<<

. 21
( 27)



>>

1980b]. The clearest presentation was in [Michor, 1980c, section 13].
Proof. Let us ¬x an embedding i ∈ Emb(M, N ). Let g be a ¬xed Riemannian
metric on N , and let expN be its exponential mapping. Then let p : N (i) ’ M
be the normal bundle of i, de¬ned in the following way: For x ∈ M let N (i)x :=
(Tx i(Tx M ))⊥ ‚ Ti(x) N be the g-orthogonal complement in Ti(x) N . Then we have
an injective vector bundle homomorphism over i:
w
¯±
N (i) TN
πN
u u
p = pi

w N.
i
M
44.1
44.1 44. Principal bundles with structure group a di¬eomorphism group 475

Now let U i = U be an open neighborhood of the zero section of N (i) which is so
small that (expN —¦¯)|U : U ’ N is a di¬eomorphism onto its image which describes
±
a tubular neighborhood of the submanifold i(M ). Let us consider the mapping

„ = „ i := (expN —¦¯)|U : N (i) ⊃ U ’ N,
±

a di¬eomorphism onto its image, and the open set in Emb(M, N ) which will serve
us as a saturated chart,

U(i) := {j ∈ Emb(M, N ) : j(M ) ⊆ „ i (U i ), j ∼ i},

where j ∼ i means that j = i o¬ some compact set in M . Then by (41.10) the set
U(i) is an open neighborhood of i in Emb(M, N ). For each j ∈ U(i) we de¬ne

•i (j) : M ’ U i ⊆ N (i),
•i (j)(x) := („ i )’1 (j(x)).

Then •i = ((„ i )’1 )— : U(i) ’ C ∞ (M, N (i)) is a smooth mapping which is bijective
onto the open set

V(i) := {h ∈ C∞ (M, N (i)) : h(M ) ⊆ U i , h ∼ 0}

in C ∞ (M, N (i)). Its inverse is given by the smooth mapping „— : h ’ „ i —¦ h.
i

i i
We have „— (h—¦f ) = „— (h)—¦f for those f ∈ Di¬(M ) which are so near to the identity
that h —¦ f ∈ V(i). We consider now the open set

{h —¦ f : h ∈ V(i), f ∈ Di¬(M )} ⊆ C∞ (M, U i ).

Obviously, we have a smooth mapping from this set into Cc (M ← U i ) — Di¬(M )
given by h ’ (h —¦ (p —¦ h)’1 , p —¦ h), where Cc (M ← U i ) is the space of sections


with compact support of U i ’ M . So if we let Q(i) := „— (Cc (M ← U i ) © V(i)) ‚
i

Emb(M, N ) we have

W(i) := U(i) —¦ Di¬(M ) ∼ Q(i) — Di¬(M ) ∼ (Cc (M ← U i ) © V(i)) — Di¬(M ),
=∞
=

since the action of Di¬(M ) on i is free. Furthermore, the restriction π|Q(i) : Q(i) ’
Emb(M, N )/ Di¬(M ) is bijective onto an open set in the quotient.
We now consider •i —¦ (π|Q(i))’1 : π(Q(i)) ’ C ∞ (M ← U i ) as a chart for the
quotient space. In order to investigate the chart change, let j ∈ Emb(M, N ) be
such that π(Q(i)) © π(Q(j)) = …. Then there is an immersion h ∈ W(i) © Q(j)
and hence there exists a unique f0 ∈ Di¬(M ) (given by f0 = p —¦ •i (h)) such that
’1 ’1
h —¦ f0 ∈ Q(i). If we consider j —¦ f0 instead of j and call it again j, we have
Q(i) © Q(j) = …, and consequently U(i) © U(j) = …. Then the chart change is given
as follows:

•i —¦ (π|Q(i))’1 —¦ π —¦ („ j )— : Cc (M ← U j ) ’ Cc (M ← U i )
∞ ∞

s ’ „ j —¦ s ’ •i („ j —¦ s) —¦ (pi —¦ •i („ j —¦ s))’1 .

44.1
476 Chapter IX. Manifolds of mappings 44.3

This is of the form s ’ β —¦ s for a locally de¬ned di¬eomorphism β : N (j) ’ N (i)
which is not ¬ber respecting, followed by h ’ h —¦ (pi —¦ h)’1 . Both composants are
smooth by the general properties of manifolds of mappings. Therefore, the chart
change is smooth.
We show that the quotient space B(M, N ) = Emb(M, N )/ Di¬(M ) is Hausdor¬.
Let i, j ∈ Emb(M, N ) with π(i) = π(j). Then i(M ) = j(M ) in N for otherwise
put i(M ) = j(M ) =: L, a submanifold of N; the mapping i’1 —¦ j : M ’ L ’ M
is then a di¬eomorphism of M and j = i —¦ (i’1 —¦ j) ∈ i —¦ Di¬(M ), so π(i) = π(j),
contrary to the assumption.
Now we distinguish two cases.
Case 1. We may ¬nd a point y0 ∈ i(M ) \ j(M ), say, which is not a cluster point of
j(M ). We choose an open neighborhood V of y0 in N and an open neighborhood
W of j(M ) in N such that V © W = …. Let V := {k ∈ Emb(M, N ) : k(M ) ‚ V }
W := {k ∈ Emb(M, N ) : k(M ) ‚ W }. Then V is obviously open in Emb(M, N ),
and V is even open in the coarser compact-open topology. Both V and W are
Di¬(M ) saturated, i ∈ W, j ∈ V, and V © W = …. So π(V) and π(W) separate π(i)
and π(j) in B(M, N ).
Case 2. Let i(M ) ‚ j(M ) and j(M ) ‚ i(M ). Let y ∈ i(X), say. Let (V, v) be a
chart of N centered at y which maps i(M )©V into a linear subspace, v(i(M )©V ) ⊆
Rm © v(V ) ‚ Rn , where m = dim M , n = dim N . Since j(M ) ⊆ i(M ) we conclude
that we also have v((i(M ) ∪ j(M )) © V ) ⊆ Rm © v(V ). So we see that L :=
i(M ) ∪ j(M ) is a submanifold of N of the same dimension as N . Let (WL , pL , L)
be a tubular neighborhood of L. Then WL |i(M ) is a tubular neighborhood of i(M )
and WL |j(M ) is one of j(M ).

