ńņš. 21 |

Thus, the fundamental problem was to classify the āprimitive transitive continuous

transformation groupsā.

When the algebra of inļ¬nitesimal generators of G was ļ¬nite dimensional, Lie and his

coworkers made good progress. Their work culminated in the work of Cartan and Killing,

classifying the ļ¬nite dimensional simple Lie groups. (Interestingly enough, they did not

then go on to solve the extension problem and so classify all Lie groups. Perhaps they

regarded this as a problem of lesser order. Or, more likely, the classiļ¬cation turned out to

be messy, uninteresting, and ultimately intractable.)

They found that the simple groups fell into two types. Besides the special linear

groups, such as SL(n, R), SL(n, C) and other complex analogs; orthogonal groups, such as

SO(p, q) and its complex analogs; and symplectic groups, such as Sp(n, R) and its complex

analogs (which became known as the classical groups), there were ļ¬ve āexceptionalā types.

This story is quite long, but very interesting. The āļ¬nite dimensional Lie groupsā went on

L.6.10 109

to become an essential part of the foundation of modern diļ¬erential geometry. A complete

account of this classiļ¬cation (along with very interesting historical notes) can be found in

[He].

However, when the algebra of inļ¬nitesimal generators of G was inļ¬nite dimensional,

the story was not so complete. Lie himself identiļ¬ed four classes of these āinļ¬nite dimen-

sional primitive transitive transformation groupsā. They were

ā¢ In every dimension n, the full diļ¬eomorphism group, Diff(Rn ).

ā¢ In every dimension n, the group of diļ¬eomorphisms which preserve a ļ¬xed volume

form Āµ, denoted by SDiff(Āµ).

ā¢ In every even dimension 2n, the group of diļ¬eomorphisms which preserve the standard

symplectic form

ā„¦n = dx1 ā§ dy 1 + Ā· Ā· Ā· + dxn ā§ dy n ,

denoted by Sp(ā„¦n ).

ā¢ In every odd dimension 2n + 1, the group of diļ¬eomorphisms which preserve, up to a

scalar function multiple, the 1-form

Ļn = dz + x1 dy 1 + Ā· Ā· Ā· + xn dy n .

This āgroupā was known as the contact group and I will denote it by Ct(Ļn ).

However, Lie and his coworkers were never able to discover any others, though they

searched diligently. (By the way, Lie was aware that there were also holomorphic analogs

acting in Cn , but, at that time, the distinction between real and complex was not generally

made explicit. Apparently, an educated reader was supposed to know or be able to guess

what the generalizations to the complex category were.)

Ā“

In a series of four papers spanning from 1902 to 1910, Elie Cartan reformulated Lieā™s

problem in terms of systems of partial diļ¬erential equations and, under the hypothesis

of analyticity (real and complex were not carefully distinguished), he proved that Lieā™s

classes were essentially all of the inļ¬nite dimensional primitive transitive transformation

groups. The slight extension was that SDiff(Āµ) had a companion extension to RĀ·SDiff(Āµ),

the diļ¬eomorphisms which preserve Āµ up to a constant multiple and that Sp(ā„¦n ) had

a companion extension to R Ā· Sp(ā„¦n ), the diļ¬eomorphisms which preserve ā„¦n up to a

constant multiple. Of course, there were also the holomorphic analogues of these. Notice

the remarkable fact that there are no āexceptional inļ¬nite dimensional primitive transitive

transformation groupsā.

These papers are remarkable, not only for their results, but for the wealth of concepts

which Cartan introduced in order to solve his problem. In these papers, Cartan introduces

the notion of G-structures (of all orders), principal bundles and their connections, jet

bundles, prolongation (both of group actions and exterior diļ¬erential systems), and a host

of other ideas which were only appreciated much later. Perhaps because of its originality,

Cartanā™s work in this area was essentially ignored for many years.

L.6.11 110

In the 1950ā™s, when algebraic varieties were being explored and developed as complex

manifolds, it began to be understood that complex manifolds were to be thought of as

manifolds with an atlas of coordinate charts whose āoverlapsā were holomorphic. Gener-

alizing this example, it became clear that, for any collection Ī“ of local diļ¬eomorphisms of

Rn which satisļ¬ed the following deļ¬nition, one could deļ¬ne a category of Ī“-manifolds as

manifolds endowed with an atlas A of coordinate charts whose overlaps lay in A.

