46.15. The weak symplectic structure on the space of Jacobi ¬elds on the

Virasoro Lie algebra. Since the Korteweg - de Vries equation has local solutions

depending smoothly on the initial conditions (and global solutions if a = 0), the

space of all Jacobi ¬elds exists and is isomorphic to (R —ω X(S 1 )) — (R —ω X(S 1 )).

The weak symplectic structure is given by (46.9):

y z y zt yt z u y z

’

σ , = , , + , ,

b c b ct bt c a b c

y u z y z u

’ ’

, , , ,

b a b b c a

(yzt ’ yt z + 2u(yzx ’ yx z)) dx

=

S1 or R

+ b(ct + ω(z, u)) ’ (bt + ω(y, u)) ’ aω(y, z)

(yzt ’ yt z + 2u(yzx ’ yx z)) dx

(1) =

S1 or R

+ bC1 ’ B1 c ’ ay z a dx.

S1 or R

46.15

510 Chapter IX. Manifolds of mappings

Complements to Manifolds of Mappings

For a compact smooth ¬nite dimensional manifold M , some results on the topolog-

ical type of the di¬eomorphism group Di¬(M ) are available in the literature. In

[Smale, 1959] it is shown that Di¬(S 2 ) is homotopy equivalent to O(3, R). This

result has been extended in [Hatcher, 1983] where it is shown that Di¬(S 3 ) is ho-

motopy equivalent to O(4, R). The component group π0 (Di¬ + (S n )) of the group of

orientation preserving di¬eomorphisms on the sphere S n is isomorphic to the group

of homotopy spheres of dimension n + 1 for n > 4, see [Kervaire, Milnor, 1963].

If M is a product of spheres then π0 (Di¬(M )) has been computed by [Browder,

1967] and [Turner, 1969]. For a simply connected orientable compact manifold M

in [Sullivan, 1978] it is shown that π0 (Di¬(M )) is commensurable to an arithmetic

group, where two groups are said to be commensurable if there is a ¬nite sequence

of homomorphisms of groups between them with ¬nite kernel and cokernel. By

[Borel, Harish-Chandra, 1962], any arithmetic group is ¬nitely presented, and by

[Borel, Serre, 1973] it is even of ¬nite type, which means that its classifying space is

homotopy equivalent to a CW-complex with ¬nitely many cells in each dimension.

Two di¬eomorphisms f, g of M are called pseudo-isotopic if there is a di¬eomor-

phism F : M — I ’ M — I restricting to f and g at the two ends of M — I, respec-

tively. Let D(M ) be the group of pseudo-isotopy classes of di¬eomorphisms of M ,

a quotient of Di¬(M )/ Di¬ 0 (M ), where Di¬ 0 (M ) is the connected component. By

[Cerf, 1970], if M is simply connected then π0 (Di¬(M )) = D(M ), whereas for non

simply connected M there is in general an abelian kernel A in an exact sequence

0 ’ A ’ π0 (Di¬(M )) ’ D(M ) ’ 0,

and A has been computed by [Hatcher, Wagoner, 1973] and [Igusa, 1984]. In

particular, A is ¬nitely generated if π0 (M ) is ¬nite.

In [Trianta¬llou, 1994] the following result is announced: If M is a smooth compact

orientable manifold of dimension ≥ 5 with ¬nite fundamental group, then D(M ) is

commensurable to an arithmetic group. Moreover, π0 (M ) is of ¬nite type.

511

Chapter X

Further Applications

47. Manifolds for Algebraic Topology . . . . . . . .... . . . . . . 512

48. Weak Symplectic Manifolds ......... .... . . . . . . 522

49. Applications to Representations of Lie Groups . .... . . . . . . 528

50. Applications to Perturbation Theory of Operators ... . . . . . . 536

51. The Nash-Moser Inverse Function Theorem .. .... . . . . . . 553

In section (47) we show how to treat direct limit manifolds like S ∞ = ’ S n or

lim

’

SO(∞, R) = lim SO(n, R) as real analytic manifolds modeled on R∞ = R(N) =

’’

N R. As topological spaces these are often used in algebraic topology, in partic-

ular the Grassmannians as classifying spaces of the groups SO(k). The di¬erential

calculus is very well applicable, and the groups GL(∞, R), SO(∞, R) turn out to

be regular Lie groups, where the exponential mapping is even locally a di¬eomor-

phism onto a neighborhood of the identity, since it factors over ¬nite dimensional

exponential mappings.

