»1 , . . . , О»2n }. The linear transformation
в€љ
AAв€— is completely determined by its eп¬Ђect on each eigenspaceв€љ AAв€— :
of
on the eigenspace corresponding to the eigenvalue О»k , the map AAв€— is
в€љ
deп¬Ѓned to be multiplication by О»k .

2. If (Vt , в„¦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, в„¦), we could also proceed as follows. Given a
symplectic basis e1 , . . . , en , f1 , . . . , fn (i.e., в„¦(ei , ej ) = в„¦(fi , fj ) = 0 and
70 12 ALMOST COMPLEX STRUCTURES

в„¦(ei , fj ) = Оґij ), one can deп¬Ѓne Jej = fj and Jfj = в€’ej . This is a com-
patible complex structure on (V, в„¦). Moreover, given в„¦ 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 в„¦ such that
J is в„¦-compatible: pick any positive inner product G such that J в€— = в€’J
and take в„¦(u, v) = G(Ju, v).

в™¦

12.3 Compatible Structures

Deп¬Ѓnition 12.4 An almost complex structure on a manifold M is a smooth
п¬Ѓeld of complex structures on the tangent spaces:
2
x в€’в†’ Jx : Tx M в†’ Tx M Jx = в€’Id .
linear, and

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

Deп¬Ѓnition 12.5 Let (M, П‰) be a symplectic manifold. An almost complex
structure J on M is called compatible (with П‰ or П‰-compatible) if the as-
signment

x в€’в†’ gx : Tx M Г— Tx M в†’ R
gx (u, v) := П‰x (u, Jx v)

is a riemannian metric on M .

For a manifold M ,

=в‡’ x в€’в†’ П‰x : Tx M Г— Tx M в†’ R is bilinear,
П‰ is a symplectic form
nondegenerate, skew-symmetric;
=в‡’ x в€’в†’ gx : Tx M Г— Tx M в†’ R
g is a riemannian metric
is a positive inner product;
=в‡’ x в€’в†’ Jx : Tx M в†’ Tx M
J almost complex structure
is linear and J 2 = в€’Id .

The triple (П‰, g, J) is called a compatible triple when g(В·, В·) = П‰(В·, JВ·).

Proposition 12.6 Let (M, П‰) 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 (В·, В·) := П‰(В·, JВ·) = g(В·, В·). в™¦
Since riemannian metrics always exist, we conclude:

Corollary 12.7 Any symplectic manifold has compatible almost complex struc-
tures.

вЂ“ How diп¬Ђerent can compatible almost complex structures be?

Proposition 12.8 Let (M, П‰) be a symplectic manifold, and J0 , J1 two almost
complex structures compatible with П‰. Then there is a smooth family J t , 0 в‰¤ t в‰¤
1, of compatible almost complex structures joining J0 to J1 .

Proof. By compatibility, we get

g0 (В·, В·) = П‰(В·, J0 В·)
П‰, J0
two riemannian metrics on M .
g1 (В·, В·) = П‰(В·, J1 В·)
П‰, J1

Their convex combinations

gt (В·, В·) = (1 в€’ t)g0 (В·, В·) + tg1 (В·, В·) , 0в‰¤tв‰¤1,

form a smooth family of riemannian metrics. Apply the polar decomposition to
(П‰, gt ) to obtain a smooth family of Jt вЂ™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 в„¦(V ) and J(V ) be the spaces of symplectic forms and complex struc-
tures on the vector space V , respectively. Take в„¦ в€€ в„¦(V ) and J в€€ J(V ).
Let GL(V ) be the group of all isomorphisms of V , let Sp(V, в„¦) be the
group of symplectomorphisms of (V, в„¦), and let GL(V, J) be the group of
complex isomorphisms of (V, J).
Show that

в„¦(V ) GL(V )/Sp(V, в„¦) and J(V ) GL(V )/GL(V, J) .
Hint: The group GL(V ) acts on в„¦(V ) by pullback. What is the stabilizer of
a given в„¦?

