u u u

=

wC y wC

¦

T C∞ (M, N ) ∞ ∞

c (M, T N ) (M, T N ).

∼

=

‚

The vertical mappings on the right hand side are ‚t 0 f = T f —¦ (‚t — 0M )|(0 — M ).

‚

The middle one is surjective since f (x) = ‚t 0 exp(h(t).f (x)) for suitable h, and h

can be chosen uniformly for f in a piece of a smooth curve into C∞ (M, T N ). By

construction the top isomorphism factors to a bijection ¦.

The mapping ¦ is smooth by (28.13) since ¦—¦δ factors over •, which maps the space

C ∞ (R2 , C∞ (M, N )) to Clc,c (R — M, T N ) ∼ C ∞ (R, C∞ (M, T N )). The inverse of

∞

= c

¦ is smooth by a similar argument, using again (28.13).

42.18. Corollary. Some tangent mappings. For f ∈ C ∞ (M1 , M2 ) and g ∈

C ∞ (N1 , N2 ) we have

T C∞ (M2 , N ) ∼ C∞ (M2 , T N ) T C∞ (M, N1 ) ∼ C∞ (M, T N1 )

=c =c

T C∞ (f, N ) C∞ (f, T N ) T C∞ (M, g) C∞ (M, T g)

u u u u

T C (M1 , N ) ∼ T C (M, N2 ) ∼

∞

C∞ (M1 , T N ) ∞

C∞ (M, T N2 ).

= =

c c

The tangent mapping of the composition

comp : C∞ (M, N ) — C∞ (P, M ) ’ C∞ (P, N )

prop

at (f, g) in direction of (X, Y ) ∈ Cc (M ← f — T N ) — Cc (P ← g — T M ) is given by

∞ ∞

T(f,g) comp .(X, Y ) = T f —¦ Y + X —¦ g ∈ Cc (P ← (f —¦ g)— T N ).

∞

42.18

42.20 42. Manifolds of mappings 449

The tangent mapping of the evaluation ev : C∞ (M, N ) — M ’ N at (f, x) in

direction of (X, ξ) ∈ Cc (M ← f — T N )—Tx M is given by T(f,x) ev .(X, ξ) = Tx f.ξ +

∞

X(x) ∈ Tf (x) N .

Proof. By (42.17), we may take a tangent vector X ∈ Tf (0, ) C∞ (M, N1 ) of the

form X = ‚t 0 f (t, ) ∈ Cc (M ← f — T N1 ), where f ∈ Clc (R — M, N1 ). Then we

∞ ∞

‚

‚ ‚

have (Tf (g— ).X)(x) = ‚t 0 g(f (t, x)) = T g. ‚t 0 f (t, x) = T g.X(x).

T (g — ) = g — is similar but easier, and the tangent mappings of the composition and

the evaluation can be computed either from the partial derivatives, or directly by

a variational computation as above.

42.19. The tangent mapping T : C∞ (M, N ) ’ C∞ (T M, T N ) is not smooth,

since the condition (42.5.1) is not preserved. But it is smooth as a mapping

T : C∞ (M, N ) ’ C∞ (M, L(T M, T N )), and its tangent mapping is given by

w

∼

=

T C∞ (M, N ) C∞ (M, T N )

c

u u

(κN )— —¦ T

T (T )

T C∞ (M, L(T M, T N )) ⊆ C ∞ (T M, T 2 N ),

where κN : T 2 N ’ T 2 N is the canonical ¬‚ip mapping, compare with (29.10).

‚

For the tangent mapping of the tangent mapping we consider ξx = ‚s |0 c(s) ∈ Tx M ,

and X ∈ Tf (0, ) C∞ (M, N ) of the form X = ‚t 0 f (t, ) ∈ Cc (M ← f — T N ) as

∞

‚

in the beginning of the proof. Then we have

‚

‚s |0 f (t, c(s))

T (f (t, )).ξx =

‚ ‚

(Tf (0, ) (T ).X)(ξx ) = ‚t 0 T (f (t, )) (ξx ) = ‚t 0 T (f (t, )).ξx

‚ ‚ ‚ ‚

‚t 0 ‚s |0 f (t, c(s)) = κN ‚s |0 ‚t 0 f (t, c(s))

=

‚

κN ‚s |0 X(c(s)) = κN .T X.ξx .

