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of the variables” which was preserved by G).
Thus, the fundamental problem was to classify the “primitive transitive continuous
transformation groups”.

When the algebra of in¬nitesimal generators of G was ¬nite dimensional, Lie and his
coworkers made good progress. Their work culminated in the work of Cartan and Killing,
classifying the ¬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 classi¬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 ¬ve “exceptional” types.
This story is quite long, but very interesting. The “¬nite dimensional Lie groups” went on

L.6.10 109
to become an essential part of the foundation of modern di¬erential geometry. A complete
account of this classi¬cation (along with very interesting historical notes) can be found in
[He].
However, when the algebra of in¬nitesimal generators of G was in¬nite dimensional,
the story was not so complete. Lie himself identi¬ed four classes of these “in¬nite dimen-
sional primitive transitive transformation groups”. They were
• In every dimension n, the full di¬eomorphism group, Diff(Rn ).
• In every dimension n, the group of di¬eomorphisms which preserve a ¬xed volume
form µ, denoted by SDiff(µ).
• In every even dimension 2n, the group of di¬eomorphisms which preserve the standard
symplectic form
„¦n = dx1 § dy 1 + · · · + dxn § dy n ,
denoted by Sp(„¦n ).
• In every odd dimension 2n + 1, the group of di¬eomorphisms which preserve, up to a
scalar function multiple, the 1-form

ωn = dz + x1 dy 1 + · · · + xn dy n .

This “group” was known as the contact group and I will denote it by Ct(ωn ).
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.)

´
In a series of four papers spanning from 1902 to 1910, Elie Cartan reformulated Lie™s
problem in terms of systems of partial di¬erential equations and, under the hypothesis
of analyticity (real and complex were not carefully distinguished), he proved that Lie™s
classes were essentially all of the in¬nite dimensional primitive transitive transformation
groups. The slight extension was that SDiff(µ) had a companion extension to R·SDiff(µ),
the di¬eomorphisms which preserve µ up to a constant multiple and that Sp(„¦n ) had
a companion extension to R · Sp(„¦n ), the di¬eomorphisms which preserve „¦n up to a
constant multiple. Of course, there were also the holomorphic analogues of these. Notice
the remarkable fact that there are no “exceptional in¬nite dimensional primitive transitive
transformation groups”.
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 di¬erential systems), and a host
of other ideas which were only appreciated much later. Perhaps because of its originality,
Cartan™s work in this area was essentially ignored for many years.


L.6.11 110
In the 1950™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 “overlaps” were holomorphic. Gener-
alizing this example, it became clear that, for any collection “ of local di¬eomorphisms of
Rn which satis¬ed the following de¬nition, one could de¬ne a category of “-manifolds as
manifolds endowed with an atlas A of coordinate charts whose overlaps lay in A.
De¬nition 3: A local di¬eomorphism of Rn is a pair (U, φ) where U ‚ Rn is an open
set and φ: U ’ Rn is a one-to-one di¬eomorphism onto its image. A set “ of local
di¬eomorphisms of Rn is said to form a pseudo-group on Rn if it satis¬es the following
three properties:
(1) (Composition and Inverses) If (U, φ) and (V, ψ) are in “, then (φ’1 (V ), ψ —¦ φ) and
(φ(U), φ’1 ) also belong to “.
(2) (Localization and Globalization) If (U, φ) is in “, and W ‚ U is open, then (W, φ|W )
is also in “. Moreover, if (U, φ) is a local di¬eomorphism of Rn such that U can be
written as the union of open subsets W± for which (W± , φ|W± ) is in “ for all ±, then
(U, φ) is in “.
(3) (Non-triviality) (Rn , id) is in “.

As it turned out, the pseudo-groups “ of interest in geometry were exactly the ones
which could be characterized as the (local) solutions of a system of partial di¬erential
equations, i.e., they were Lie™s transformation groups. This caused a revival of interest
in Cartan™s work. Consequently, much of Cartan™s work has now been redone in modern
language. In particular, Cartan™s classi¬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
“natural” 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 ‚ R2n (with its standard symplectic structure) is
the annulus described by the relations r < |x| < R, then there cannot be a symplectic
di¬eomorphism φ: Ar,R ’ As,S that “exchanges the boundaries”. (Hint: Show that if φ
existed one would be able to construct a symplectic structure on S 2n .) Conclude that one
cannot na¨±vely de¬ne connected sum in the category of symplectic manifolds. (The “na¨ ±ve”
de¬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 di¬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 de¬nite symmetric matrices. Show that
n
the map σ: S+ ’ S+ de¬ned by σ(s) = s2 is a one-to-one di¬eomorphism of S+ onto
n n n
itself. Conclude that every element of Sn has a unique positive de¬nite square root
+

and that√ map s ’ s is a smooth mapping. Show also that, for any r ∈ O(n),
the √
we have trar = tr a r, so that the square root function is O(n)-equivariant.
(ii) Let A• denote the space of n-by-n invertible anti-symmetric matrices. Show that, for
n
a ∈ √• , the matrix ’a2 is symmetric and positive de¬nite. Show that the matrix
An
b = ’a2 is the unique symmetric positive de¬nite matrix that satis¬es b2 = ’a2

and moreover that b commutes with a. Check also that the mapping a ’ ’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 ¬bers
and any smooth, invertible skew-symmetric bundle mapping A: E ’ E, there exists
a unique smooth positive de¬nite symmetric bundle mapping B: E ’ E that satis¬es
B 2 = ’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 (n’1)! v where
v ∈ 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 ¬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 „¦0 is a symplectic structure on a compact manifold M, then there is an
open neighborhood U in H 2 (M, R) of [„¦0 ], such that, for all u ∈ U, there is a symplectic
structure „¦u on M with [„¦u ] = u. (Hint: Since M is compact, for any closed 2-form Υ,
the 2-form „¦ + tΥ is non-degenerate for all su¬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 µ0 and µ1 di¬er by an oriented dif-
feomorphism (i.e., there exists an orientation preserving di¬eomorphism φ: M ’ M that
satis¬es φ— (µ1 ) = µ0 ) if and only if