2. Let (R2n , в„¦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 в€€ GL(2n; R) | в„¦0 (Au, Av) = в„¦0 (u, v) for all u, v в€€ R2n } .

Identifying the complex n Г— n matrix X + iY with the real 2n Г— 2n matrix
X в€’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 В
d В
d В
Sp(2n)
d В
d В
d В
d U(n) В  В
d
dВ  В
d
В
d В
GL(n; C) O(2n)
d
В d В
d
В  d В
d
dВ  d В
dВ  dВ

72
73
HOMEWORK 8

3. Let (V, в„¦) be a symplectic vector space of dimension 2n, and let J : V в†’
V , J 2 = в€’Id, be a complex structure on V .

(a) Prove that, if J is в„¦-compatible and L is a lagrangian subspace of
(V, в„¦), then JL is also lagrangian and JL = LвЉҐ , where вЉҐ denotes
orthogonality with respect to the positive inner product GJ (u, v) =
в„¦(u, Jv).
(b) Deduce that J is в„¦-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 в„¦(ei , ej ) = в„¦(fi , fj ) = 0 and в„¦(ei , fj ) = Оґij .
13 Compatible Triples

13.1 Compatibility

Let (M, П‰) be a symplectic manifold. As shown in the previous lecture, compat-
ible almost complex structures always exist on (M, П‰). We also showed that the
set of all compatible almost complex structures on (M, П‰) is path-connected. In
fact, the set of all compatible almost complex structures is even contractible.
(This is important for deп¬Ѓning invariants.) Let J (Tx M, П‰x ) be the set of all
compatible complex structures on (Tx M, П‰x ) for x в€€ M .

Proposition 13.1 The set J (Tx M, П‰x ) is contractible, i.e., there exists a ho-
motopy
ht : J (Tx M, П‰x ) в€’в†’ J (Tx M, П‰x ) , 0 в‰¤ t в‰¤ 1 ,
starting at the identity h0 = Id,
п¬Ѓnishing at a trivial map h1 : J (Tx M, П‰x ) в†’ {J0 },
and п¬Ѓxing J0 (i.e., ht (J0 ) = J0 , в€Ђt) for some J0 в€€ J (Tx M, П‰x ).

Proof. Homework 9.
Consider the п¬Ѓber bundle J в†’ M with п¬Ѓber

Jx := J (Tx M, П‰x ) over x в€€ M .

A compatible almost complex structure J on (M, П‰) is a section of J . The
space of sections of J is contractible because the п¬Ѓbers are contractible.
Remarks.

вЂў We never used the closedness of П‰ to construct compatible almost complex
structures. The construction holds for an almost symplectic manifold
(M, П‰), that is, a pair of a manifold M and a nondegenerate 2-form П‰, not
necessarily closed.

вЂў Similarly, we could deп¬Ѓne a symplectic vector bundle to be a vector
bundle E в†’ M over a manifold M equipped with a smooth п¬Ѓeld П‰ of
п¬Ѓberwise nondegenerate skew-symmetric bilinear maps

П‰x : Ex Г— Ex в€’в†’ R .

The existence of such a п¬Ѓeld П‰ 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.

в™¦

74
75
13.2 Triple of Structures

13.2 Triple of Structures
If (П‰, J, g) is a compatible triple, then any one of П‰, J or g can be written in
terms of the other two:
g(u, v) = П‰(u, Jv)
П‰(u, v) = g(Ju, v)
J(u) = g в€’1 (П‰(u))
where
в€’в†’ T в€— M u в€’в†’ П‰(u, В·)
П‰ : TM
в€’в†’ T в€— M u в€’в†’ g(u, В·)
g : TM
are the linear isomorphisms induced by the bilinear forms П‰ and g.
The relations among П‰, J and g can be summarized in the following table.
The last column lists diп¬Ђerential equations these structures are usually asked to
satisfy.