=

42.20. Theorem. Let M and N be smooth ¬nite dimensional manifolds, and

let q : N ’ M be smooth. Then the set C∞ (q) of all smooth sections of q

is a splitting smooth submanifold of C∞ (M, N ), whose tangent space is given by

T C∞ (q) = C∞ (M, ker(T q)) ‚ C∞ (M, T N ). If q : E ’ M is a ¬nite dimensional

c c

∞

vector bundle, the convenient vector space Cc (M ← E) is a splitting smooth sub-

manifold of C∞ (M, E).

Let now M and N be real analytic ¬nite dimensional manifolds with M compact,

and let q : N ’ M be real analytic. Then the set C ω (q) of all real analytic sec-

tions of q is a splitting real analytic submanifold of C ω (M, N ), and also C∞ (q)

is a is a splitting real analytic submanifold of C∞ (M, N ). If q : E ’ M is a

real analytic ¬nite dimensional vector bundle with M compact, the convenient vec-

tor space C ω (M ← E) is a splitting real analytic submanifold of C ω (M, E), and

C ∞ (M ← E) is a splitting real analytic submanifold of C∞ (M, E).

It is possible to extend this result at least to the case of a ¬ber bundle p : E ’ M

with in¬nite dimensional standard ¬ber by requiring certain properties. We do not

42.20

450 Chapter IX. Manifolds of mappings 42.21

present it here since in the only possible application (42.21) we have a simpler direct

proof.

Proof. If a smooth section s : M ’ N of q exists, then q, restricted to an open

neighborhood of s(M ), is a surjective submersion. Thus, there exists an open

neighborhood Ws of s(M ) in N such that ps := s —¦ q|Ws : Ws ’ s(M ) is a

surjective submersion, and we may assume that Ws is a tubular neighborhood, so

that ps : Ws ’ s(M ) is a vector bundle. Since C∞ (M, Ws ) is open in C∞ (M, N ),

we may replace N by Ws or assume that q : N ’ M is a vector bundle, and that

s is the zero section.

Claim. There exists a local addition ± : T N ’ N such that

(1) ± restricts to a local addition T 0(M ) ’ 0(M ) on the zero section.

(2) On each ¬ber Nx the local addition ± restricts to the addition T Nx ∼=

Nx — Nx ’ Nx .

In fact, choose a second vector bundle E ’ M such that N • E = M — Rk is

trivial, choose a local addition ±M on M , and let ±k be the addition on Rk . Then

±M — ±k restricts to a local addition on N with the required properties.

Now we consider the atlas for C∞ (M, N ) induced by ±, as in (42.4), i.e., we use the

formulas of (42.1) with exp replaced by ±. In particular, for the zero section s = 0

and for g ∈ U0 ‚ C∞ (M, N ) we have

u0 (g) = (IdM , (πN , ±)’1 —¦ (0, g)) ∈ Cc (M ← 0— T N ) ∼

∞

=

∼ C ∞ (M ← T M • N ) ∼ C ∞ (M ← T M ) — C ∞ (M ← N ),

=c =c c

∞

so that u0 (g) ∈ 0 — Cc (M ← N ) if and only if g is a section of the vector bundle.

∞

Moreover, Cc (M ← N ) ‚ U0 , so the second statement follows.

The statement about T C∞ (q) follows from (42.17) by noting that the derivative

of smooth curves in C∞ (q) are precisely sections s : M ’ ker(T q) such that s =

0 —¦ πE —¦ s o¬ some compact set in M .

This proof also works in the real analytic cases.

42.21. Theorem. Let (p : P ’ M, G) be a principal ¬ber bundle with ¬nite

dimensional base manifold M and a possibly in¬nite dimensional Lie group G as

structure group.

Then the gauge group Gau(P ) = C∞ (M ← P [G, conj]) from (37.17) carries the

∞

structure of a smooth Lie group modeled on Cc (P [g, Ad]).

