∞

N 1 1

’ ’

(t’ 1 )2 (t’ 1 )2

fN (t) = hn (t)e , f (t) = hn (t)e .

3 3

n=0 n=0

1’ 41

1 n

Then f ≥ 0 is a smooth function which in (0, 3)

has zeros exactly at t = 3

1

and which is 0 for t ∈ (0, 3 ). In every neighborhood of f lies a function fN which

/

44.18

44.19 44. Principal bundles with structure group a di¬eomorphism group 481

has only ¬nitely many of the zeros of f and is identically zero in the interval

[(1 ’ 4N1+1 )/3, 1/3]. All di¬eomorphisms in the isotropy subgroup of f are also

contained in the isotropy subgroup of fN , but the latter group contains additionally

all di¬eomorphisms of S 1 which have support only on [(1 ’ 4N1+1 )/3, 1/3]. This

contradicts (44.17.5).

44.19. Counter-example. [Michor, Schichl, 1997]. The action of the gauge group

Gau(E) on Conn(E) does not admit slices, for dim M ≥ 2.

We will construct locally a connection, which satis¬es that in any neighborhood

there exist connections which have a bigger isotropy subgroup. Let n = dim S,

and let h : Rn ’ R be a smooth nonnegative bump function, which satis¬es

carr h = {s ∈ Rn | s ’ s0 < 1}. Put hr (s) := rh(s0 + 1 (s ’ s0 )), then carr hr =

r

n s1

{s ∈ R | s ’ s0 < r}. Then put hr (s) := h(s ’ (s1 ’ s0 )) which implies

carr hs1 = {s ∈ Rn | s ’ s1 < r}. Using these functions, we can de¬ne new

r

functions fk for k ∈ N as

1

fk (s) = k hsk /2k (s),

z

4

k

s∞ ’s0 1 1

for some s∞ ∈ Rn and sk := s0 + z(2 ’1’

where z := ). Further

l=0 2l 2k

3

set

N

1

’

f N (s) := e f (s) := lim f N (s).

s’s∞ 2 fk (s),

N ’∞

k=0

The functions f N and f are smooth, respectively, since all the functions fk are

smooth, on every point s at most one summand is nonzero, and the series is in each

derivative uniformly convergent on a neighborhood of s∞ . The carriers are given by

∞

N

carr f N = k=0 {s ∈ Rn | s ’ sk < 21 z } and carr f = k=0 {s ∈ Rn | s ’ sk <

k

1

z }. The functions f N and f vanish in all derivatives in all xk , and f vanishes

2k

in all derivatives in s∞ .

∼

=

Let ψ : E|U ’ U — S be a ¬ber bundle chart of E with a chart u : U ’ Rm on M ,

’

∼

= ∞

and let v : V ’ Rn be a chart on S. Choose g ∈ Cc (M, R) with … = supp(g) ‚ U

’

and dg § du1 = 0 on an open dense subset of supp(g). Then we can de¬ne a

Christo¬el form as in (37.5) by

“ := g du1 — f (v)‚v1 ∈ „¦1 (U, X(S)).

This de¬nes a connection ¦ on E|U which can be extended to a connection ¦ on

E by the following method. Take a smooth functions k1 , k2 ≥ 0 on M satisfying

k1 + k2 = 1 and k1 = 1 on supp(g) and supp(k1 ) ‚ U and any connection ¦ on

E, and set ¦ = k1 ¦“ + k2 ¦ , where ¦“ denotes the connection which is induced

locally by “. In any neighborhood of ¦ there exists a connection ¦N de¬ned by

“N := g du1 — f N (s)‚v1 ∈ „¦1 (U, X(S)),

and extended like ¦.

44.19

482 Chapter IX. Manifolds of mappings 44.19

Claim: There is no slice at ¦.

