999

A(R) — point

u 9 A(m)

∼

=

w A(R — R)

A(Id —»)

A(R — point)

We may investigate now the di¬erence between A(R) = point and A(R) = point.

In the latter case for » = 0 we have A(») = A(0) since multiplication by A(»)

equals A(m» ) which is a di¬eomorphism for » = 0 and factors over a one pointed

space for » = 0. So for A(R) = point which we assume from now on, the group

homomorphism » ’ A(») from R into A(R) is actually injective.

This de¬nition of C ∞ -algebras is due to [Lawvere, 1967], for a thorough account

see [Moerdijk, Reyes, 1991], for a discussion from the point of view of functional

analysis see [Kainz, Kriegl, Michor, 1987]. In particular there on a C ∞ -algebra A

the natural topology is de¬ned as the ¬nest locally convex topology on A such that

for all a = (a1 , . . . , an ) ∈ An the evaluation mappings µa : C ∞ (Rn , R) ’ A are

continuous. In [Kainz, Kriegl, Michor, 1987, 6.6] one ¬nds a method to recognize

C ∞ -algebras among locally-m-convex algebras. In [Michor, Vanˇura, 1996] one

z

¬nds a characterization of the algebras of smooth functions on ¬nite dimensional

algebras among all C ∞ -algebras.

31.15. Theorem. Let F : Mf¬n ’ Mf be a product preserving functor. Then

either F (R) is a point or F (R) is a C ∞ -algebra. If • : F1 ’ F2 is a natural

transformation between two such functors, then •R : F1 (R) ’ F2 (R) is an algebra

homomorphism.

If F has property ((31.13.1)) then the natural topology on F (R) is ¬ner than the

given manifold topology and thus is Hausdor¬ if the latter is it.

If F has property ((31.13.2)) then F (R) is a local algebra with an algebra homo-

morphism π = πR : F (R) ’ R whose kernel is the maximal ideal.

31.15

31.16 31. Product preserving functors on manifolds 315

Proof. By de¬nition F restricts to a product preserving functor from the category

of all Rn ™s and smooth mappings between them, thus it is a C ∞ -algebra.

If F has property ((31.13.1)) then for all a = (a1 , . . . , an ) ∈ F (R)n the evaluation

mappings are given by

µa = eva —¦F : C ∞ (Rn , R) ’ C ∞ (F (R)n , F (R)) ’ F (R)

and thus are even smooth.

If F has property ((31.13.2)) then obviously πR = π : F (R) ’ R is an algebra

homomorphism. It remains to show that the kernel of π is the largest ideal. So if

a ∈ A has π(a) = 0 ∈ R then we have to show that a is invertible in A. Since the

following diagram is a pullback,

F (i)

F (R \ {0}) ’ ’ ’ F (R)

’’

¦ ¦

¦ ¦

π π

i

R \ {0} ’’’

’’ R

we may assume that a = F (i)(b) for a unique b ∈ F (R \ {0}). But then 1/i : R \

{0} ’ R is smooth, and F (1/i)(b) = a’1 , since F (1/i)(b).a = F (1/i)(b).F (i)(b) =

F (m)F (1/i, i)(b) = F (1)(b) = 1, compare (31.14).

31.16. Examples. Let A be an augmented local C ∞ -algebra with maximal ideal

N . Then A is quotient of a free C ∞ -algebra C¬n (RΛ ) of smooth functions on some

∞

large product RΛ , which depend globally only on ¬nitely many coordinates, see

[Moerdijk, Reyes, 1991] or [Kainz, Kriegl, Michor, 1987]. So we have a short exact

sequence

•

∞

0 ’ I ’ C¬n (RΛ ) ’ A ’ 0.

’

Then I is contained in the codimension 1 maximal ideal •’1 (N ), which is easily

∞

seen to be {f ∈ C¬n (R» ) : f (x0 ) = 0} for some x0 ∈ RΛ . Then clearly • factors

over the quotient of germs at x0 . If A has Hausdor¬ natural topology, then • even

factors over the Taylor expansion mapping, by the argument in [Kainz, Kriegl,

∞

Michor, 1987, 6.1], as follows. Let f ∈ C¬n (RΛ ) be in¬nitely ¬‚at at x0 . We shall

show that f is in the closure of the set of all functions with germ 0 at x0 . Let

x0 = 0 without loss. Note ¬rst that f factors over some quotient RΛ ’ RN , and

we may replace RΛ by RN without loss. De¬ne g : RN — RN ’ RN ,

if |x| ¤ |y|,

0

g(x, y) =

(1 ’ |y|/|x|)x if |x| > |y|.