In section (48) we consider a manifold with a closed 2-form σ inducing an injective

(but in general not surjective) mapping σ : T M ’ T — M . This is called a weak

symplectic manifold, and there are di¬culties in de¬ning the Poisson bracket for

general smooth functions. We describe a natural subspace of functions for which

the Poisson bracket makes sense, and which admit Hamiltonian vector ¬elds.

For a (unitary) representation of a (¬nite dimensional) Lie group G in a Hilbert

space H one wishes to have the in¬nitesimal representation of the Lie algebra at

disposal. Classically this is given by unbounded operators and o¬ers analytical

di¬culties. We show in (49.5) and (49.10) that the dense subspaces H∞ and Hω of

smooth and real analytic vectors are invariant convenient vector spaces on which

the action G — H∞ ’ H∞ is smooth (resp. real analytic). These are well known

results. Our proofs are transparent and surprisingly simple; they use, however, the

uniform boundedness principles (5.18) and (11.12). Using this and the results from

section (48), we construct the moment mapping of any unitary representation.

Section (50) on perturbation theory of operators is devoted to the background and

proof of theorem (50.16) which says that a smooth curve of unbounded selfadjoint

operators on Hilbert space with compact resolvent admits smooth parameteriza-

tions of its eigenvalues and eigenvectors, under some condition. The real analytic

version of this theorem is due to [Rellich, 1940], see also [Kato, 1976, VII, 3.9], with

formally stronger notions of real analyticity which are quite di¬cult to handle.

512 Chapter X. Further Applications 47.2

Again the power of convenient calculus shows in the ease with which this result is

derived.

In section (51) we present a version of one of the hard implicit function theorems,

which is applicable to some non-linear partial di¬erential equations. Its origins

are the result of John Nash about the existence of isometric embeddings of Rie-

mannian manifolds into Rn ™s, see [Nash, 1956]. It was then identi¬ed by [Moser,

1961], [Moser, 1966] as an abstract implicit function theorem, and found the most

elaborate presentations in [Hamilton, 1982], and [Gromov, 1986]. But the original

application about the existence of isometric embeddings was ¬nally reproved in a

very simple way by [G¨nther, 1989, 1990], who composed the nonlinear perturba-

u

tion problem with the inverse of a Laplace operator and then applied the Banach

¬xed point theorem. This is characteristic for applications of hard implicit function

theorems: Each serious application is incredibly complicated, and ¬nally a simple ad

hoc method solves the problem. To our knowledge of the original applications only

two have not yet found direct simpler proofs: The result by Hamilton [Hamilton,

1982], that a compact Riemannian 3-manifold with positive Ricci curvature also

admits a metric with constant scalar curvature; and the application on the small

divisor problem in celestial mechanics. We include here the hard implicit function

theorem of Nash and Moser in the form of [Hamilton, 1982], in full generality and

without any loss, in condensed form but with all details.

47. Manifolds for Algebraic Topology

47.1. Convention. In this section the space R(N) of all ¬nite sequences with the

direct sum topology plays a an important role. It is also denoted by R∞ , mainly in

in algebraic topology. It is a convenient vector space. We consider it equipped with

the weak inner product x, y := xi yi , which is bilinear and bounded, therefore

smooth. It is called weak, since it is non degenerate in the following sense: the

associated linear mapping R(N) ’ (R(N) ) = RN is injective but far from being

surjective. We will also use the weak Euclidean distance |x| := x, x , whose

square is a smooth function.

47.2. Example: The sphere S ∞ . This is the set {x ∈ R(N) : x, x = 1},

the usual in¬nite dimensional sphere used in algebraic topology, the topological

inductive limit of S n ‚ S n+1 ‚ . . . . The inductive limit topology coincides with

the subspace topology since clearly lim S n ’ S ∞ ‚ R(N) is continuous, S ∞ as

’’

∞

(N)

closed subset of R with the c -topology is compactly generated, and since each

compact set is contained in a step of the inductive limit.

We show that S ∞ is a smooth manifold by describing an explicit smooth atlas, the

stereographic atlas. Choose a ∈ S ∞ (”south pole”). Let

x’ x,a a

U+ := S ∞ \ {a}, u+ : U+ ’ {a}⊥ , u+ (x) = 1’ x,a ,

x’ x,a a

U’ := S ∞ \ {’a}, u’ : U’ ’ {a}⊥ , u’ (x) = 1+ x,a .

47.2

47.2 47. Manifolds for algebraic topology 513

From an obvious drawing in the 2-plane through 0, x, and a it is easily seen that

u+ is the usual stereographic projection. We also get

|y|2 ’1

u’1 (y) = for y ∈ {a}⊥ \ {0}

2

|y|2 +1 a + |y|2 +1 y

+

and (u’ —¦u’1 )(y) = |y|2 . The latter equation can directly be seen from the drawing

y

+

using the intersection theorem.