Proof: We have to consider the isotropy subgroups of ¦ and ¦N . Since the con-

nections ¦ and ¦N coincide outside of U , we may investigate them locally on

W = {u : k1 (u) = 1} ‚ U . The curvature of ¦ is given locally on W by (37.5) as

X(S)

= dg § du1 — f (v)‚v1 ’ 0.

RU := d“ ’ 1 [“, “]§

(1) 2

For every element of the gauge group Gau(E) which is in the isotropy group

Gau(E)¦ the local representative over W which looks like γ : (u, v) ’ (u, γ(u, v))

˜

by (37.5) satis¬es

)).“(ξu , v) = “(ξu , γ(u, v)) ’ Tu (γ( , v)).ξu ,

(2) Tv (γ(u,

‚γ 1 ‚γ i j

1

‚ i = g(u)du1 — f (γ(u, v))‚v1 ’

g(u)du — f (v) du — ‚vi .

iv ‚uj

‚v

i i,j

Comparing the coe¬cients of duj — ‚vi we get for γ over W the equations

‚γ i

= 0 for (i, j) = (1, 1),

‚uj

‚γ 1 ‚γ 1

g(u)f (v) 1 = g(u)f (γ(u, v)) ’

(3) .

‚u1

‚v

Considering next the transformation γ — RU = RU of the curvature (37.4.3), we get

˜

Tv (γ(u, )).RU (ξu , ·u , v) = RU (ξu , ·u , γ(u, v)),

‚γ 1

1

‚vi = dg § du1 — f (γ(u, v))‚v1 .

dg § du — f (v)

(4)

‚v i

i

Another comparison of coe¬cients yields the equations

‚γ 1

f (v) i = 0 for i = 1,

‚v

‚γ 1

(5) f (v) 1 = f (γ(u, v)),

‚v

whenever dg § du1 = 0, but this is true on an open dense subset of supp(g). Finally,

putting (5) into (3) shows

‚γ i

= 0 for all i, j.

‚uj

Collecting the results on supp(g), we see that γ has to be constant in all directions

of u. Furthermore, wherever f is nonzero, γ 1 is a function of v 1 only and γ has to

map zero sets of f to zero sets of f .

Replacing “ by “N we get the same results with f replaced by f N . Since f = f N

wherever f N is nonzero or f vanishes, γ in the isotropy group of ¦ obeys all these

equations not only for f but also for f N on supp f N ∪ f ’1 (0). On carr f \ carr f N

the gauge transformation γ is a function of v 1 only, hence it cannot leave the

44.19

44.21 44. Principal bundles with structure group a di¬eomorphism group 483

zero set of f N by construction of f and f N . Therefore, γ obeys all equations for

f N whenever it obeys all equations for f , thus every gauge transformation in the

isotropy subgroup of ¦ is in the isotropy subgroup of ¦N .

On the other hand, any γ with support in carr f \ carr f N which changes only

in the v 1 direction and does not keep the zero sets of f invariant, de¬nes a gauge

transformation in the isotropy subgroup of ¦N which is not in the isotropy subgroup

of ¦.

Therefore, there exists in every neighborhood of ¦ a connection ¦N whose isotropy

subgroup is bigger than the isotropy subgroup of ¦. Thus, by property (44.17.5)

no slice exists at ¦.

44.20. Counter-example. [Michor, Schichl, 1997]. The action of the gauge group

Gau(E) on Conn(E) also admits no slices for dim M = 1, i.e. for M = S 1 .

The method of (44.19) is not applicable in this situation, since dg § du1 = 0 is not

possible, any connection ¦ on E is ¬‚at. Hence, the horizontal bundle is integrable,

the horizontal foliation induced by ¦ exists and determines ¦. Any gauge trans-

formation leaving ¦ invariant also has to map leaves of the horizontal foliation to

other leaves of the horizontal foliation.

We shall construct connections ¦» near ¦» such that the isotropy groups in Gau(E)

look radically di¬erent near the identity, contradicting (44.17.5).

