V of e with v(e) = 0 ∈ W where W is open in the Lie algebra g. Then put

T G ⊇ U := g∈G T (µg )V ∼ G — V and let ± : U ’ G be given by ±(ξ) :=

=

’1

πG (ξ).v (T (µπ(ξ)’1 ).ξ) be the local addition.

If a manifold N admits a local addition ±, then it admits a ˜spray™, thus a torsionfree

covariant derivative on T N . Recall from [Ambrose, Palais, Singer, 1960] or [Lang,

1972] that a spray is a vector ¬eld S on T M such that πT M —¦S = IdT M , T (πM )—¦S =

IdT M , so that in induced local charts as in (29.9) and (29.10) we have S(x, y) =

(x, y; y, “x (y)), where ¬nally it is also required that y ’ “x (y) is quadratic. In

‚

order to see this, let •(X) := ‚t 0 ±(tX). Then • : T M ’ T M is a vector bundle

automorphism with inverse (in local charts) •’1 (x, y) = ‚t 0 (pr1 , ±)’1 (x, x + ty).

‚

d2 ’1

dt2 |t=0 ±(t•

Then one checks easily that S(X) := (X)) is a spray.

Theorem. Let M be a smooth ¬nite dimensional manifold, and let N be a smooth

manifold, possibly in¬nite dimensional, which admits a smooth local addition ±.

42.4

442 Chapter IX. Manifolds of mappings 42.7

Then the space C∞ (M, N ) of all smooth mappings from M to N is a smooth mani-

fold, modeled on spaces Cc (M ← f — T N ) of smooth sections with compact support

∞

of pullback bundles along f : M ’ N over M .

Let us remark again that for a compact smooth manifold M and a convenient vector

space E the smooth manifold C∞ (M, E) is di¬eomorphic to the convenient vector

space C ∞ (M, E), which is a special case of (30.1) for a trivial bundle with ¬nite

dimensional base.

42.5. Lemma. Smooth curves in C∞ (M, N ). Let M and N be smooth man-

ifolds with M ¬nite dimensional and N admitting a smooth local addition. Then

the smooth curves c in C∞ (M, N ) correspond exactly to the smooth mappings

c§ ∈ C ∞ (R — M, N ) which satisfy the following property:

(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.

In particular, the identity induces a smooth mapping C∞ (M, N ) ’ C ∞ (M, N ) into

the Fr¨licher space C ∞ (M, N ) discussed in (23.2.3), which is a di¬eomorphism if

o

and only if M is compact or N is discrete.

Proof. Since R is locally compact, property (1) is equivalent to

(2) For each t ∈ R there is an open neighborhood U of t in R and a compact

K ‚ M such that the restriction has the property that c§ (t, x) is constant

in t ∈ U for all x ∈ M \ K.

Since this is a local condition on R, and since smooth curves in C∞ (M, N ) locally

take values in charts as in the proof of theorem (42.1), it su¬ces to describe the

∞

smooth curves in the space Cc (M ← E) of sections with compact support of a

vector bundle (p : E ’ M, V ) with ¬nite dimensional base manifold M , with the

structure described in (30.4). This was done in (30.9).

42.6. Theorem. C ω -manifold structure of C ω (M, N ). Let M and N be real

analytic manifolds, let M be compact, and let N be either ¬nite dimensional, or let

us assume that N admits a real analytic local addition in the sense of (42.4).

Then the space C ω (M, N ) of all real analytic mappings from M to N is a real

analytic manifold, modeled on spaces C ω (M ← f — T N ) of real analytic sections of

pullback bundles along f : M ’ N over M .

Proof. The proof is a variant of the proof of (42.4), using a real analytic Riemann-

ian metric if N is ¬nite dimensional, and the given real analytic local addition oth-

erwise. For ¬nite dimensional N a detailed proof can be found in [Kriegl, Michor,

1990].

42.7. Lemma. Let M , N be real analytic ¬nite dimensional manifolds. Then

the space C ω (M, N ) of all real analytic mappings is dense in C ∞ (M, N ), in the

Whitney C ∞ -topology.

This is not true in the manifold topology of C∞ (M, N ) used in (42.1), if M is not

compact, because of the compact support condition used there.