Since f is ¬‚at at 0, the mapping y ’ (x ’ fy (x) := f (g(x, y)) is a continuous

mapping RN ’ C ∞ (RN , R) with the property that f0 = f and fy has germ 0 at 0

for all y = 0.

Thus the augmented local C ∞ -algebras whose natural topology is Hausdor¬ are

∞

exactly the quotients of algebras of Taylor series at 0 of functions in C¬n (RΛ ).

It seems that local implies augmented: one has to show that a C ∞ -algebra which

is a ¬eld is 1-dimensional. Is this true?

31.16

316 Chapter VI. In¬nite dimensional manifolds 31.17

31.17. Chart description of functors induced by C ∞ -algebras. Let A =

R · 1 • N be an augmented local C ∞ -algebra which carries a compatible convenient

structure, i.e. A is a convenient vector space and each mapping A : C ∞ (Rn , Rm ) ’

C ∞ (An , Am ) is a well de¬ned smooth mapping. As in the proof of (31.15) one sees

that the natural topology on A is then ¬ner than the given convenient one, thus is

Hausdor¬. Let us call this an augmented local convenient C ∞ -algebra.

We want to associate to A a functor TA : Mf¬n ’ Mf from the category Mf¬n

of all ¬nite dimensional separable smooth manifolds to the category of smooth

manifolds modeled on convenient vector spaces.

Step 1. Let π = πA : A ’ A/N = R be the augmentation mapping. This is a

surjective homomorphism of C ∞ -algebras, so the following diagram commutes for

f ∈ C ∞ (Rn , Rm ):

wA

TA f

An m

u u

πn πm

wR

f

n m

R

If U ‚ Rn is an open subset we put TA (U ) := (π n )’1 (U ) = U — N n , which is open

subset in TA (Rn ) := An .

Step 2. Now suppose that f : Rn ’ Rm vanishes on some open set V ‚ Rn . We

claim that then TA f vanishes on the open set TA (V ) = (π n )’1 (V ). To see this let

x ∈ V , and choose a smooth function g ∈ C ∞ (Rn , R) with g(x) = 1 and support

in V . Then g.f = 0 thus we have also 0 = A(g.f ) = A(m) —¦ A(g, f ) = A(g).A(f ),

where the last multiplication is pointwise diagonal multiplication between A and

Am . For a ∈ An with (π n )(a) = x we get π(A(g)(a)) = g(π(a)) = g(x) = 1,

thus A(g)(a) is invertible in the algebra A, and from A(g)(a).A(f )(a) = 0 we may

conclude that A(f )(a) = 0 ∈ Am .

Step 3. Now let f : U ’ W be a smooth mapping between open sets U ⊆ Rn

and W ⊆ Rm . Then we can de¬ne TA (f ) : TA (U ) ’ TA (W ) in the following way.

For x ∈ U let fx : Rn ’ Rm be a smooth mapping which coincides with f in a

neighborhood V of x in U . Then by step 2 the restriction of A(fx ) to TA (V ) does

not depend on the choice of the extension fx , and by a standard argument one can

uniquely de¬ne a smooth mapping TA (f ) : TA (U ) ’ TA (V ). Clearly this gives us

an extension of the functor A from the category of all Rn ™s and smooth mappings

into convenient vector spaces to a functor from open subsets of Rn ™s and smooth

mappings into the category of c∞ -open (indeed open) subsets of convenient vector

spaces.