Let us assume without loss of generality that E is connected, and then, by replacing

S 1 by a ¬nite covering if necessary, that the ¬ber is connected. Then there exists

a smooth global section χ : S 1 ’ E. By an argument given in the proof of (42.20)

there exists a tubular neighborhood π : U ‚ E ’ im χ such that π = χ —¦ p|U

(i.e. a tubular neighborhood with vertical ¬bers). This tubular neighborhood then

contains an open thickened sphere bundle with ¬ber S 1 — Rn’1 , and since we

are only interested in gauge transformations near IdE , which e.g. keep a smaller

thickened sphere bundle inside the larger one, we may replace E by an S 1 -bundle.

By replacing the Klein bottle by a 2-fold covering we may ¬nally assume that the

bundle is pr1 : S 1 — S 1 ’ S 1 .

Consider now connections where the horizontal foliation is a 1-parameter subgroup

with slope » we see that the isotropy group equals S 1 if » is irrational, and equals

S 1 times the di¬eomorphism group of a closed interval if » is rational.

44.21. A classifying space for the di¬eomorphism group. Let 2 be the

Hilbert space of square summable sequences, and let S be a compact manifold.

By a slight generalization of theorem (44.1) (we use a Hilbert space instead of a

Riemannian manifold N ), the space Emb(S, 2 ) of all smooth embeddings is an

open submanifold of C ∞ (S, 2 ), and it is also the total space of a smooth principal

bundle with structure group Di¬(S) acting from the right by composition. The base

space B(S, 2 ) := Emb(S, 2 )/ Di¬(S) is a smooth manifold modeled on Fr´chet e

spaces which are projective limits of Hilbert spaces. B(S, 2 ) is a Lindel¨f space in

o

the quotient topology, and the model spaces admit bump functions, thus B(S, 2 )

44.21

484 Chapter IX. Manifolds of mappings 44.23

admits smooth partitions of unity, by (16.10). We may view B(S, 2 ) as the space

of all submanifolds of 2 which are di¬eomorphic to S, a nonlinear analog of the

in¬nite dimensional Grassmannian.

2

44.22. Lemma. The total space Emb(S, ) is contractible.

Therefore, by the general theory of classifying spaces the base space B(S, 2 ) is a

classifying space of Di¬(S). We will give a detailed description of the classifying

process in (44.24).

2 2

— [0, 1] ’

Proof. We consider the continuous homotopy A : through isome-

tries which is given by A0 = Id and by

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, ∞).

Then A1/2 (a0 , a1 , a2 , . . . ) = (a0 , 0, a1 , 0, a2 , 0, . . . ) is in 2

even and on the other hand

A1 (a0 , a1 , a2 , . . . ) = (0, a0 , 0, a1 , 0, a2 , 0, . . . ) is in 2 . The same homotopy makes

odd

∞ (N)

sense as a mapping A : R — R ’ R , and here it is easily seen to be smooth:

a smooth curve in R(N) is locally bounded and thus locally takes values in a ¬nite

dimensional subspace RN ‚ R(N) . The image under A then has values in R2N ‚

R(N) , and the expression is clearly smooth as a mapping into R2N . This is a variant

of a homotopy constructed by [Ramadas, 1982].

Given two embeddings e1 and e2 ∈ Emb(S, 2 ) we ¬rst deform e1 through embed-

dings to e1 ∈ Emb(S, 2 ), and e2 to e2 ∈ Emb(S, 2 ). Then we connect them

even odd

by te1 + (1 ’ t)e2 which is a smooth embedding for all t since the values are always

orthogonal.

44.23. We consider the smooth action ev : Di¬(S) — S ’ S and the associated

bundle Emb(S, 2 )[S, ev] = Emb(S, 2 ) —Di¬(S) S which we call E(S, 2 ), a smooth

¬ber bundle over B(S, 2 ) with standard ¬ber S. In view of the interpretation of

B(S, 2 ) as the nonlinear Grassmannian, we may visualize E(S, 2 ) as the ”univer-

sal S-bundle” as follows: E(S, 2 ) = {(N, x) ∈ B(S, 2 ) — 2 : x ∈ N } with the

di¬erentiable structure from the embedding into B(S, 2 ) — 2 .