42.7

42.10 42. Manifolds of mappings 443

Proof. By [Grauert, 1958, theorem 3], there is a real analytic embedding i : N ’

Rk on a closed submanifold, for some k. We use the constant standard inner product

on Rk to obtain a real analytic tubular neighborhood U of i(N ) with projection

p : U ’ i(N ). By [Grauert, 1958, proposition 8] applied to each coordinate of

Rk , the space C ω (M, Rk ) of real analytic Rk -valued functions is dense in the space

C ∞ (M, Rk ) of smooth functions, in the Whitney C ∞ -topology. If f : M ’ N is

smooth we may approximate i —¦ f by real analytic mappings g in C ω (M, U ), then

p —¦ g is real analytic M ’ i(N ) and approximates i —¦ f .

42.8. Theorem. C ω -manifold structure on C∞ (M, N ). Let M and N be real

analytic ¬nite dimensional manifolds, with M compact. Then the smooth manifold

C∞ (M, N ) with the structure from (42.1) is even a real analytic manifold.

Proof. For a ¬xed real analytic exponential mapping on N the charts (Uf , uf )

from (42.1) for f ∈ C ω (M, N ) form a smooth atlas for C∞ (M, N ), since C ω (M, N )

is dense in C∞ (M, N ) by (42.7)

’1

The chart changings uf —¦ u’1 = („f —¦ „g )— are real analytic by (30.10).

g

42.9. Corollary. Let Mi and Ni be smooth manifolds with Mi ¬nite dimensional

for i = 1, 2 and Ni admitting smooth local additions. Then we have:

(1) If f : N1 ’ N2 is initial (27.11) then the mapping

C∞ (M, f ) : C∞ (M, N1 ) ’ C∞ (M, N2 )

is initial, too.

(2) If f : M2 ’ M1 is ¬nal (27.15) and proper then the mapping C∞ (f, N ) :

C∞ (M1 , N ) ’ C∞ (M2 , N ) is initial.

Proof. (1) Let c : R ’ C∞ (M, N1 ) be such that f— —¦c : R ’ C∞ (M, N2 ) is smooth.

By (42.5), the associated mapping (f— —¦ c)§ = f —¦ c§ : R — M ’ N2 is smooth and

satis¬es (42.5.1). Since f is initial, c§ is smooth, and since f is injective, c§ satis¬es

(42.5.1), hence c is smooth.

Proof of (2) Since f is ¬nal between ¬nite dimensional manifolds, it is a surjective

submersion, so R — f is also a surjective submersion and thus ¬nal.

Let c : R ’ C∞ (M1 , N ) be such that f — —¦ c : R ’ C∞ (M2 , N ) is smooth. By

(42.5), the associated mapping (f — —¦ c)§ = c§ —¦ (R — f ) : R — M2 ’ N is smooth

and satis¬es (42.5.1). Since R — f is also ¬nal, c§ is smooth. Since f and thus also

R — f is proper, c§ satis¬es (42.5.1), and thus c is smooth.

42.10. Lemma. Let M and N be real analytic ¬nite dimensional manifolds with

M compact. Let (U± , u± ) be a real analytic atlas for M , and let i : N ’ Rn

be a closed real analytic embedding into some Rn . Let M be a possibly in¬nite

dimensional real analytic manifold.

Then f : M ’ C ω (M, N ) is real analytic or smooth if and only if C ω (u’1 , i) —¦ f :

±

ω n

M ’ C (u± (U± ), R ) is real analytic or smooth, respectively.

42.10

444 Chapter IX. Manifolds of mappings 42.13

Furthermore, f : M ’ C∞ (M, N ) is real analytic or smooth if and only if the

mapping C ∞ (u’1 , i) —¦ f : M ’ C ∞ (u± (U± ), Rn ) is real analytic or smooth, respec-

±

tively.

Proof. The statement that i— is initial is obvious. So we just have to treat

C ∞ (u’1 , N ). The corresponding statement for spaces of sections of vector bundles

±

are (30.6) for the real analytic case and (30.1) for the smooth case. So if f takes val-

ues in a chart domain Ug of C ∞ (M, N ) for a real analytic g : M ’ N , the result fol-

lows. Recall from the proof of (42.1) that Ug = {h ∈ C β (M, N ) : (g(x), h(x)) ∈ V }

where V is a ¬xed open neighborhood of the diagonal in N — N , and where β = ∞

or ω. Let f (z0 ) ∈ Ug for z0 ∈ M. Since M is covered by ¬nitely many of its charts

U± , and since by assumption f (z)|U± is near f (z0 )|U± for z near z0 , so f (z) ∈ Ug

for z near z0 in M. So f takes values locally in charts, and the result follows.