The two stereographic charts above can be extended to charts on open sets in R(N)

in such a way that S ∞ becomes a splitting submanifold of R(N) :

u+ : R(N) \ [0, +∞)a ’ a⊥ + (’1, +∞)a

˜

z

u+ (z) := u+ ( ) + (|z| ’ 1)a

˜

|z|

= (1 + z, a )u’1 (z ’ z, a a)

+

Since the model space R(N) of S ∞ has the bornological approximation property by

(28.6), and is re¬‚exive, by (28.7) the operational tangent bundle of S ∞ equals the

kinematic one: DS ∞ = T S ∞ .

We claim that T S ∞ is di¬eomorphic to {(x, v) ∈ S ∞ — R(N) : x, v = 0}.

The Xx ∈ Tx S ∞ are exactly of the form c (0) for a smooth curve c : R ’ S ∞

with c(0) = x by (28.13). Then 0 = dt |0 c(t), c(t) = 2 x, Xx . For v ∈ x⊥ we use

d

v

c(t) = cos(|v|t)x + sin(|v|t) |v| .

The construction of S ∞ works for any positive de¬nite bounded bilinear form on

any convenient vector space.

The sphere is smoothly contractible, by the following argument: We consider the

homotopy A : R(N) — [0, 1] ’ R(N) through isometries which is given by A0 = Id

and by (44.22)

At (a0 , a1 , a2 , . . . ) = (a0 , . . . , an’2 , an’1 cos θn (t), an’1 sin θn (t),

an cos θn (t), an sin θn (t), an+1 cos θn (t), an+1 sin θn (t), . . . )

for n+1 ¤ t ¤ n , where θn (t) = •(n((n + 1)t ’ 1)) π for a ¬xed smooth function

1 1

2

• : R ’ R which is 0 on (’∞, 0], grows monotonely to 1 in [0, 1], and equals 1

on [1, ∞). The mapping A is smooth since it maps smooth curves (which locally

map into some RN ) to smooth curves (which then locally have values in R2N ).

(N)

Then A1/2 (a0 , a1 , a2 , . . . ) = (a0 , 0, a1 , 0, a2 , 0, . . . ) is in Reven , and on the other

(N)

hand A1 (a0 , a1 , a2 , . . . ) = (0, a0 , 0, a1 , 0, a2 , 0, . . . ) is in Rodd . This is a variant of

a homotopy constructed by [Ramadas, 1982]. Now At |S ∞ for 0 ¤ t ¤ 1/2 is a

(N)

smooth isotopy on S ∞ between the identity and A1/2 (S ∞ ) ‚ Reven . The latter set

is contractible in a chart.

One may prove in a simpler way that S ∞ is contractible with a real analytic homo-

topy with one corner: roll all coordinates one step to the right and then contract

in the stereographic chart opposite to (1, 0, . . . ).

47.2

514 Chapter X. Further Applications 47.4

47.3. Example. The Grassmannians and the Stiefel manifolds. The

Grassmann manifold G(k, ∞; R) = G(k, ∞) is the set of all k-dimensional lin-

ear subspaces of the space of all ¬nite sequences R(N) . The Stiefel manifold of

orthonormal k-frames O(k, ∞; R) = O(k, ∞) is the set of all linear isometries

Rk ’ R(N) , where the latter space is again equipped with the standard weak inner

product described at the beginning of (47.2). The Stiefel manifold of all k-frames

GL(k, ∞; R) = GL(k, ∞; R) is the set of all injective linear mappings Rk ’ R(N) .

)t : L(Rk , R(N) ) ’ L(R(N) , Rk ) which

There is a canonical transposition mapping (

is given by

incl A

At : R(N) ’ ’ RN = R(N) ’ (Rk ) = Rk

’ ’

and satis¬es At (x), y = x, A(y) . The transposition mapping is bounded and

linear, so it is real analytic. Then we have

GL(k, ∞) = {A ∈ L(Rk , R(N) ) : At —¦ A ∈ GL(k)},

since At —¦ A ∈ GL(k) if and only if Ax, Ay = At Ax, y = 0 for all y implies

x = 0, which is equivalent to A injective. So in particular GL(k, ∞) is open in

L(Rk , R(N) ). The Lie group GL(k) acts freely from the right on the space GL(k, ∞).

Two elements of GL(k, ∞) lie in the same orbit if and only if they have the same

image in R(N) . We have a surjective mapping π : GL(k, ∞) ’ G(k, ∞), given by

π(A) = A(Rk ), where the inverse images of points are exactly the GL(k)-orbits.