The tangent bundle T E(S, 2 ) is then the space of all (N, x, ξ, v) where N ∈

B(S, 2 ), x ∈ N , ξ is a vector ¬eld along and normal to N in 2 , and v ∈ Tx 2 such

that the part of v normal to Tx N equals ξ(x). This follows from the description

of the principal ¬ber bundle Emb(S, 2 ) ’ B(S, 2 ) given in (44.1) combined with

(42.17). Obviously, the vertical bundle V E(S, 2 ) consists of all (N, x, v) with x ∈ N

and v ∈ Tx N . The orthonormal projection p(N,x) : 2 ’ Tx N de¬nes a connection

¦class : T E(S, 2 ) ’ V E(S, 2 ) which is given by ¦class (N, x, ξ, v) = (N, x, p(N,x) v).

It will be called the classifying connection for reasons to be explained in the next

theorem.

44.23

44.24 44. Principal bundles with structure group a di¬eomorphism group 485

44.24. Theorem. Classifying space for Di¬(S).

The ¬ber bundle (E(S, 2 ) ’ B(S, 2 ), S) is classifying for S-bundles and ¦class is

a classifying connection:

For each ¬nite dimensional bundle (p : E ’ M, S) and each connection ¦ on E

there is a smooth (classifying) mapping f : M ’ B(S, 2 ) such that (E, ¦) is iso-

morphic to (f — E(S, 2 ), f — ¦class ). Homotopic maps pull back isomorphic S-bundles

and conversely (the homotopy can be chosen smooth). The pulled back connection

d

is invariant under a homotopy H if and only if i(C class T(x,t) H.(0x , dt ))Rclass = 0

where C class is the horizontal lift of ¦class , and Rclass is its curvature .

Since S is compact the classifying connection ¦class is complete, and its parallel

transport Ptclass has the following classifying property:

— class

˜ ˜

f —¦ Ptf ¦ (c, t) = Ptclass (f —¦ c, t) —¦ f ,

where f : E ∼ f — E(S, 2 ) ’ E(S, 2 ) is the ¬berwise di¬eomorphic which covers

˜ =

the classifying mapping f : M ’ B(S, 2 ).

Proof. We choose a Riemannian metric g1 on the vector bundle V E ’ E and

a Riemannian metric g2 on the manifold M . We can combine these two into the

Riemannian metric g := (T p| ker ¦)— g2 • g1 on the manifold E, for which the

horizontal and vertical spaces are orthogonal. By the theorem of [Nash, 1956], see

also [G¨nther, 1989] for an easy proof, there is an isometric embedding h : E ’ RN

u

for N large enough. We then embed RN into the Hilbert space 2 and consider

f : M ’ B(S, 2 ), given by f (x) = h(Ex ). Then

w E(S,

˜

f =(f,h) 2

E )

p

u u

w B(S,

f 2

M )

is ¬berwise a di¬eomorphism, so the diagram is a pullback and f — E(S, 2 ) = E.

Since T (f, h) maps horizontal and vertical vectors to orthogonal ones we have

(f, h)— ¦class = ¦. If Pt denotes the parallel transport of the connection ¦ and

c : [0, 1] ’ M is a (piecewise) smooth curve we have for u ∈ Ec(0)

˜ ˜‚

¦class f (Pt(c, t, u)) = ¦class .T f . ‚t 0 Pt(c, t, u)

‚

‚t 0

˜ ‚

= T f .¦. ‚t 0 Pt(c, t, u) = 0, so

˜ ˜

f (Pt(c, t, u)) = Ptclass (f —¦ c, t, f (u)).