42.11. Corollary. Let M and N be ¬nite dimensional real analytic manifolds

with M compact. Then the inclusion C ω (M, N ) ’ C∞ (M, N ) is real analytic.

Proof. Use the chart description and lemma (11.3).

42.12. Lemma. Curves in spaces of mappings. Let M and N be ¬nite

dimensional real analytic manifolds with M compact.

(1) A curve c : R ’ C ω (M, N ) is real analytic if and only if the associated

mapping c§ : R — M ’ N is real analytic.

The curve c : R ’ C ω (M, N ) is smooth if and only if c§ : R — M ’ N

satis¬es the following condition:

For each n there is an open neighborhood Un of R—M in R—MC

and a (unique) C n -extension c : Un ’ NC such that c(t, ) is

˜ ˜

holomorphic for all t ∈ R.

(2) The curve c : R ’ C∞ (M, N ) is real analytic if and only if c§ satis¬es the

following condition:

For each n there is an open neighborhood Un of R — M in C — M

and a (unique) C n -extension c : Un ’ NC such that c( , x) is

˜ ˜

holomorphic for all x ∈ M .

Note that the two conditions are in fact local in R. We need N ¬nite dimensional

since we need an extension NC of N to a complex manifold.

Proof. This follows from the corresponding statement (30.8) for spaces of sections

of vector bundles, and from the chart structure on C ω (M, N ) and C∞ (M, N ).

42.13. Theorem. Smoothness of composition. If M , N are smooth mani-

folds with M ¬nite dimensional and N admitting a smooth local addition, then the

evaluation mapping ev : C∞ (M, N ) — M ’ N is smooth.

42.13

42.14 42. Manifolds of mappings 445

If P is another smooth ¬nite dimensional manifold, then the composition mapping

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

prop

is smooth, where C∞ (P, M ) denotes the space of all proper smooth mappings

prop

P ’ M (i.e. compact sets have compact inverse images). This space is open

in C∞ (P, M ).

In particular, f— : C∞ (M, N ) ’ C∞ (M, N ) and g — : C∞ (M, N ) ’ C∞ (P, N ) are

smooth for f ∈ C ∞ (N , N ) and g ∈ C∞ (P, M ).

prop

The corresponding statement for real analytic mappings can be shown along similar

lines, using (42.12). But we will give another proof in (42.15) below.

Proof. Using the description of smooth curves in C∞ (M, N ) given in (42.5), we

immediately see that (ev —¦(c1 , c2 ))(t) = c§ (t, c2 (t)) is smooth for each smooth

1

∞

(c1 , c2 ) : R ’ C (M, N ) — M , so ev is smooth as claimed.

The space of proper mappings C∞ (P, M ) is open in the manifold C∞ (P, M ) since

prop

changing a mapping only on a compact set does not change its property of being

proper. Let (c1 , c2 ) : R ’ C∞ (M, N ) — C∞ (P, M ) be a smooth curve. Then we

prop

§ §

have (comp —¦(c1 , c2 ))(t)(p) = c1 (t, c2 (t, p)), and one may check that this has again

property (44.5.1), so it is a smooth curve in C∞ (P, N ). Thus, comp is smooth.

42.14. Theorem. Exponential law. Let M, M , and N be smooth manifolds

with M ¬nite dimensional and N admitting a smooth local addition.

Then we have a canonical injection

C ∞ (M, C∞ (M, N )) ⊆ C ∞ (M — M, N ),

where the image in the right hand side consists of all smooth mappings f : M—M ’

N which satisfy the following property

(1) If M is locally metrizable then for each point x ∈ M there is an open

neighborhood U and a compact set K ‚ M such that f (x, y) is constant in

x ∈ U for all y ∈ M \ K.

(2) For general M: For each c ∈ C ∞ (R, M) and each t ∈ R there exists

a neighborhood U of t and a compact set K ‚ M such that f (c(s), y) is

constant in s ∈ U for each y ∈ M \ K.