Similarly, we have

O(k, ∞) = {A ∈ L(Rk , R(N) ) : At —¦ A = Idk }.

The Lie group O(k) of all isometries of Rk acts freely from the right on the space

O(k, ∞). Two elements of O(k, ∞) lie in the same orbit if and only if they have

the same image in R(N) . The projection π : GL(k, ∞) ’ G(k, ∞) restricts to a

surjective mapping π : O(k, ∞) ’ G(k, ∞), and the inverse images of points are

now exactly the O(k)-orbits.

47.4. Lemma. Iwasawa decomposition. Let T (k; R) = T (k) be the group

of all upper triangular k — k-matrices with positive entries on the main diagonal.

Then each B ∈ GL(k, ∞) can be written in the form B = p(B) —¦ q(B), with unique

p(B) ∈ O(k, ∞) and q(B) ∈ T (k). The mapping q : GL(k, ∞) ’ T (k) is real

analytic, and p : GL(k, ∞) ’ O(k, ∞) ’ GL(k, ∞) is real analytic, too.

Proof. We apply the Gram Schmidt orthonormalization procedure to the vectors

B(e1 ), . . . , B(ek ) ∈ R(N) . The coe¬cients of this procedure form an upper trian-

gular k — k-matrix q(B) whose entries are rational functions of the inner products

B(ei ), B(ej ) and are positive on the main diagonal. So (B —¦ q(B)’1 )(e1 ), . . . , (B —¦

q(B)’1 )(ek ) is the orthonormalized frame p(B)(e1 ), . . . , p(B)(ek ).

47.4

47.5 47. Manifolds for algebraic topology 515

47.5. Theorem. The following are real analytic principal ¬ber bundles:

(π : O(k, ∞; R) ’ G(k, ∞; R), O(k, R)),

(π : GL(k, ∞; R) ’ G(k, ∞; R), GL(k, R)),

(p : GL(k, ∞; R) ’ O(k, ∞; R), T (k; R)).

The last one is trivial. The embeddings Rn ’ R(N) induce real analytic embeddings,

which respect the principal right actions of all the structure groups

O(k, n) ’ O(k, ∞),

GL(k, n) ’ GL(k, ∞),

G(k, n) ’ G(k, ∞).

All these cones are inductive limits in the category of real analytic (and smooth)

manifolds. All manifolds are smoothly paracompact.

Proof. Step 1. G(k, ∞) is a real analytic manifold.

For A ∈ O(k, ∞) we consider the open subset WA := {B ∈ GL(k, ∞) : At —¦ B ∈

GL(k)} of L(Rk , R(N) ), and we let VA := WA © O(k, ∞) = {B ∈ O(k, ∞) : At —¦ B ∈

GL(k)}. Obviously, VA is invariant under the action of O(k) and VAU = VA for

U ∈ O(k). So we may denote Uπ(A) := π(VA ). Let P := π(A) = A(Rk ) ∈ G(k, ∞).

We de¬ne the mapping

vA : VA ’ L(P, P ⊥ ),

vA (B) := B(At B)’1 At ’ AAt |P

= (IdR(N) ’AAt )B(At B)’1 At |P.

In order to visualize this de¬nition note that A —¦ At is the orthonormal projection

R(N) ’ P , and that the image of B in R(N) = P • P ⊥ is the graph of vA (B). It is

easily checked that vA (B) ∈ L(P, P ⊥ ) and that vA (BU ) = vA (B) = vAU (B) for all

U ∈ O(k). So we may de¬ne

uP : UP ’ L(P, P ⊥ ),

uP (π(B)) := vA (B).

For C ∈ L(P, P ⊥ ) the mapping A+C—¦A is a parameterization of the graph of C, it is

in GL(k, ∞), and we have (using p from lemma (47.3)) that u’1 (C) = π(p(A+CA)),

P

since for B ∈ VA the image of B equals the graph of C = uP (π(B)), which in turn

is equal to (A + CA)(Rk ) = (A + CA) q(A + CA)’1 (Rk ) = p(A + CA)(Rk ).

⊥

Now we check the chart changes: Let P1 = π(A1 ), P2 = π(A2 ), and C ∈ L(P1 , P1 ),

then we have

’1

uP2 —¦ u’1 (C) = (IdR(N) ’A2 At ) p(A1 + C A1 ) At p(A1 + C A1 ) At |P2 ,

2 2 2

P1

47.5

516 Chapter X. Further Applications 47.6

⊥

which is de¬ned on the open set of all C ∈ L(P1 , P1 ) for which At p(A1 + C A1 ) is

2

in GL(k) and which is real analytic there.