Now let H be a continuous homotopy M — I ’ B(S, 2 ). Then we may approx-

imate H by smooth mappings with the same H0 and H1 , if they are smooth,

see [Br¨cker, J¨nich, 1973], where the in¬nite dimensionality of B(S, 2 ) does not

o a

disturb. Then we consider the bundle H — E(S, 2 ) ’ M — I, equipped with the

connection H — ¦class , whose curvature is H — Rclass . Let ‚t be the vector ¬eld tan-

gential to all {x} — I on M — I. Parallel transport along the lines t ’ (x, t) with

44.24

486 Chapter IX. Manifolds of mappings 44.27

respect H — ¦class is given by the ¬‚ow of the horizontal lift (H — C class )(‚t ) of ‚t . Let

us compute its action on the connection H — ¦class whose curvature is H — Rclass by

(37.4.3). By lemma (44.25) below we have

—

(H — C class )(‚t )

H — ¦class = ’ 1 i(H — C class )(‚t ) (H — Rclass )

‚

Flt

‚t 2

1

= ’ H — i(C class T(x,t) H.(0x , dt ))Rclass ,

d

2

which implies the result.

44.25. Lemma. Let ¦ be a connection on a ¬nite dimensional ¬ber bundle (p :

E ’ M, S) with curvature R and horizontal lift C. Let X ∈ X(M ) be a vector ¬eld

on the base.

Then for the horizontal lift CX ∈ X(E) we have

(FlCX )— ¦ = [CX, ¦] = ’ 1 iCX R.

‚

LCX ¦ = t

‚t 0 2

(FlCX )— ¦ = [CX, ¦]. From (35.9.2)

‚

Proof. From (35.14.5) we get LCX ¦ = t

‚t 0

we have

iCX R = iCX [¦, ¦]

= [iCX ¦, ¦] ’ [¦, iCX ¦] + 2i[¦,CX] ¦

= ’2¦[CX, ¦].

The vector ¬eld CX is p-related to X, and ¦ ∈ „¦1 (E; T E) is p-related to 0 ∈

„¦1 (M ; T M ), so by (35.13.7) the form [CX, ¦] ∈ „¦1 (E; T E) is also p-related to

0 = [X, 0] ∈ „¦1 (M ; T M ). So T p.[CX, ¦] = 0, [CX, ¦] has vertical values, and

[CX, ¦] = ¦[CX, ¦].

44.26. A consequence of theorem (43.7) is that the classifying spaces of Di¬(S)

and Di¬(S, µ0 ) are homotopy equivalent. So their classifying spaces are homotopy

equivalent, too.

We now sketch a smooth classifying space for Di¬ µ0 . Consider the space B1 (S, 2 )

of all submanifolds of 2 of type S and total volume 1 in the volume form induced

from the inner product on 2 . It is a closed splitting submanifold of codimension

1 of B(S, 2 ) by the Nash-Moser inverse function theorem (51.17). This theorem is

applicable if we use 2 as image space, because the modeling spaces are then tame

Fr´chet spaces in the sense of (51.9). It is not applicable directly for R(N) as image

e

space.

44.27. Theorem. Classifying space for Di¬ ω (S). Let S be a compact real

analytic manifold. Then the space Embω (S, 2 ) of real analytic embeddings of S

into the Hilbert space 2 is the total space of a real analytic principal ¬ber bundle

with structure group Di¬ ω (S) and real analytic base manifold B ω (S, 2 ), which is

44.27

45.1 45. Manifolds of Riemannian metrics 487

a classifying space for the Lie group Di¬ ω (S). It carries a universal Di¬ ω (S)-

connection.

In other words:

Embω (S, N ) —Di¬ ω (S) S ’ B ω (S, 2

)

classi¬es real analytic ¬ber bundles with typical ¬ber S and carries a universal

(generalized) connection.

The proof is similar to that of (44.24) with the appropriate changes to C ω .