If G is a regular Lie group then Gau(P ) is regular, too. If G admits an exponen-

tial mapping then Gau(P ) also admits an exponential mapping. If G is compact

then Gau(P ) is di¬eomorphic to the splitting submanifold C∞ (P, G)G of all G-

equivariant smooth mappings in C∞ (P, G).

If, moreover, M is compact and (p : P ’ M, G) is a real analytic principal bundle

with real analytic Lie group G, possibly in¬nite dimensional, then Gauω (P ) :=

42.21

42.21 42. Manifolds of mappings 451

C ω (M ← P [G, conj]) is a real analytic Lie group with the corresponding properties

as above.

Proof. The associated bundle P [G, conj] = P —(G,conj) G is a group bundle over

M with typical ¬ber G. It admits transition functions with values in Aut(G).

Therefore, the multiplication in G induces a smooth ¬berwise group multipli-

cation µ : P [G, conj] —M P [G, conj] ’ P [G, conj], also the ¬berwise inversion

ν : P [G, conj] ’ P [G, conj] is smooth.

The associated bundle P [g, Ad] = P —(G,Ad) g is a bundle of Lie algebras with

the same cocycle of transition functions. Thus, the bracket in g induces a smooth

∞

¬berwise bilinear mapping [ , ] : P [g, Ad]—M P [g, Ad] ’ P [g, Ad] and Cc (M ←

P [gAd]) is a convenient Lie algebra.

We shall use the canonical mappings q : P — g ’ P —G g = p[g, Ad] from (37.12.1),

„ G : P —M P ’ G from (37.8), and „ g : P —M P [g, Ad] ’ g from (37.12). We

also recall the the bijection C ∞ (P, g)G ∼ C ∞ (M ← P [g]) from (37.16), denoted by

=

f ’ sf and given by sf (p(u)) = q(u, f (u)), with inverse s ’ fs = „ g —¦ (IdP , s —¦ p).

Let u : U ’ V ⊆ g be a chart of G centered at e. Note that any model space of a

Lie group is isomorphic to the Lie algebra.

U± := {χ ∈ Gau(P ) : „ (±(z), χ(z)) ∈ U for all z ∈ P

and p({x : ±(x) = χ(x)}) has compact closure in M },

u± : U± ’ C ∞ (P, g)G ,

¯

u± (χ) = u —¦ „ —¦ (±, χ),

¯

˜ ∞

u± : U± ’ V := {s ∈ Cc (M ← P [g, Ad]) : „ g (z, s(p(z))) ∈ V for all z ∈ P }

∞

⊆ Cc (M ← P [g, Ad]),

u± (χ) = su± (χ) = su—¦„ —¦(±,χ) ,

¯

u’1 (s)(z) = ±(z).u’1 („ g (z, s(p(z)))),

±

u’1 (s) = ±.(u’1 .„ g (IdP , s —¦ p)).

±

For the chart change we see that for s ∈ uβ (U± © Uβ ) we have

(u± —¦ u’1 )(s)(p(z)) = q z, u(„ P (±(z), β(z).u’1 („ g (z, s(p(z)))))) .

β

By (30.9.1), the space of smooth curves C ∞ (R, Cc (M ← P [g, Ad])) consists of all

∞

sections c such that c§ : R — M ’ P [g, Ad] is smooth and the following condition

holds:

(1) For each compact interval [a, b] ‚ R there is a compact subset K ‚ M such

that c§ (t, x) is constant in t ∈ [a, b] for all x ∈ M \ K.

Obviously, the chart change respects the set of smooth curves and is smooth. Thus,

the atlas (U± , u± ) describes the structure of a smooth manifold which we denote by

Gau(P ) ∼ C∞ (M ← P [G, conj]), and we also see that the space of smooth curves

=

C ∞ (R, C∞ (M ← P [G, conj])) consists of all sections c such that the associated

42.21

452 Chapter IX. Manifolds of mappings 42.22

mapping c§ : R — M ’ P [G, conj] is smooth and condition (1) holds. Composition

and inversion are smooth on Gau(P ) since these correspond just to push forwards

of sections via the smooth ¬berwise group multiplication and inversion described at

the beginning of the proof. The Lie algebra of C∞ (M ← P [G, conj]) is Cc (M ←

∞

P [g, Ad]).