44.2. Result. [Cervera, Mascaro, Michor, 1991]. Let M and N be smooth mani-
folds. Then the di¬eomorphism group Di¬(M ) acts smoothly from the right on the
manifold Immprop (M, N ) of all smooth proper immersions M ’ N , which is an
open subset of C∞ (M, N ).
Then the space of orbits Immprop (M, N )/ Di¬(M ) is Hausdor¬ in the quotient
topology.
Let Immfree, prop (M, N ) be set of all proper immersions, on which Di¬(M ) acts
freely. Then this is open in C∞ (M, N ) and it is the total space of a smooth principal
¬ber bundle

Immfree,prop (M, N ) ’ Immfree,prop (M, N )/ Di¬(M ).


44.3. Theorem (Principal bundle of real analytic embeddings). [Kriegl,
Michor, 1990, section 6]. Let M and N be real analytic ¬nite dimensional manifolds,
connected and second countable without boundary such that dim M ¤ dim N , with
M compact. Then the set Embω (M, N ) of all real analytic embeddings M ’ N is an
open submanifold of C ω (M, N ). It is the total space of a real analytic principal ¬ber
bundle with structure group Di¬ ω (M ), whose real analytic base manifold B ω (M, N )
is the space of all real analytic submanifolds of N of type M .

44.3
44.5 44. Principal bundles with structure group a di¬eomorphism group 477

Proof. The proof of (44.1) is valid with the obvious changes. One starts with
a real analytic Riemannian metric and uses its exponential mapping. The space
of embeddings is open, since embeddings are open in C ∞ (M, N ), which induces a
coarser topology.

44.4. The nonlinear frame bundle of a ¬ber bundle. [Michor, 1988], [Michor,
1991]. Let now (p : E ’ M, S) be a ¬ber bundle, and let us ¬x a ¬ber bundle atlas
(U± ) with transition functions ψ±β : U±β — S ’ S. By (42.14) we have

C ∞ (U±β , C∞ (S, S)) ⊆ C ∞ (U±β — S, S)

with equality if and only if S is compact. Let us therefore assume from now on
that S is compact. Then we assume that the transition functions ψ±β : U±β ’
Di¬(S, S).
Now we de¬ne the nonlinear frame bundle of (p : E ’ M, S) as follows. We consider
the set Di¬{S, E} := x∈M Di¬(S, Ex ) and equip it with the in¬nite dimensional
di¬erentiable structure which one gets by applying the functor Di¬(S, ) to the
cocycle of transition functions (ψ±β ). Then the resulting cocycle of transition func-
tions for Di¬{S, E} induces the structure of a smooth principal bundle over M with
structure group Di¬(M ). The principal action is just composition from the right.
We can now consider the smooth action ev : Di¬(S) — S ’ S and the associated
bundle Di¬{S, E}[S, ev] = Di¬{S,E}—S . The mapping ev : Di¬{S, E} — S ’ E
Di¬(S)
is invariant under the Di¬(S)-action and factors therefore to a smooth mapping
Di¬{S, E}[S, ev] ’ E as in the following diagram:

w Di¬{S, E} — S
pr
Di¬{S, E} — S
Di¬(S)
ev
u
u
E Di¬{S, E}[S, ev].

The bottom mapping is easily seen to be a di¬eomorphism. Thus, the bundle
Di¬{S, E} may in full right be called the (nonlinear) frame bundle of E.

44.5. Let now ¦ ∈ „¦1 (E; T E) be a connection on E, see (37.2). We want to lift
¦ to a principal connection on Di¬{S, E}, and for this we need a good description
of the tangent space T Di¬{S, E}. With the method of (42.17) one can easily show
that

{f ∈ C ∞ (S, T E|Ex ) : T p —¦ f = one point
T Di¬{S, E} =
x∈M
in Tx M and πE —¦ f ∈ Di¬(S, Ex )}.

Starting from the connection ¦ we can then consider ω(f ) := T (πE —¦ f )’1 —¦ ¦ —¦ f :
S ’ T E ’ V E ’ T S for f ∈ T Di¬{S, E}. Then ω(f ) is a vector ¬eld on S, and
we have:

44.5
478 Chapter IX. Manifolds of mappings 44.10

Lemma. ω ∈ „¦1 (Di¬{S, E}; X(S)) is a principal connection, and the induced con-
nection on E = Di¬{S, E}[S, ev] coincides with ¦.

Proof. The fundamental vector ¬eld ζX on Di¬{S, E} for X ∈ X(S) is given by
ζX (g) = T g —¦ X. Then ω(ζX (g)) = T g ’1 —¦ ¦ —¦ T g —¦ X = X since T g —¦ X has vertical
values. Hence, ω reproduces fundamental vector ¬elds.
Now let h ∈ Di¬(S), and denote by rh the principal right action. Then we have

((rh )— ω)(f ) = ω(T (rh )f ) = ω(f —¦ h) = T (πE —¦ f —¦ h)’1 —¦ ¦ —¦ f —¦ h
= T h’1 —¦ ω(f ) —¦ h = AdDi¬(S) (h’1 )ω(f ).

44.6. Theorem. Let (p : E ’ M, S) be a ¬ber bundle with compact standard ¬ber
S. Then connections on E and principal connections on Di¬{S, E} correspond to
each other bijectively, and their curvatures are related as in (37.24). Each principal
connection on Di¬{S, E} admits a global parallel transport. The holonomy groups
and the restricted holonomy groups are equal as subgroups of Di¬(S).

Proof. This follows directly from (37.24) and (37.25). Each connection on E is
complete since S is compact, and the lift to Di¬{S, E} of its parallel transport is
the global parallel transport of the lift of the connection, so the two last assertions
follow.