Step 2. The principal bundles.

We ¬x A ∈ O(k, ∞) and consider the section

sA : Uπ(A) ’ VA ,

sA (Q) := p(A + uπ(A) (Q) A)

and the principal ¬ber bundle chart

ψA : VA ’ Uπ(A) — O(k),

ψA (B) := π(B), sA (π(B))t B ,

’1

ψA (Q, U ) = sA (Q) U.

Clearly, these charts give a principal ¬ber bundle atlas with cocycle of transition

functions Q ’ sA2 (Q)t sA1 (Q) ∈ O(k).

The same formulas (for A still in O(k, ∞)) give ¬ber bundle charts ψA : WA ’

Uπ(A) — GL(k) for GL(k, ∞) ’ G(k, ∞).

The injection O(k, ∞) ’ GL(k, ∞) is a real analytic section of the real analytic

projection p : GL(k, ∞) ’ O(k, ∞), which by lemma (47.4) gives a trivial principal

¬ber bundle with structure group T (k). This fact implies that O(k, ∞) is a splitting

real analytic submanifold of the convenient vector space L(Rk , R(N) ).

Since R(N) is the inductive limit of the direct summands Rn in the category of

convenient vector spaces and real analytic (smooth) mappings, and since the chart

constructions above restrict to the usual ones on the ¬nite dimensional Grassman-

nians and bundles, the assertion on the inductive limits follows.

All these manifolds are smoothly paracompact. For R(N) this is in (16.10), so it holds

for L(Rk , R(N) ) and for the closed subspace O(k, ∞), see (47.3). Then it follows

for G(k, ∞) since O(k, ∞) ’ G(k, ∞) is a principal ¬ber bundle with compact

structure group O(k), by integrating the members of the partition over the ¬ber.

Then we get the result for GL(k, ∞) by bundle argumentation on GL(k, ∞) ’

G(k, ∞), since the ¬ber GL(k) is ¬nite dimensional, so the product is well behaved

by (4.16).

47.6. Theorem. The principal bundle (O(k, ∞), π, G(k, ∞)) is classifying for

¬nite dimensional principal O(k)-bundles and carries a universal real analytic O(k)-

connection ω ∈ „¦1 (O(k, ∞), o(k)).

This means: For each ¬nite dimensional smooth or real analytic principal O(k)-

bundle P ’ M with principal connection ωP there is a smooth or real analytic

mapping f : M ’ G(k, ∞) such that the pullback O(k)-bundle f — O(k, ∞) is iso-

morphic to P and the pullback connection f — ω equals ωP via this isomorphism.

For ∞ replaced by a large N and bundles where the dimension of the base is

bounded this is due to [Schla¬‚i, 1980].

47.6

47.6 47. Manifolds for algebraic topology 517

Proof. Step 1. The tangent bundle of O(k, ∞) is given by

T O(k, ∞) = {(A, X) ∈ O(k, ∞) — L(Rk , R(N) ) : X t A ∈ o(k)}.

We have O(k, ∞) = {A : At A = Idk }, thus TA O(k, ∞) ⊆ {X : X t A + At X = 0}.

Since At A = Idk is an equation of constant rank when restricted to GL(k, n) for

¬nite n, we have equality by the implicit function theorem.

Another argument which avoids the implicit function theorem is the following.

By theorem (47.5) the vertical tangent space {AZ : Z ∈ t(k)} at A ∈ O(k, ∞)

of the bundle GL(k, ∞) ’ O(k, ∞) is transversal to TA O(k, ∞), where t(k) is

the Lie algebra of T (k). We have equality since an easy computation shows that

{AZ : Z ∈ t(k)} © {X : X t A + At X = 0} = 0.

Step 2. The inner product on Rk and the weak inner product on R(N) induce a

bounded weak inner product on the space L(Rk , R(N) ) by X, Y = Trace(X t Y ) =

Trace(XY t ), where the second trace makes sense since XY t has ¬nite dimensional

range. With respect to this inner product we consider the orthonormal projection

¦A : TA O(k, ∞) ’ VA O(k, ∞) onto the vertical tangent space VA O(k, ∞) = {AY :

Y ∈ o(k)} of O(k, ∞) ’ G(k, ∞). Its kernel, the horizontal space, turns out to be

{Z ∈TA O(k, ∞) : Trace(Z t AY ) = 0 for all Y ∈ o(k)} =

= {Z : Z t A both skew and symmetric}

= {Z ∈ L(Rk , R(N) ) : Z t A = 0}