Deļ¬nition 3: A local diļ¬eomorphism of Rn is a pair (U, Ļ) where U ā‚ Rn is an open

set and Ļ: U ā’ Rn is a one-to-one diļ¬eomorphism onto its image. A set Ī“ of local

diļ¬eomorphisms of Rn is said to form a pseudo-group on Rn if it satisļ¬es the following

three properties:

(1) (Composition and Inverses) If (U, Ļ) and (V, Ļ) are in Ī“, then (Ļā’1 (V ), Ļ ā—¦ Ļ) and

(Ļ(U), Ļā’1 ) also belong to Ī“.

(2) (Localization and Globalization) If (U, Ļ) is in Ī“, and W ā‚ U is open, then (W, Ļ|W )

is also in Ī“. Moreover, if (U, Ļ) is a local diļ¬eomorphism of Rn such that U can be

written as the union of open subsets WĪ± for which (WĪ± , Ļ|WĪ± ) is in Ī“ for all Ī±, then

(U, Ļ) is in Ī“.

(3) (Non-triviality) (Rn , id) is in Ī“.

As it turned out, the pseudo-groups Ī“ of interest in geometry were exactly the ones

which could be characterized as the (local) solutions of a system of partial diļ¬erential

equations, i.e., they were Lieā™s transformation groups. This caused a revival of interest

in Cartanā™s work. Consequently, much of Cartanā™s work has now been redone in modern

language. In particular, Cartanā™s classiļ¬cation was redone according to modern standards

of rigor and a very readable account of this theory can be found in [SS].

In any case, symplectic geometry, seen in this light, is one of a small handful of

ānaturalā geometries that one can impose on manifolds.

L.6.12 111

Exercise Set 6:

Symplectic Manifolds, II

1. Assume n > 1. Show that if Ar,R ā‚ R2n (with its standard symplectic structure) is

the annulus described by the relations r < |x| < R, then there cannot be a symplectic

diļ¬eomorphism Ļ: Ar,R ā’ As,S that āexchanges the boundariesā. (Hint: Show that if Ļ

existed one would be able to construct a symplectic structure on S 2n .) Conclude that one

cannot naĀØÄ±vely deļ¬ne connected sum in the category of symplectic manifolds. (The ānaĀØ Ä±veā

deļ¬nition would be to try to take two symplectic manifolds M1 and M2 of the same dimen-

sion, choose an open ball in each one, cut out a sub-ball of each and identify the resulting

annuli by an appropriate diļ¬eomorphism that was chosen to be a symplectomorphism.)

2. This exercise completes the proof of Proposition 1.

(i) Let S+ denote the space of n-by-n positive deļ¬nite symmetric matrices. Show that

n

the map Ļ: S+ ā’ S+ deļ¬ned by Ļ(s) = s2 is a one-to-one diļ¬eomorphism of S+ onto

n n n

itself. Conclude that every element of Sn has a unique positive deļ¬nite square root

+

ā

and thatā map s ā’ s is a smooth mapping. Show also that, for any r ā O(n),

the ā

we have trar = tr a r, so that the square root function is O(n)-equivariant.

(ii) Let Aā¢ denote the space of n-by-n invertible anti-symmetric matrices. Show that, for

n

a ā āā¢ , the matrix ā’a2 is symmetric and positive deļ¬nite. Show that the matrix

An

b = ā’a2 is the unique symmetric positive deļ¬nite matrix that satisļ¬es b2 = ā’a2

ā

and moreover that b commutes with a. Check also that the mapping a ā’ ā’a2 is

O(n)-equivariant.

(iii) Now verify the claim made in the proof of Proposition 1 that, for any smooth vector

bundle E over a manifold M endowed with a smooth inner product on the ļ¬bers

and any smooth, invertible skew-symmetric bundle mapping A: E ā’ E, there exists

a unique smooth positive deļ¬nite symmetric bundle mapping B: E ā’ E that satisļ¬es

B 2 = ā’A2 and that commutes with A.