44.7. Remark on the holonomy Lie algebra. Let M be connected, let ρ =
’dω ’ 1 [ω, ω]X(S) be the usual X(S)-valued curvature of the lifted connection ω on
2
Di¬{S, E}. Then we consider the R-linear span of all elements ρ(ξf , ·f ) in X(S),
where ξf , ·f ∈ Tf Di¬{S, E} are arbitrary (horizontal) tangent vectors, and we call
this span hol(ω). Then by the Di¬(S)-equivariance of ρ the vector space hol(ω) is
an ideal in the Lie algebra X(S).

44.8. Lemma. Let f : S ’ Ex0 be a di¬eomorphism in Di¬{S, E}x0 . Then
f— : X(S) ’ X(Ex0 ) induces an isomorphism between hol(ω) and the R-linear span
of all g — R(CX, CY ), X, Y ∈ Tx M , and g : Ex0 ’ Ex any di¬eomorphism.

The proof is obvious.

44.9. Gauge theory for ¬ber bundles. We consider the bundle Di¬{E, E} :=
x∈M Di¬(Ex , Ex ) which bears the smooth structure described by the cocycle of
’1
transition functions Di¬(ψ±β , ψ±β ) = (ψ±β )— (ψβ± )— , where (ψ±β ) is a cocycle of
transition functions for the ¬ber bundle (p : E ’ M, S).

44.10. Lemma. The associated bundle Di¬{S, E}[Di¬(S), conj] is isomorphic to
the ¬ber bundle Di¬{E, E}.

Proof. The mapping A : Di¬{S, E} — Di¬(S) ’ Di¬{E, E}, given by A(f, g) :=
f —¦g—¦f ’1 : Ex ’ S ’ S ’ Ex for f ∈ Di¬(S, Ex ), is Di¬(S)-invariant, so it factors
to a smooth mapping Di¬{S, E}[Di¬(S)] ’ Di¬{E, E}. It is bijective and admits
locally over M smooth inverses, so it is a ¬ber respecting di¬eomorphism.


44.10
44.16 44. Principal bundles with structure group a di¬eomorphism group 479

44.11. The gauge group Gau(E) of the ¬nite dimensional ¬ber bundle (p : E ’
M, S) with compact standard ¬ber S is, by de¬nition, the group of all principal
bundle automorphisms of the Di¬(S)-bundle (Di¬{S, E} which cover the identity
of M . The usual reasoning (37.17) gives that Gau(E) equals the space of all smooth
sections of the associated bundle Di¬{S, E}[Di¬(S), conj] which by (44.10) equals
the space of sections of the bundle Di¬{E, E} ’ M . We equip it with the topology
and di¬erentiable structure described in (42.21).

44.12. Theorem. The gauge group Gau(E) = C∞ (M ← Di¬{E, E}) is a regular
Lie group. Its exponential mapping is not surjective on any neighborhood of the
identity. Its Lie algebra consists of all vertical vector ¬elds with compact support
on E (or M ) with the negative of the usual Lie bracket. The obvious embedding
Gau(E) ’ Di¬(E) is a smooth homomorphism of regular Lie groups.

Proof. The ¬rst statement has already been shown before the theorem. A curve
through the identity of principal bundle automorphisms of Di¬{S, E} ’ M is a
smooth curve through the identity in Di¬(E) consisting of ¬ber respecting map-
pings. The derivative of such a curve is thus an arbitrary vertical vector ¬eld with
compact support. The space of all these is therefore the Lie algebra of the gauge
group, with the negative of the usual Lie bracket.
The exponential mapping is given by the ¬‚ow operator of such vector ¬elds. Since
on each ¬ber it is just conjugate to the exponential mapping of Di¬(S), it has all
the properties of the latter. Gau(E) ’ Di¬(E) is a smooth homomorphism since
by (40.3) its prolongation to the universal cover of Gau(E) is smooth.

44.13. Remark. If S is not compact we may circumvent the nonlinear frame
bundle, and we may de¬ne the gauge group Gau(E) directly as the splitting closed
subgroup of Di¬(E) which consists of all ¬ber respecting di¬eomorphisms which
cover the identity of M . The Lie algebra of Gau(E) consists then of all vertical
vector ¬elds on E with compact support on E. We do not work out the details of
this approach.

44.14. The space of connections. Let J 1 (E) ’ E be the a¬ne bundle of
1-jets of sections of E ’ M . We have J 1 (E) = { ∈ L(Tx M, Tu E) : T p —¦ =
IdTx M , u ∈ E, p(u) = x}. Then a section of J 1 (E) ’ E is just a horizontal lift
mapping T M —M E ’ T E which is ¬ber linear over E, so it describes a connection
as treated in (37.2), and we may view the space of sections C ∞ (E ← J 1 (E)) as the
space of all connections.

44.15. Theorem. The action of the gauge group Gau(E) on the space of connec-
tions C ∞ (E ← J 1 (E)) is smooth.

Proof. This follows from (42.13)

44.16. We will now give a di¬erent description of the action. We view a connection
¦ again as a linear ¬ber wise projection T E ’ V E, so the space of connections

44.16
480 Chapter IX. Manifolds of mappings 44.18

is now Conn(E) := {¦ ∈ „¦1 (E; T E) : ¦ —¦ ¦ = ¦, ¦(T E) = V E}. Since S
is compact the canonical isomorphism Conn(E) ’ C ∞ (E ← J 1 (E)) is even a
di¬eomorphism. Then the action of f ∈ Gau(E) ‚ Di¬(E) on ¦ ∈ Conn(E) is
given by f— ¦ = (f ’1 )— ¦ = T f —¦ ¦ —¦ T f ’1 . Now it is very easy to describe the
in¬nitesimal action. Let X be a vertical vector ¬eld with compact support on E
and consider its global ¬‚ow FlX .
t
Then we have dt |0 (FlX )— ¦ = LX ¦ = [X, ¦], the Fr¨licher Nijenhuis bracket, by
d
o
t
(35.14.5). The tangent space of Conn(E) at ¦ is the space T¦ Conn(E) = {Ψ ∈
„¦1 (E; T E) : Ψ|V E = 0}. The ”in¬nitesimal orbit” at ¦ in T¦ Conn(E) is {[X, ¦] :

X ∈ Cc (E ← V E)}.
The isotropy subgroup of a connection ¦ is {f ∈ Gau(E) : f — ¦ = ¦}. Clearly, this
is just the group of all those f which respect the horizontal bundle HE = ker ¦.
The most interesting object is of course the orbit space Conn(E)/ Gau(E).