Under the assumption that N admits smooth functions which separate points, we

have equality if and only if M is compact, or N is discrete, or each f ∈ C ∞ (M, R)

is constant along all smooth curves into M.

If M and N are real analytic manifolds with M compact we have

C ω (M, C ω (M, N )) = C ω (M — M, N )

for each real analytic (possibly in¬nite dimensional) manifold M.

Proof. The smooth case is simple: The equivalence for general M follows directly

from the description of all smooth curves in C∞ (M, N ) given in the proof of (42.5).

42.14

446 Chapter IX. Manifolds of mappings 42.15

It remains to show that for locally metrizable M a smooth mapping f : M ’

C∞ (M, N ) satis¬es condition (1). Since f is smooth, locally it has values in a

chart, so we may assume that M is open in a Fr´chet space by (4.19), and that f

e

∞

has values in Cc (M ← E) for some vector bundle p : E ’ M .

∞

We claim that f locally factors into some CKn (E) where (Kn ) is an exhaustion

of M by compact subsets such that Kn is contained in the interior of Kn+1 . If

not there exist a (fast) converging sequence (yn ) in M and xn ∈ Kn such that

/

f (yn )(xn ) = 0. One may ¬nd a proper smooth curve e : R ’ M with e(n) = xn

and a smooth curve g : R ’ M with g(1/n) = yn . Then by (30.4), Pt(e, )— —¦ f —¦ g

∞

is a smooth curve into Cc (R, Ee(0) ). Since the latter space is a strict inductive

limit of spaces CI (R, Ee(0) ) for compact intervals I, the curve Pt(e, )— —¦ f —¦ g

∞

locally factors into some CI (R, Ee(0) ), but (e— —¦ f —¦ g)(1/n)(n) = f (yn )(xn ) = 0, a

∞

contradiction.

We check now the statement on equality: if M is compact, or if N is discrete then

(2) is automatically satis¬ed. If each f ∈ C ∞ (M, R) is constant along all smooth

curves into M, we may check global constancy in (2) by composing with smooth

functions on N which separate points there.

For the converse, we may assume that there are a function f ∈ C ∞ (M, R), a curve

c ∈ C ∞ (R, M) such that f —¦c is not constant, and an injective smooth curve e : R ’

N . Then M — M (x, y) ’ e(f (x)) is in C ∞ (M — M, N ) \ C ∞ (M, C∞ (M, N ))

since condition (2) is violated for the curve c.

Now we treat the real analytic case. Let f § ∈ C ω (M—M, N ) ‚ C ∞ (M—M, N ) =

C ∞ (M, C∞ (M, N )). So we may restrict f to a neighborhood U in M, where it takes

values in a chart Ug of C ∞ (M, N ) for g ∈ C ω (M, N ). Then f (U ) ‚ Ug ©C ω (M, N ),

one of the canonical charts of C ω (M, N ). So we may assume that f : U ’ C ω (M ←

g — T N ). For a real analytic vector bundle atlas (U± , ψ± ) of g — T N the composites

U ’ C ω (M ← g — T N ) ’ C ω (U± , Rn ) are real analytic by applying cartesian

closedness (11.18) to the mapping (x, y) ’ ψ± (πN , exp)’1 (g(y), f § (x, y)). By the

description (30.6) of the structure on C ω (M ← g — T N ), the chart representation of

f is real analytic, so f is it also.

Let conversely f : M ’ C ω (M, N ) be real analytic. Then its chart representation is

real analytic and we may use cartesian closedness in the other direction to conclude

that f § is real analytic.

42.15. Corollary. If M and N are real analytic manifolds with M compact

and N admitting a real analytic local addition, then the evaluation mapping ev :

C ω (M, N ) — M ’ N is real analytic.

If P is another compact real analytic manifold, then the composition mapping

comp : C ω (M, N ) — C ω (P, M ) ’ C ω (P, N ) is real analytic.

In particular, f— : C ω (M, N ) ’ C ω (M, N ) and g — : C ω (M, N ) ’ C ω (P, N ) are

real analytic for real analytic f : N ’ N and g ∈ C ω (P, M ).

Proof. The mapping ev∨ = IdC ω (M,N ) is real analytic, so ev too, by (42.14).

42.15

42.17 42. Manifolds of mappings 447

The mapping comp§ = ev —¦(IdC ω (M,N ) — ev) : C ω (M, N ) — C ω (P, M ) — P ’

C ω (M, N ) — M ’ N is real analytic, thus comp too.