3. This exercise requires that you know something about characteristic classes.

(i) Show that S 4n has no almost complex structure for any n. (Hint: What could the

total Chern and Pontrijagin class of the tangent bundle be?)

(Using the Bott Periodicity Theorem, it can be shown that the characteristic

class cn(E) of any complex bundle E over S 2n is an integer multiple of (nā’1)! v where

v ā H 2n (S 2n, Z) is a generator. It follows that, among the spheres, only S 2 and S 6

could have almost complex structures and it turns out that they both do. It is a long

standing problem whether or not S 6 has a complex structure.)

(ii) Using the formulas for 4-manifolds developed in the Lecture, determine how many

possibilities there are for the ļ¬rst Chern class c1 (J ) of an almost complex structure J

on M where M a connected sum of 3 or 4 copies of CP2 .

E.6.1 112

4. Show that, if ā„¦0 is a symplectic structure on a compact manifold M, then there is an

open neighborhood U in H 2 (M, R) of [ā„¦0 ], such that, for all u ā U, there is a symplectic

structure ā„¦u on M with [ā„¦u ] = u. (Hint: Since M is compact, for any closed 2-form Ī„,

the 2-form ā„¦ + tĪ„ is non-degenerate for all suļ¬ciently small t.)

5. Mimic the proof of Theorem 1 to prove another theorem of Moser: For any compact,

connected, oriented manifold M, two volume forms Āµ0 and Āµ1 diļ¬er by an oriented dif-

feomorphism (i.e., there exists an orientation preserving diļ¬eomorphism Ļ: M ā’ M that

satisļ¬es Ļā— (Āµ1 ) = Āµ0 ) if and only if

Āµ0 = Āµ1 .

M M

(This theorem is also true without the hypothesis of compactness, but the proof is slightly

more delicate.)

6. Let M be a connected, smooth oriented 4-manifold and let Āµ ā A4 (M) be a volume

form that satisļ¬es M Āµ = 1. (By the previous problem, any two such forms diļ¬er by an

oriented diļ¬eomorphism of M.) For any (smooth) ā„¦ ā A2 (M), deļ¬ne ā—(ā„¦2 ) ā C ā(M) by

the equation

ā„¦2 = ā—(ā„¦2 ) Āµ.

Now, ļ¬x a cohomology class u ā HdR (M) satisfying u2 = r[Āµ] where r = 0. Deļ¬ne the

2

functional F : u ā’ R

F (ā„¦) = ā—(ā„¦2 ) ā„¦2 for ā„¦ ā u.

M

Show that any F -critical 2-form ā„¦ ā u is a symplectic form satisfying ā—(ā„¦2 ) = r and that

F has no critical values other than r2 . Show also that F (ā„¦) ā„ r2 for all ā„¦ ā u.

This motivates deļ¬ning an invariant of the class u by

I(u) = inf F (ā„¦).

ā„¦āu

Gromov has suggested (private communication) that perhaps I(u) = r2 for all u, even

when the inļ¬mum is not attained.

7. Let P ā‚ M be a closed submanifold and let U ā‚ M be an open neighborhood of P in

M that can be retracted onto P , i.e., there exists a smooth map R: U Ć— [0, 1] ā’ U so that

R(u, 1) = u for all u ā U, R(p, t) = p for all p ā P and t ā [0, 1], and R(u, 0) lies in P for

all u ā U. (Every closed submanifold of M has such a neighborhood.)

Show that if Ī¦ is a closed k-form on U that vanishes at every point of P , then there

exists a (kā’1)-form Ļ on U that vanishes on P and satisļ¬es dĻ = Ī¦. (Hint: Mimic

PoincarĀ“ā™s Homotopy Argument: Let Ī„ = Rā— (Ī¦) and set Ļ… = ā‚ Ī„. Then, using the fact

e ā‚t

that Ļ…(u, t) can be regarded as a (kā’1)-form at u for all t, deļ¬ne

1

Ļ(u) = Ļ…(u, t) dt.

0

Now verify that Ļ has the desired properties.)

E.6.2 113

8. Show that Theorem 2 implies Darbouxā™ Theorem. (Hint: Take P to be a point in a

symplectic manifold M.)

