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