µ0 = µ1 .
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 µ ∈ A4 (M) be a volume
form that satis¬es M µ = 1. (By the previous problem, any two such forms di¬er by an
oriented di¬eomorphism of M.) For any (smooth) „¦ ∈ A2 (M), de¬ne —(„¦2 ) ∈ C ∞(M) by
the equation
„¦2 = —(„¦2 ) µ.
Now, ¬x a cohomology class u ∈ HdR (M) satisfying u2 = r[µ] where r = 0. De¬ne the
2

functional F : u ’ R
F („¦) = —(„¦2 ) „¦2 for „¦ ∈ u.
M
Show that any F -critical 2-form „¦ ∈ u is a symplectic form satisfying —(„¦2 ) = r and that
F has no critical values other than r2 . Show also that F („¦) ≥ r2 for all „¦ ∈ u.
This motivates de¬ning an invariant of the class u by
I(u) = inf F („¦).
„¦∈u

Gromov has suggested (private communication) that perhaps I(u) = r2 for all u, even
when the in¬mum is not attained.

7. Let P ‚ M be a closed submanifold and let U ‚ M be an open neighborhood of P in
M that can be retracted onto P , i.e., there exists a smooth map R: U — [0, 1] ’ U so that
R(u, 1) = u for all u ∈ U, R(p, t) = p for all p ∈ P and t ∈ [0, 1], and R(u, 0) lies in P for
all u ∈ U. (Every closed submanifold of M has such a neighborhood.)
Show that if ¦ is a closed k-form on U that vanishes at every point of P , then there
exists a (k’1)-form φ on U that vanishes on P and satis¬es dφ = ¦. (Hint: Mimic
Poincar´™s Homotopy Argument: Let Υ = R— (¦) and set … = ‚ Υ. Then, using the fact
e ‚t
that …(u, t) can be regarded as a (k’1)-form at u for all t, de¬ne
1
φ(u) = …(u, t) dt.
0
Now verify that φ has the desired properties.)


E.6.2 113
8. Show that Theorem 2 implies Darboux™ 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, „¦) be a symplectic
manifold. Show that the following description of the ¬‚ux homomorphism is valid. Let p be
an e-based path in Sp(„¦). Thus, p: [0, 1] — M ’ M satis¬es p— („¦) = „¦ for all 0 ¤ t ¤ 1.
t

Show that p („¦) = „¦ + •§dt for some 1-form • on [0, 1] — M. Let ιt : M ’ [0, 1] — M be
the “t-slice inclusion”: ιt (m) = (t, m), and set •t = ι— (•).
t
Show that •t is closed for all 0 ¤ t ¤ 1. Show that if we set

1
˜
¦(p) = •t dt,
0


then the cohomology class [¦(p)] ∈ HdR (M, R) depends only on the homotopy class of p
˜ 1

and hence de¬nes a map ¦: Sp0 („¦) ’ HdR (M, R). Verify that this map is the same as the
1

¬‚ux homomorphism de¬ned in Exercise 5.10.
Use this description to show that if p is in the kernel of ¦, then p is homotopic to a
path p for which the forms •t are all exact. This shows that the kernel of ¦ is actually
connected.

10. The point of this exercise is to show that any symplectic vector bundle over a sym-
plectic manifold (M, „¦) can occur as the symplectic normal bundle for some symplectic
embedding M into some other symplectic manifold.
Let (M, „¦) be a symplectic manifold and let π: E ’ M be a symplectic vector bundle
over M of rank 2n. (I.e., E comes equipped with a section B of Λ2 (E — ) that restricts
to each ¬ber Em to be a symplectic structure Bm .) Show that there exists a symplectic
structure Ψ on an open neighborhood in E of the zero section of E that satis¬es the
condition that Ψ0m = „¦m + Bm under the natural identi¬cation T0m E = Tm M • Em .
(Hint: Choose a locally ¬nite open cover U = {U± | ± ∈ A} of M so that, if we de¬ne
E± = π ’1 (U± ), then there exists a symplectic trivialization „± : E± ’ R2n (where R2n is
given its standard symplectic structure „¦0 = dxi §dy i ). Now let {»± | ± ∈ A} be a partition
of unity subordinate to the cover U. Show that the form

Ψ = π — („¦) + —
d »± „± (xi dy i )
±


has the desired properties.)

11. Show that if E is a symplectic vector bundle over M and L ‚ E is a Lagrangian
subbundle, then E is isomorphic to L • L— as a symplectic bundle. (The symplectic
bundle structure Υ on L • L— is the one that, on each ¬ber satis¬es

Υ (v, ±), (w, β) = ±(w) ’ β(v). )

E.6.3 114
(Hint: First choose a complementary subbundle F ‚ E so that E = L • F . Show that
F is naturally isomorphic to L— abstractly by using the fact that the symplectic structure
on E is non-degenerate. Then show that there exists a bundle map A: F ’ L so that

˜
F = {v + Av | v ∈ F }

is also a Lagrangian subbundle of E that is complementary to L and isomorphic to L— via
some bundle map ±: L— ’ F . Now show that id • ±: L • L— ’ L • F E is a symplectic
˜ ˜
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 , „¦) be
a symplectic manifold and let f = (f 1 , . . . , f n ): M ’ Rn be a smooth submersion with

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