42.16. Lemma. Let Mi and Ni be ¬nite dimensional real analytic manifolds with

Mi compact. Then for f ∈ C ∞ (N1 , N2 ) the push forward f— : C∞ (M, N1 ) ’

C∞ (M, N2 ) is real analytic if and only if f is real analytic. For f ∈ C ∞ (M2 , M1 )

the pullback f — : C∞ (M1 , N ) ’ C∞ (M2 , N ) is, however, always real analytic.

Proof. If f is real analytic and if g ∈ C ω (M, N1 ), then the mapping uf —¦g —¦f— —¦u’1 : g

∞ — ∞ —

C (M ← g T N1 ) ’ C (M ← (f —¦ g) T N2 ) is a push forward by the real analytic

mapping (pr1 , (π, expN2 )’1 —¦ (f —¦ g —¦ pr1 , f —¦ expN1 —¦ pr2 )) : g — T N1 ’ (f —¦ g)— T N2 ,

which is real analytic by (30.10).

The canonical mapping evx : C∞ (M, N2 ) ’ N2 is real analytic since evx |Ug =

expN2 —¦ evx —¦ug : Ug ’ C ∞ (M ← g — T N2 ) ’ Tg(x) N2 ’ N2 , where the second evx

is linear and bounded. Furthermore, const : N1 ’ C∞ (M, N1 ) is real analytic since

the mapping ug —¦ const : y ’ (x ’ (πN1 , expN1 )’1 (g(x), y)) is locally real analytic

into C ω (M ← g — T N1 ) and hence into C ∞ (M ← g — T N1 ).

If f— is real analytic, also f = evx —¦f— —¦ const is.

For the second statement choose real analytic atlas (U± , ui ) of Mi such that

i

±

f (U± ) ⊆ U± and a closed real analytic embedding j : N ’ Rn . Then the dia-

2 1

gram

w

f—

∞

C∞ (M2 , N )

C (M1 , N )

C∞ ((u1 )’1 , j) C∞ ((u2 )’1 , j)

u u

± ±

wC

(u2 —¦ f —¦ (u1 )’1 )—

± ±

C∞ (u1 (U± ), Rn ) ∞

1

(u2 (U± ), Rn )

2

± ±

commutes, the bottom arrow is a continuous and linear mapping, so it is real

analytic. Thus, by (42.10), the mapping f — is real analytic.

42.17. Theorem. Let M and N be smooth manifolds with M compact and N

admitting a local addition. Then the in¬nite dimensional smooth vector bundles

T C∞ (M, N ) and C∞ (M, T N ) ‚ C∞ (M, T N ) over C∞ (M, N ) are canonically iso-

c

morphic. The same assertion is true for C ω (M, N ) if M is compact.

Here by C∞ (M, T N ) we denote the space of all smooth mappings f : M ’ T N

c

such that f (x) = 0πM f (x) for x ∈ Kf , a suitable compact subset of M (equivalently,

/

such that the associated section of the pull back bundle (πM —¦ f )— T N ’ M has

compact support).

One can check directly that the atlas from (42.1) for C∞ (M, N ) induces an atlas

for T C∞ (M, N ), which is equivalent to that for C∞ (M, T N ) via some natural iden-

ti¬cations in T T N . This is carried out in great detail in [Michor, 1980c, 10.13].

We shall give here a simpler proof, using (42.5).

42.17

448 Chapter IX. Manifolds of mappings 42.18

Proof. Recall from (28.13) the diagram

u eC

C ∞ (R, C∞ (M, N ))/ ∼ ∞

(R, C∞ (M, N ))

δ ee

ee

∼δ

u h

e u

ev0

=

wC

T C∞ (M, N ) ∞

(M, N ),

πC∞ (M,N )

From (42.5) we see that C ∞ (R, C∞ (M, N )) corresponds to the space Clc (R—M, N )

∞

of all mappings g § ∈ C ∞ (R — M, N ) satisfying

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

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

Now we consider the diagram

wC y wC

∼

=

C ∞ (R, C∞ (M, N )) ∞ ∞

— M, N ) (R — M, N )

lc (R

•

u

δ

‚ ‚

C ∞ (R, C∞ (M, N ))/ ∼ ‚t 0 ‚t 0