Let us now suppose that the Lie group G is regular with evolution operator evolG :

C ∞ (R, g) ’ G. Since the smooth group bundle P [G, conj] is described by a cocycle

of transition functions with values in the group of (inner) automorphisms of G and

since by (38.4) we have evolG —¦ •— = • —¦ evolG for any automorphism • of G, there

is an induced ¬berwise evolution operator

evol : P [C ∞ (R, g), Ad— ] ∼ C ∞ (R, p[g, Ad]) ’ P [G, conj],

=

which, by push forward on sections, induces

evolGau(P ) : C ∞ (R, Cc (M ← P [g, Ad])) ’ Cc (M ← P [G, conj]).

∞ ∞

This maps smooth curves to smooth curves and is the evolution operator of Gau(P ).

The remaining assertions are easy to check.

42.22. Manifolds of holomorphic mappings. It is a natural question whether

the methods of this section carry over to spaces of holomorphic mappings between

complex manifolds. The situation is described in the following result.

Lemma.

(1) Each ¬nite dimensional Stein manifold M admits holomorphic local addi-

tions T M ⊃ U ’ M in the sense of (42.4).

(2) Complex projective spaces do not admit holomorphic local additions.

Proof. (1) A Stein manifold M is biholomorphically embedded as a closed complex

submanifold of some Cn (where n = 2 dimC M + 2 su¬ces), see [Gunning, Rossi,

1965, p. 224], and there exists a holomorphic tubular neighborhood p : V ’ M in

Cn , see [Gunning, Rossi, 1965, p. 257], by an application of Cartan™s theorem B that

a coherent sheaf on a Stein manifold is acyclic. The a¬ne addition • : T Cn ’ Cn ,

given by •(z, Z) := z + Z then gives a local addition p —¦ •|T M : T M ⊃ U ’ M

for a suitable open neighborhood U of 0 in T M .

(2) First we show that CP1 does not admit a holomorphic local addition. The usual

a¬ne charts u0 [z0 : z1 ] = z1 and u1 [z0 : z1 ] = z0 have as chart change mapping

z0 z1

z ’ 1/z on C \ {0}. Its tangent mapping is (z, w) ’ ( z , ’ z12 w). A local addition

1

would be given by two holomorphic mappings ±i : T C ⊃ U ’ C on an open

neighborhood U of the zero section {(z, 0) : z ∈ C} with

1

±0 ( z , ’ z12 w) =

1

for z = 0,

±1 (z, w)

‚

‚w |w=0 ±i (z, w)

±i (z, 0) = z, = 0 for all z.

42.22

42.22 42. Manifolds of mappings 453

‚2

‚

of 1 = ±0 ( z , ’ z12 w)±1 (z, w) are in turn

1

‚w |w=0 ‚w2 |w=0

The derivatives and

1 1

0 = z (‚2 ±1 (z, 0) ’ ‚2 ±0 ( z , 0)),

12 12

2 1 1

0 = z ‚2 ±1 (z, 0) ’ z 2 ‚2 ±0 ( z , 0).‚2 ±1 (z, 0) + z 3 ‚2 ±0 ( z , 0)

22 2 2

1 1

z 3 (z ‚2 ±1 (z, 0) ’ z(‚2 ±1 (z, 0)) + ‚2 ±0 ( z , 0)).

=

1

2 2

Hence, limz’0 ‚2 ±0 ( z , 0) = 0, and consequently ‚2 ±0 (z, 0) = 0 for all z since

it is an entire function on C which vanishes at in¬nity. But then we get that

1

z ’ ‚2 ±1 (z, 0) = z (‚2 ±1 (z, 0))2 ’ 0 has a pole at 0, a contradiction.

2

Now we treat CPn . Suppose that a holomorphic local addition ± : T (CPn ) ⊃ U ’

CPn exists. Let us consider CP1 ‚ CPn , given by [z0 : z1 ] ’ [z0 : z1 : 0 : · · · : 0].