44.17. Slices. [Palais, Terng, 1988] Let M be a smooth manifold, G a Lie group,
G — M ’ M a smooth action, x ∈ M, and let Gx = {g ∈ G : g.x = x} denote
the isotropy group at x. A contractible subset S ⊆ M is called a slice at x, if it
contains x and satis¬es
(1) If g ∈ Gx then g.S = S.
(2) If g ∈ G with g.S © S = … then g ∈ Gx .
(3) There exists a local continuous section χ : G/Gx ’ G de¬ned on a neigh-
borhood V of the identity coset such that the mapping F : V — S ’ M,
de¬ned by F (v, s) := χ(v).s is a homeomorphism onto a neighborhood of x.
This is a local version of the usual de¬nition in ¬nite dimensions, which is too
narrow for the in¬nite dimensional situation. However, in ¬nite dimensions the
de¬nition above is equivalent to the usual one where a subset S ⊆ M is called a
slice at x, if there is a G-invariant open neighborhood U of the orbit G.x and a
smooth equivariant retraction r : U ’ G.x such that S = r’1 (x). In the general
case we have the following properties:
(4) For y ∈ F (V — S) © S we get Gy ‚ Gx , by (2).
(5) For y ∈ F (V — S) the isotropy group Gy is conjugate to a subgroup of Gx ,
by (3) and (4).
44.18. Counter-example. [Cerf, 1970], [Michor, Schichl, 1997]. The right action
of Di¬(S 1 ) on C ∞ (S 1 , R) does not admit slices.

Let h(t) : S 1 = (R mod 1) ’ R be a smooth bump function with h(t) = 0 for
t ∈ [0, 1 ] and h(t) > 0 for t ∈ (0, 1 ). Then put hn (t) = 41 h(4n (t ’ (1 ’ 41 )/3))
/ n n
4 4
which is is nonzero in the interval (1 ’ 41 )/3, (1 ’ 4n+1 )/3 , and consider
1
n



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 .


45. Manifolds of Riemannian Metrics

The usual metric on the space of all Riemannian metrics was considered by [Ebin,
1970], who used it to construct a slice for the action of the group of di¬eomorphism
on the space of all metrics. It was then reconsidered by [Freed, Groisser, 1989],
and by [Gil-Medrano, Michor, 1991] for noncompact M . The results in this section
are largely taken from the last paper and from [Gil-Medrano, Michor, Neuwirther,
1992].

45.1. Bilinear structures. Throughout this section let M be a smooth second
countable ¬nite dimensional manifold. Let —2 T — M denote the vector bundle of all
0 —
2 -tensors on M , which we canonically identify with the bundle L(T M, T M ). Let
GL(T M, T — M ) denote the non degenerate ones. For any b : Tx M ’ Tx M we let

b—
——
’—
the transposed be given by bt : Tx M ’ Tx M ’ Tx M . As a bilinear structure b
is skew symmetric if and only if bt = ’b, and b is symmetric if and only if bt = b.
In the latter case a frame (ej ) of Tx M can be chosen in such a way that in the dual

frame (ej ) of Tx M we have

b = e1 — e1 + · · · + ep — ep ’ ep+1 — ep+1 ’ ep+q — ep+q ;

b has signature (p, q) and is non degenerate if and only if p + q = n, the dimension
of M . In this case, q alone will be called the signature.
A section b ∈ C ∞ (GL(T M, T — M )) will be called a non degenerate bilinear struc-
ture on M , and we will denote the space of all such structures by B(M ) = B :=
C ∞ (GL(T M, T — M )). It is open in the space of sections C ∞ (L(T M, T — M )) for the
Whitney C ∞ -topology, in which the latter space is, however, not a topological vec-
tor space, as explained in detail in (41.13). The space Bc := Cc (L(T M, T — M )) of


sections with compact support is the largest topological vector space contained in
the topological group (C ∞ (L(T M, T — M )), +), and the trace of the Whitney C ∞ -
topology on it induces the convenient vector space structure described in (30.4).
So we declare the path components of B = C ∞ (GL(T M, T — M )) for the Whitney
C ∞ -topology also to be open; these are open in a¬ne subspaces of the form b + Bc
for some b ∈ B and we equip them with the translates of the c∞ -topology on Bc .
The resulting topology is ¬ner than the Whitney topology and will be called the
natural topology, similar as in (42.1).

45.1
488 Chapter IX. Manifolds of mappings 45.3

45.2. The metrics. The tangent bundle of the space B = C ∞ (GL(T M, T — M ))
of bilinear structures is T B = B — Bc = C ∞ (GL(T M, T — M )) — Cc (L(T M, T — M )).


