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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 .

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