Then we have a holomorphic retraction r : V ’ CP1 given by r[z0 : · · · : zn ] = [z0 :

z1 ] for V = {[z0 : · · · : zn ] : (z0 , z1 ) = (0, 0)}. But then r —¦ ±|(U © T (CP1 )) is a

holomorphic local addition on CP1 , a contradiction.

Results. From the argument given in (42.4) follows that a complex manifold ad-

mitting a holomorphic local addition also admits a holomorphic spray and thus a

holomorphic linear connection on T M . Existence of the latter has been investigated

in [Atiyah, 1957]. Let us sketch the relevant results. For a complex manifold M let

T M be the complex tangent bundle, let GL(Cm , T M ) be the linear frame bundle.

Then using the local description from (29.9) and (29.10) we get in turn:

GL(Cm , T M ) (x, s) ∈ U — GL(m, C),

T (GL(Cm , T M )) (x, s, ξ, σ) ∈ U — GL(m, C) — Cm — gl(m, C),

{f ∈ LC (Cm , T (πM )’1 )(ξ) :

T (GL(Cm , T M )) =

ξ∈T M

πT M —¦ f ∈ GL(Cm , T M )},

(x, Id, ξ, A) = (x, s —¦ s’1 , ξ, σ —¦ s’1 ),

T (GL(Cm , T M ))/GL(m, C)

T (GL(Cm , T M ))

{f ∈ LC (Tx M, T (πM )’1 )(ξ) :

=

GL(m, C)

x∈M ξ∈Tx M

πT M —¦ f = IdTx M },

which turns out to be a holomorphic vector bundle over M . Then we have the

following exact sequence of holomorphic vector bundles over M :

vlT M —¦(Id, ) T (πM )

0 ’ L(T M, T M ) ’ ’ ’ ’ ’ T (GL(Cm , T M ))/GL(m, C) ’ ’ ’ M ’ 0.

’ ’ ’ ’’ ’’

T

A holomorphic splitting of this sequence is exactly a holomorphic linear connection

on T M . This sequence de¬nes an extension of the bundle T M by L(T M, T M ), i.e.,

an element b(T M ) in the sheaf cohomology H 1 (M ; T — M — L(T M, T M )). Thus:

[Atiyah, 1957, from theorems 2 and 5]. A complex manifold M admits a holomor-

phic linear connection if and only if b(T M ) vanishes.

Note that via Cartan™s theorem B this again implies that Stein manifolds admit

holomorphic local additions. Moreover, Atiyah proved the following results:

42.22

454 Chapter IX. Manifolds of mappings 43.1

Result. [Atiyah, 1957, theorem 6]. If M is a compact K¨hler manifold, then the

a

k-th Chern class of T M is given by

√

ck (T M ) = (’2π ’1)’k Sk [b(T M )],

where Sk is the k characteristic coe¬cient gl(m, C) ’ C.

Note that this also implies that CPn does not admit local additions.

[Atiyah, 1957, proposition 22]. Even if all characteristic classes of M vanish, M

need not admit a holomorphic connection.

43. Di¬eomorphism Groups

43.1. Theorem. Di¬eomorphism group. For a smooth manifold M the group

Di¬(M ) of all smooth di¬eomorphisms of M is an open submanifold of C∞ (M, M ),

composition and inversion are smooth. It is a regular Lie group in the sense of

(38.4).

The Lie algebra of the smooth in¬nite dimensional Lie group Di¬(M ) is the conve-

∞

nient vector space Cc (M ← T M ) of all smooth vector ¬elds on M with compact

support, equipped with the negative of the usual Lie bracket. The exponential map-

ping exp : Cc (M ← T M ) ’ Di¬ ∞ (M ) is the ¬‚ow mapping to time 1, and it is

∞

smooth.

Proof. We ¬rst show that Di¬(M ) is open in C∞ (M, M ). Let c : R ’ C∞ (M, M )

be a smooth curve such that c(0) is a di¬eomorphism. We have to show that then

c(t) also is a di¬eomorphism for small t. The mapping c(t) stays in the WO1 -open

(and thus open by (42.1)) subset of immersions for t near 0, see (41.10).