Then b ∈ B induces two ¬berwise bilinear forms on L(T M, T — M ) which are given by
(h, k) ’ tr(b’1 hb’1 k) and (h, k) ’ tr(b’1 h) tr(b’1 k). We split each endomorphism
tr(H)
H = b’1 h : T M ’ T M into its trace free part H0 := H ’ dim M Id and its
trace part which simpli¬es some formulas later on. Thus, we have tr(b’1 hb’1 k) =
tr((b’1 h)0 (b’1 k)0 )+ dim M tr(b’1 h) tr(b’1 k). The structure b also induces a volume
1

density on the base manifold M by the local formula

| det(bij )| |dx1 § · · · § dxn |.
vol(b) =

For each real ± we have a smooth symmetric bilinear form on B, given by

(tr((b’1 h)0 (b’1 k)0 ) + ± tr(b’1 h) tr(b’1 k)) vol(b).
G± (h, k) =
b
M

It is invariant under the action of the di¬eomorphism group Di¬(M ) on the space B
of bilinear structures. The integral is de¬ned since h and k have compact supports.
For n = dim M we have
1/n
tr(b’1 hb’1 k) vol(b),
Gb (h, k) := Gb (h, k) =
M

which for positive de¬nite b is the usual metric on the space of all Riemannian
metrics. We will see below in (45.3) that for ± = 0 it is weakly non degenerate,
i.e. G± de¬nes a linear injective mapping from the tangent space Tb B = Bc =
b
Cc (L(T M, T — M )) into its dual Cc (L(T M, T — M )) , the space of distributional
∞ ∞

densities with values in the dual bundle. This linear mapping is, however, never
surjective. So we have a one parameter family of pseudo Riemannian metrics on the
in¬nite dimensional space B which obviously is smooth in all appearing variables.

45.3. Lemma. For h, k ∈ Tb B we have

tr(b’1 h)b, k),
G± (h, k) = Gb (h + ±n’1
b n
tr(b’1 h)b, k), if ± = 0,
Gb (h, k) = G± (h ’ ±n’1
b ±n2

where n = dim M . The pseudo Riemannian metric G± is weakly non degenerate
for all ± = 0.

Proof. The ¬rst equation is an obvious reformulation of the de¬nition, the sec-
ond follows since h ’ h ’ ±n’1 tr(b’1 h)b is the inverse of the transform h ’
±n2
h + n tr(b h)b. Since tr(b’1 hx (b’1 hx )t,g ) > 0 if hx = 0, where t,g is the
’1
±n’1
x x
transposed of a linear mapping with respect to an arbitrary ¬xed Riemannian met-
ric g, we have

Gb (h, b(b’1 h)t,g ) = tr(b’1 h(b’1 h)t,g ) vol(b) > 0
M

if h = 0. So G is weakly non degenerate, and by the second equation G± is weakly
non degenerate for ± = 0.


45.3
45.6 45. Manifolds of Riemannian metrics 489

45.4. Remark. Since G± is only a weak pseudo Riemannian metric, all objects
which are only implicitly given a priori lie in the Sobolev completions of the relevant
spaces. In particular, this applies to the formula

2G± (ξ, ±
=ξG± (·, ζ) + ·G± (ζ, ξ) ’ ζG± (ξ, ·)
· ζ)
+ G± ([ξ, ·], ζ) + G± ([·, ζ], ξ) ’ G± ([ζ, ξ], ·),

which a priori gives only uniqueness but not existence of the Levi Civita covariant
derivative. We will show that it exists and we use it in the form explained in (37.28).

45.5. Lemma. For x ∈ M the pseudo metric on GL(Tx M, Tx M ) given by

γbx (hx , kx ) := tr((b’1 hx )0 (b’1 kx )0 ) + ± tr(b’1 hx ) tr(b’1 kx )
±
x x x x

n(n’1)
has signature (the number of negative eigenvalues) for ± > 0 and has sig-
2
nature ( n(n’1) + 1) for ± < 0.
2

Proof. In the framing H = b’1 hx and K = b’1 kx we have to determine the
x x
signature of the symmetric bilinear form (H, K) ’ tr(H0 K0 )+± tr(H) tr(K). Since
the signature is constant on connected components we have to determine it only
1 1
for ± = n and ± = n ’ 1.
1
For ± = n we note ¬rst that on the space of matrices (H, K) ’ tr(HK t ) is positive
de¬nite, and since the linear isomorphism K ’ K t has the space of symmetric
matrices as eigenspace for the eigenvalue 1 and the space of skew symmetric matrices
as eigenspace for the eigenvalue ’1, we conclude that the signature is n(n’1) in
2
this case.
1
For ± = n ’ 1 we proceed as follows: On the space of matrices with zeros on
the main diagonal the signature of (H, K) ’ tr(HK) is n(n’1) by the argument
2
above and the form (H, K) ’ ’ tr(H) tr(K) vanishes. On the space of diagonal
matrices which we identify with Rn the whole bilinear form is given by x, y =
ii i i n
ix y ’( i x )( i y ). Let (ei ) denote the standard basis of R , and put a1 :=
1
n (e1 + · · · + en ) and

1
(e1 + · · · + ei’1 ’ (i ’ 1)ei )
ai :=
1)2
i ’ 1 + (i ’
1
for i > 1. Then a1 , a1 = ’1 + n, and for i > 1 we get ai , aj = δi,j . So the
signature is 1 in this case.

45.6. Let t ’ b(t) be a smooth curve in B. So b : R — M ’ GL(T M, T — M ) is
smooth, and by the choice of the topology on B made in (45.1) the curve b(t) varies
only in a compact subset of M , locally in t, by (30.9). Then its energy is given by
a2
a2
G± (bt , bt )dt
1
Ea1 (b) := b
2
a1
a2
tr((b’1 bt )0 (b’1 bt )0 ) + ± tr(b’1 bt )2 vol(b) dt,
1
= 2
a1 M

45.6
490 Chapter IX. Manifolds of mappings 45.8


where bt = ‚t b(t).
Now we consider a variation of this curve, so we assume that (t, s) ’ b(t, s) is
smooth in all variables and locally in (t, s) it only varies within a compact subset
in M ” this is again the e¬ect of the topology chosen in (45.1). Note that b(t, 0)
is the original b(t) above.