Data Condition/Technique Consequence Question

П‰(Ju, Jv) = П‰(u, v) g(u, v) := П‰(u, Jv)
П‰, J (g п¬‚at?)
П‰(u, Ju) > 0, u = 0 is positive inner product

g(Ju, Jv) = g(u, v) П‰(u, v) := g(Ju, v)
g, J П‰ closed?
(i.e., J is orthogonal) is nondeg., skew-symm.

П‰, 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 П‰0 , П‰1 Then П‰0 , П‰1 are deformation-
equivalent, that is, there exists a smooth family П‰t , 0 в‰¤ t в‰¤ 1, of symplectic
forms joining П‰0 to П‰1 .

Proof. Take П‰t = (1 в€’ t)П‰0 + tП‰1 , 0 в‰¤ t в‰¤ 1. Then:
вЂў П‰t is closed.
вЂў П‰t is nondegenerate, since
gt (В·, В·) := П‰t (В·, JВ·) = (1 в€’ t)g0 (В·, В·) + tg1 (В·, В·)
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 :

П‰t = cos ПЂt dx1 dy1 + sin ПЂt dx1 dy2 + sin ПЂt dy1 dx2 + cos ПЂt dx2 dy2 , 0 в‰¤ t в‰¤ 1 .

There is no J in R4 compatible with both П‰0 and П‰1 . в™¦

Deп¬Ѓnition 13.3 A submanifold X of an almost complex manifold (M, J) is an
almost complex submanifold when J(T X) вЉ† T X, i.e., for all x в€€ X, v в€€
Tx X, we have Jx v в€€ Tx X.

Proposition 13.4 Let (M, П‰) 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, П‰).

Proof. Let i : X в†’ M be the inclusion. Then iв€— П‰ is a closed 2-form on X.
Nondegeneracy:

в€Ђx в€€ X , в€Ђu, v в€€ Tx X .
П‰x (u, v) = gx (Jx u, v) ,

Since gx |Tx X is nondegenerate, so is П‰x |Tx X . Hence, iв€— П‰ is symplectic.
вЂ“ 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.
в™¦
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, в„¦) be a symplectic vector space, and let J (V, в„¦) be the space of
all complex structures on (V, в„¦) which are в„¦-compatible; i.e., given a complex
structure J on V we have

J в€€ J (V, в„¦) в‡ђв‡’ GJ (В·, В·) := в„¦(В·, JВ·) is a positive inner product on V .

Fix a lagrangian subspace L0 of (V, в„¦). Let L(V, в„¦, L0 ) be the space of all
lagrangian subspaces of (V, в„¦) which intersect L0 transversally. Let G(L0 ) be
the space of all positive inner products on L0 .
Consider the map

ОЁ : J (V, в„¦) в†’ L(V, в„¦, L0 ) Г— G(L0 )
в†’ (JL0 , GJ |L0 )
J

Show that:

1. ОЁ is well-deп¬Ѓned.
2. ОЁ is a bijection.
Hint: Given (L, G) в€€ L(V, в„¦, L0 ) Г— G(L0 ), deп¬Ѓne J in the following manner:
For v в€€ L0 , v вЉҐ = {u в€€ L0 | G(u, v) = 0} is a (n в€’ 1)-dimensional space of L0 ;
its symplectic orthogonal (v вЉҐ )в„¦ is (n+1)-dimensional. Check that (v вЉҐ )в„¦ в€©L is
1-dimensional. Let Jv be the unique vector in this line such that в„¦(v, Jv) = 1.
Check that, if we take vвЂ™s in some G-orthonormal basis of L0 , this deп¬Ѓnes the
required element of J (V, в„¦).

3. L(V, в„¦, L0 ) is contractible.
Hint: Prove that L(V, в„¦, L0 ) can be identiп¬Ѓed with the vector space of all
symmetric n Г— 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 в†’ 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, в„¦) is contractible.

77
14 Dolbeault Theory

14.1 Splittings

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