9. This exercise assumes that you have done Exercise 5.10. Let (M, ā„¦) be a symplectic

manifold. Show that the following description of the ļ¬‚ux homomorphism is valid. Let p be

an e-based path in Sp(ā„¦). Thus, p: [0, 1] Ć— M ā’ M satisļ¬es pā— (ā„¦) = ā„¦ for all 0 ā¤ t ā¤ 1.

t

ā—

Show that p (ā„¦) = ā„¦ + Ļ•ā§dt for some 1-form Ļ• on [0, 1] Ć— M. Let Ī¹t : M ā’ [0, 1] Ć— M be

the āt-slice inclusionā: Ī¹t (m) = (t, m), and set Ļ•t = Ī¹ā— (Ļ•).

t

Show that Ļ•t is closed for all 0 ā¤ t ā¤ 1. Show that if we set

1

Ė

Ī¦(p) = Ļ•t dt,

0

then the cohomology class [Ī¦(p)] ā HdR (M, R) depends only on the homotopy class of p

Ė 1

and hence deļ¬nes a map Ī¦: Sp0 (ā„¦) ā’ HdR (M, R). Verify that this map is the same as the

1

ļ¬‚ux homomorphism deļ¬ned in Exercise 5.10.

Use this description to show that if p is in the kernel of Ī¦, then p is homotopic to a

path p for which the forms Ļ•t are all exact. This shows that the kernel of Ī¦ is actually

connected.

10. The point of this exercise is to show that any symplectic vector bundle over a sym-

plectic manifold (M, ā„¦) can occur as the symplectic normal bundle for some symplectic

embedding M into some other symplectic manifold.

Let (M, ā„¦) be a symplectic manifold and let Ļ: E ā’ M be a symplectic vector bundle

over M of rank 2n. (I.e., E comes equipped with a section B of Ī2 (E ā— ) that restricts

to each ļ¬ber Em to be a symplectic structure Bm .) Show that there exists a symplectic

structure ĪØ on an open neighborhood in E of the zero section of E that satisļ¬es the

condition that ĪØ0m = ā„¦m + Bm under the natural identiļ¬cation T0m E = Tm M ā• Em .

(Hint: Choose a locally ļ¬nite open cover U = {UĪ± | Ī± ā A} of M so that, if we deļ¬ne

EĪ± = Ļ ā’1 (UĪ± ), then there exists a symplectic trivialization Ļ„Ī± : EĪ± ā’ R2n (where R2n is

given its standard symplectic structure ā„¦0 = dxi ā§dy i ). Now let {Ī»Ī± | Ī± ā A} be a partition

of unity subordinate to the cover U. Show that the form

ĪØ = Ļ ā— (ā„¦) + ā—

d Ī»Ī± Ļ„Ī± (xi dy i )

Ī±

has the desired properties.)

11. Show that if E is a symplectic vector bundle over M and L ā‚ E is a Lagrangian

subbundle, then E is isomorphic to L ā• Lā— as a symplectic bundle. (The symplectic

bundle structure Ī„ on L ā• Lā— is the one that, on each ļ¬ber satisļ¬es

Ī„ (v, Ī±), (w, Ī²) = Ī±(w) ā’ Ī²(v). )

E.6.3 114

(Hint: First choose a complementary subbundle F ā‚ E so that E = L ā• F . Show that

F is naturally isomorphic to Lā— abstractly by using the fact that the symplectic structure

on E is non-degenerate. Then show that there exists a bundle map A: F ā’ L so that

Ė

F = {v + Av | v ā F }

is also a Lagrangian subbundle of E that is complementary to L and isomorphic to Lā— via

some bundle map Ī±: Lā— ā’ F . Now show that id ā• Ī±: L ā• Lā— ā’ L ā• F E is a symplectic

Ė Ė

bundle isomorphism.)

12. Action-Angle Coordinates. Proposition 4 can be used to show the existence of so-

called action angle coordinates in the neighborhood of a compact level set of a completely

integrable Hamiltonian system. (See Lecture 5). Here is how this goes: Let (M 2n , ā„¦) be

a symplectic manifold and let f = (f 1 , . . . , f n ): M ā’ Rn be a smooth submersion with

ńņš. 21 |