45.7. Lemma. In the setting of (45.6), we have the ¬rst variation formula

, s)) = G± (bt , bs )|t=a1 +
± a1

‚s |0 E(G )a0 (b( t=a0
b
a1
1 1
G(’btt + bt b’1 bt +tr(b’1 bt b’1 bt )b ’ tr(b’1 bt )bt +
+
4 2
a0
1
+ ± (’ tr(b’1 btt ) ’ tr(b’1 bt )2 + tr(b’1 bt b’1 bt ))b, bs ) dt =
4
= G± (bt , bs )|t=a1 +
t=a0
b
a1
1 1
tr(b’1 b’1 bt )b+
G± (’btt + bt b’1 bt ’ tr(b’1 bt )bt +
+ t
2 4±n
a0
±n ’ 1
tr(b’1 bt )2 b, bs ) dt.
+ 2
4±n


Proof. We may interchange ‚s |0 with the ¬rst integral describing the energy in
(45.6) since this is ¬nite dimensional analysis, and we may interchange it with the
second one, since M is a continuous linear functional on the space of all smooth
densities with compact support on M , by the chain rule. Then we use that tr— is
linear and continuous, d(vol)(b)h = 2 tr(b’1 h) vol(b), and that d(( )’1 )— (b)h =
1

’b’1 hb’1 , and partial integration.

45.8. The geodesic equation. By lemma (45.7), the curve t ’ b(t) is a geodesic
if and only if we have
±n ’ 1
1 1
btt = bt b’1 bt ’ tr(b’1 bt )bt + tr(b’1 bt b’1 bt )b + tr(b’1 bt )2 b.
4±n2
2 4±n
= “b (bt , bt ),

where the G± -Christo¬el symbol “± : B — Bc — Bc ’ Bc is given by symmetrization
1 ’1 1 1 1
hb k + kb’1 h ’ tr(b’1 h)k ’ tr(b’1 k)h+
“± (h, k) =
b
2 2 4 4
±n ’ 1
1
tr(b’1 hb’1 k)b + tr(b’1 h) tr(b’1 k)b.
+
4±n2
4±n
The sign of “± is chosen in such a way that the horizontal subspace of T 2 B is
parameterized by (x, y; z, “x (y, z)). If instead of the obvious framing we use T B =
(b, h) ’ (b, b’1 h) =: (b, H) ∈ {b} — Cc (L(T M, T M )), the Christo¬el

B — Bc
symbol looks like
1 1 1
±
(HK + KH) ’ tr(H)K ’ tr(K)H
“b (H, K) =
2 4 4
±n ’ 1
1
+ tr(HK) Id + tr(H) tr(K),
4±n2
4±n
45.8
45.11 45. Manifolds of Riemannian metrics 491

and the G± -geodesic equation for B(t) := b’1 bt becomes
±n ’ 1
1 1
’1
tr(B)2 Id .

tr(BB) Id ’ tr(B)B +
Bt = ‚t (b bt ) =
4±n2
4±n 2

45.9. The curvature. For another manifold N , for vector ¬elds X, Y ∈ X(N )
and a vector ¬eld s : N ’ T M along f : N ’ M we have

])s = (K —¦ T K ’ K —¦ T K —¦ κT M ) —¦ T 2 s —¦ T X —¦ Y,
R(X, Y )s = ( ’[ X, Y
[X,Y ]

where K : T T M ’ M is the connector (37.28), κT M is the canonical ¬‚ip T T T M ’
T T T M (29.10), and where the second formula in local coordinates reduces to the
usual formula

R(h, k) = d“(h)(k, ) ’ d“(k)(h, ) ’ “(h, “(k, )) + “(k, “(h, )),
(1)

see [Kainz, Michor, 1987] or [Kol´ˇ, Michor, Slovak, 1993, 37.15].
ar

45.10. Theorem. The curvature for the pseudo Riemannian metric G± on the
manifold B of all non degenerate bilinear structures is given by
1 1
b’1 R± (h, k)l = [[H, K], L] + (’ tr(HL)K + tr(KL)H)+
b
4 16±
4±n ’ 3±n2 + 4n ’ 4
(tr(H) tr(L)K ’ tr(K) tr(L)H)+
+
16±n2
4±2 n2 ’ 4±n + ±n2 + 3
(tr(HL) tr(K) Id ’ tr(KL) tr(H) Id),
+
16±n2
where H = b’1 h, K = b’1 k and L = b’1 l.

Proof. This is a long but direct computation starting from (45.9.1).

The geodesic equation can be solved explicitly, and we have

45.11. Theorem. Let b0 ∈ B and h ∈ Tb0 B = Bc . Then the geodesic for the
metric G± in B starting at b0 in the direction of h is the curve

exp±0 (th) = b0 e(a(t) Id +b(t)H0 ) ,
b

where H0 is the traceless part of H := (b0 )’1 h (i.e. H0 = H ’ tr(H) Id), and where
n

a(t) = a±,H (t) and b(t) = b±,H (t) in C (M, R) are de¬ned as follows:
’1
2 t 2±
2 2
a±,H (t) = log (1 + tr(H)) + t tr(H0 ) ,
n 4 16
±
2
t ±’1 tr(H0 )
4
for ±’1 tr(H0 ) > 0
2

 ±’1 tr(H 2 ) arctan


4 + t tr(H)

0




 2
t ’±’1 tr(H0 )
4
b±,H (t) = for ±’1 tr(H0 ) < 0
2
artanh
4 + t tr(H)
2
 ’±’1 tr(H0 )




t

 2
for tr(H0 ) = 0.


t
1 + 4 tr(H)


45.11
492 Chapter IX. Manifolds of mappings 45.11

Here arctan is taken to have values in (’ π , π ) for the points of the base manifold,
22
where tr(H) ≥ 0, and on a point where tr(H) < 0 we de¬ne

 arctan in [0, π ) for t ∈ [0, ’ tr(H) )
4
±
2
2
t ±’1 tr(H0 )


π 4
for t = ’ tr(H)
arctan = 2
4 + t tr(H) 
 arctan in ( π , π) for t ∈ (’ 4 , ∞).

2 tr(H)

To describe the domain of de¬nition of the exponential mapping we consider the
sets

Z h := {x ∈ M : 2
1
trx (H0 ) = 0 and trx (H) < 0},
±
Gh := {x ∈ M : 0 > trx (H0 ) > ’ trx (H)2 and trx (H) < 0}
2
1
±
γ ± (h, h)
= {x ∈ M : ±γ(h, h) 0 for ± 0, trx (H) < 0},
E h := {x ∈ M : ’ trx (H)2 = 2
1
trx (H0 ) and trx (H) < 0}
±
= {x ∈ M : γ ± (h, h) = 0 and trx (H) < 0},
Lh := {x ∈ M : ’ trx (H)2 > 2
1
trx (H0 )}
±
= {x ∈ M : γ ± (h, h) 0 for ± 0},

where γ(h, h) = trx (H 2 ), and γ ± (h, h) = trx (H0 ) + ± trx (H)2 , see (45.5), are the
2

integrands of Gb0 (h, h) and G±0 (h, h), respectively. Then we consider the numbers
b

4
z h := inf ’ : x ∈ Zh ,
trx (H)
2
’± trx (H) ’ ’± trx (H0 )
h
: x ∈ Gh
g := inf 4 ,
2 ) + ± tr(H)2
trx (H0
2
eh := inf ’ : x ∈ Eh ,
trx (H)
2
’± trx (H) ’ ’± trx (H0 )
h
: x ∈ Lh
l := inf 4 ,
2 ) + ± tr(H)2
trx (H0

if the corresponding set is not empty, with value ∞ if the set is empty. Denote
mh := inf{z h , g h , eh , lh }. Then exp±0 (th) is maximally de¬ned for t ∈ [0, mh ).
b

The second representations of the sets Gh , Lh , and E h clari¬es how to take care of
timelike, spacelike, and lightlike vectors, respectively.

Proof. Using X(t) := g ’1 gt the geodesic equation reads as

±n ’ 1
1 1
tr(X 2 ) Id + tr(X)2 Id,
X = ’ tr(X)X + 2
2 4±n 4±n
and it is easy to see that a solution X satis¬es
1
X0 = ’ tr(X)X0 .
2
45.11
45.12 45. Manifolds of Riemannian metrics 493

Then X(t) is in the plane generated by H0 and Id for all t and the solution has the
form g(t) = b0 exp(a(t) Id +b(t)H0 ). Since gt = g(t)(a (t) Id +b (t)H0 ) we have

X(t) = a (t) Id +b (t)H0 and
X (t) = a (t) Id +b (t)H0 ,

and the geodesic equation becomes

1
a (t) Id +b (t)H0 = ’ na (t)(a (t) Id +b (t)H0 )+
2
1
(na (t)2 + b (t)2 tr(H0 )) Id +
2
+
4±n
±n ’ 1 2
(n a (t)2 ) Id .
+ 2
4±n

We may assume that Id and H0 are linearly independent; if not H0 = 0 and b(t) = 0.
Hence, the geodesic equation reduces to the di¬erential equation

2
±
n tr(H0 )
2
(b )2
 a = ’ (a ) +

4 4±n
 b = ’na b

2
tr(H)
with initial conditions a(0) = b(0) = 0, a (0) = n, and b (0) = 1.
If we take p(t) = exp( n a) it is easy to see that then p should be a solution of p =0
2
and from the initial conditions

t2
t
p(t) = 1 + tr(H) + (tr(H)2 + ±’1 tr(H0 )).
2
2 16

Using that the second equation becomes b = p’1 , and then b is obtained just
by computing the integral. The solutions are de¬ned in [0, mh ) where mh is the
in¬mum over the support of h of the ¬rst positive root of the polynomial p, if it
exists, and ∞ otherwise. The description of mh is now a technical fact.

45.12. The exponential mapping. For b0 ∈ GL(Tx M, Tx M ) and H = (b0 )’1 h


let Cb0 be the subset of L(Tx M, Tx M ) given by the union of the sets (compare with
Z h , Gh , E h , Lh from (45.11))

2
{h : tr(H0 ) = 0, tr(H) ¤ ’4},
2
’± tr(H) ’ ’± tr(H0 )
2 2
1
> ’ tr(H) , 4 ¤ 1, tr(H) < 0 ,
h:0> tr(H0 ) 2
± tr(H0 ) + ± tr(H)2
h : ’ tr(H)2 = 2
1
tr(H0 ), tr(H) < ’2 ,
±
closure
2
’± tr(H) ’ ’± tr(H0 )
h : ’ tr(H)2 > 2
1
¤1
tr(H0 ), 4 ,
2
± ± tr(H)2
tr(H0 ) +

45.12
494 Chapter IX. Manifolds of mappings 45.12

which by some limit considerations coincides with the union of the following two
sets:
closure
2
’± tr(H) ’ ’± tr(H0 )
tr(H0 ) > ’ tr(H)2 , 4
2
1
¤ 1, tr(H) < 0
h:0> ,
2
± ± tr(H)2
tr(H0 ) +
closure
2
’± tr(H) ’ ’± tr(H0 )
h : ’ tr(H)2 > 2
1
¤1
tr(H0 ), 4 .
2
± ± tr(H)2
tr(H0 ) +

So Cb0 is closed. We consider the open sets Ub0 := L(Tx M, Tx M ) \ Cb0 , Ub0 :=
{(b0 )’1 h : h ∈ Ub0 } ‚ L(Tx M, Tx M ), and ¬nally the open sub ¬ber bundles over
GL(T M, T — M )

{b0 } — Ub0 : b0 ∈ GL(T M, T — M ) ‚ GL(T M, T — M ) —M L(T M, T — M ),
U :=

{b0 } — Ub0 : b0 ∈ GL(T M, T — M ) ‚ GL(T M, T — M ) —M L(T M, T M ).
U :=

Then we consider the mapping ¦ : U ’ GL(T M, T — M ) which is given by the
following composition
Id —M exp

U ’ U ’ GL(T M, T — M ) —M L(T M, T M ) ’ ’ ’ ’
’ ’ ’’’
Id —M exp
’ ’ ’ ’ GL(T M, T — M ) —M GL(T M, T M ) ’ GL(T M, T — M ),
’’’ ’
where (b0 , h) := (b0 , (b0 )’1 h) is a ¬ber respecting di¬eomorphism, (b0 , H) := b0 H
is a di¬eomorphism for ¬xed b0 , and where the other two mappings will be discussed
below.
The usual ¬berwise exponential mapping
exp : L(T M, T M ) ’ GL(T M, T M )
is a di¬eomorphism near the zero section on the ball of radius π centered at zero
in a norm on the Lie algebra for which the Lie bracket is sub multiplicative, for
example. If we ¬x a symmetric positive de¬nite inner product g, then exp restricts
to a global di¬eomorphism from the linear subspace of g-symmetric endomorphisms
onto the open subset of matrices which are positive de¬nite with respect to g. If g
has signature this is no longer true since then g-symmetric matrices may have non
real eigenvalues.
On the open set of all matrices whose eigenvalues » satisfy | Im »| < π, the expo-
nential mapping is a di¬eomorphism, see [Varadarajan, 1977].
The smooth mapping • : U ’ GL(T M, T — M ) —M L(T M, T M ) is given by
•(b0 , H) := (b0 , a±,H (1) Id +b±,H (1)H0 ) (see theorem (45.11)). It is a di¬eomor-
phism onto its image with the following inverse:
√ ’1
±
2
± tr(H0 )
tr(H)
 4 e 4 cos ’ 1 Id +
n 4



√ ’1


2
± tr(H0 )
tr(H)
ψ(H) := 4
+ √ ’1 2
e 4 sin H0 if tr(H0 ) = 0
4
2
± tr(H0 )




tr(H)

4
e 4 ’ 1 Id otherwise,

n


45.12
45.15 45. Manifolds of Riemannian metrics 495

where cos is considered as a complex function, cos(iz) = i cosh(z).
The mapping (pr1 , ¦) : U ’ GL(T M, T — M ) —M GL(T M, T — M ) is a di¬eomor-
phism on an open neighborhood of the zero section in U .

45.13. Theorem. In the setting of (45.12) the exponential mapping exp±0 for the
b
±
metric G is a real analytic mapping de¬ned on the open subset

Ub0 := {h ∈ Cc (L(T M, T — M )) : (b0 , h)(M ) ‚ U },




and it is given by
expb0 (h) = ¦ —¦ (b0 , h).

The mapping (πB , exp) : T B ’ B—B is a real analytic di¬eomorphism from an open
neighborhood of the zero section in T B onto an open neighborhood of the diagonal
in B — B. Ub0 is the maximal domain of de¬nition for the exponential mapping.

Proof. Since B is a disjoint union of chart neighborhoods, it is trivially a real
analytic manifold, even if M is not supposed to carry a real analytic structure.
From the consideration in (45.12) it follows that exp = ¦— and (πM , exp) are just
push forwards by real analytic ¬ber respecting mappings of sections of bundles. So
by (30.10) they are smooth, and this applies also to their inverses.
To show that these mappings are real analytic, by (10.3) it remains to check that
they map real analytic curves into real analytic curves. These are described in
(30.15). It is clear that ¦ has a ¬berwise extension to a holomorphic germ since ¦
is ¬ber respecting from an open subset in a vector bundle and is ¬berwise a real
analytic mapping. So the push forward ¦— preserves the description (30.15) and
maps real analytic curves to real analytic curves.

45.14. Submanifolds of pseudo Riemannian metrics. We denote by Mq
the space of all pseudo Riemannian metrics on the manifold M of signature (the
dimension of a maximal negative de¬nite subspace) q. It is an open set in a closed
locally a¬ne subspace of B and thus a splitting submanifold of it with tangent
bundle T Mq = Mq — Cc (M ← S 2 T — M ).



We consider a geodesic c(t) = c0 e(a(t) Id +b(t)H0 ) for the metric G± in B starting
at c0 in the direction of h as in (45.11). If c0 ∈ Mq then h ∈ Tc0 Mq if and
only if H = (c0 )’1 h ∈ Lsym,c0 (T M, T M ) is symmetric with respect to the pseudo
Riemannian metric c0 . But then e(a(t) Id +b(t)H0 ) ∈ Lsym,c0 (T M, T M ) for all t in
the domain of de¬nition of the geodesic, so c(t) is a curve of pseudo Riemannian
metrics and thus of the same signature q as c0 . Thus, we have

45.15. Theorem. For each q ¤ n = dim M the submanifold Mq of pseudo
Riemannian metrics of signature q on M is a geodesically closed submanifold of
(B, G± ) for each ± = 0.


45.15
496 Chapter IX. Manifolds of mappings 45.17

1
Remark. The geodesics of (M0 , G± ) have been studied for ± = n , in [Freed,
Groisser, 1989], [Gil-Medrano, Michor, 1991] and from (45.15) and (45.11) we see
that they are completely analogous for every positive ±.
For ¬xed x ∈ M there exists a family of homothetic pseudo metrics on the ¬nite
2—
dimensional manifold S+ Tx M whose geodesics are given by the evaluation of the
geodesics of (M0 , G± ) (see [Gil-Medrano, Michor, 1991] for more details). When
± is negative, it is not di¬cult to see, from (45.15) and (45.11) again, that a
geodesic of (M0 , G± ) is de¬ned for all t if and only if the initial velocity h satis¬es
γ ± (h, h) ¤ 0 and tr H > 0 at each point of M , and then the same is true for all
2—
the pseudo metrics on S+ Tx M. These results appear already in [DeWitt, 1967] for
n = 3.

45.16. The local signature of G± . Since G± operates in in¬nite dimensional
spaces, the usual de¬nition of signature is not applicable. But for ¬xed g ∈ Mq the
signature of
’1 ’1 ’1 ’1
±
γgx (hx , kx ) = tr((gx hx )0 (gx kx )0 ) + ± tr(gx kx ) tr(gx kx )
2— —
on Tg (Sq Tx M ) = S 2 Tx M is independent of x ∈ M and the special choice of
g ∈ Mq . We will call it the local signature of G± .

45.17. Lemma. The signature of the quadratic form of (45.16) is

<<

. 21
( 27)



>>