Step 4. Let M be a smooth ¬nite dimensional manifold, let (U± , u± : U± ’

u± (U± ) ‚ Rm ) be a smooth atlas of M with chart changings u±β := u± —¦ u’1 :

β

∞

uβ (U±β ) ’ u± (U±β ). Then by step 3 we get smooth mappings between c -open

31.17

31.18 31. Product preserving functors on manifolds 317

subsets of convenient vector spaces

wT

TA (u±β )

TA (uβ (U±β )) A (u± (U±β ))

u u

π π

w u (U

u±β

uβ (U±β ) ±β )

±

form again a cocycle of chart changings and we may use them to glue the c∞ -open

’1

sets TA (u± (U± )) = πRm (u± (U± )) ‚ Am in order to obtain a smooth manifold which

we denote by TA M . By the diagram above we see that TA M will be the total space

of a ¬ber bundle T (πA , M ) = πA,M : TA M ’ M , since the atlas (TA (U± ), TA (u± ))

constructed just now is already a ¬ber bundle atlas. So if M is Hausdor¬ then also

TA M is Hausdor¬, since two points xi can be separated in one chart if they are in

the same ¬ber, or they can be separated by inverse images under πA,M of open sets

in M separating their projections.

This construction does not depend on the choice of the atlas. For two atlas have a

common re¬nement and one may pass to this.

If f ∈ C ∞ (M, M ) for two manifolds M , M , we apply the functor TA to the local

representatives of f with respect to suitable atlas. This gives local representatives

which ¬t together to form a smooth mapping TA f : TA M ’ TA M . Clearly we

again have TA (f —¦g) = TA f —¦TA g and TA (IdM ) = IdTA M , so that TA : Mf¬n ’ Mf

is a covariant functor.

31.18. Theorem. Main properties. Let A = R · 1 • N be a local augmented

convenient C ∞ -algebra. Then we have:

(1) The construction of (31.17) de¬nes a covariant functor TA : Mf¬n ’ Mf

such that πA : TA M ’ M is a smooth ¬ber bundle with standard ¬ber N m

if dim M = m. For any f ∈ C ∞ (M, M ) we have a commutative diagram

wT

TA f

TA M AM

πA,M πA,M

u u

wM.

f

M

Thus, (TA , πA ) is a bundle functor on Mf¬n whose ¬bers may be in¬nite

dimensional. It gives a vector bundle functor on Mf if and only if N is

nilpotent of order 2.

(2) The functor TA : Mf ’ Mf is multiplicative: It respects products and pre-

serves the same classes of smooth mappings as in (31.7.2): Embeddings of

(splitting) submanifolds, surjective smooth mappings admitting local smooth

sections, ¬ber bundle projections. For ¬xed manifolds M and M the map-

ping TA : C ∞ (M, M ) ’ C ∞ (TA M, TA M ) is smooth.

(3) Any bounded algebra homomorphism • : A ’ B between augmented conve-

nient C ∞ -algebras induces a natural transformation T (•, ) = T• : TA ’

TB . If • is split injective, then T (•, M ) : TA M ’ TB M is a split embedding

31.18

318 Chapter VI. In¬nite dimensional manifolds 31.19

for each manifold M . If • is split surjective, then T (•, M ) is a ¬ber bundle

projection for each M . So we may view T as a co-covariant bifunctor from

the category of augmented convenient C ∞ -algebras algebras times Mf¬n to

Mf .

Proof. (1) is clear from (31.17). The ¬ber bundle πA,M : TA M ’ M is a vector

bundle if and only if the transition functions TA (u±β ) are ¬ber linear N — E± ’

N — Eβ . So only the ¬rst derivatives of u±β should act on N , so any product of

two elements in N must be 0, thus N has to be nilpotent of order 2.

(2) The functor TA respects ¬nite products in the category of c∞ -open subsets of

convenient vector spaces by (31.5), step 3 and 5. All the other assertions follow by

looking again at the chart structure of TA M and by taking into account that f is

part of TA f (as the base mapping).

(3) We de¬ne T (•, Rn ) := •n : An ’ B n . By (31.17), step 3, this restricts to a

natural transformation TA ’ TB on the category of open subsets of Rn ™s, and by

gluing we may extend it to a functor on the category Mf . Obviously T is a co-

covariant bifunctor on the indicated categories. Since πB —¦ • = πA (• respects the

identity), we have T (πB , M ) —¦ T (•, M ) = T (πA , M ), so T (•, M ) : TA M ’ TB M is

¬ber respecting for each manifold M . In each ¬ber chart it is a linear mapping on

m m

the typical ¬ber NA ’ NB .

So if • is split injective, T (•, M ) is ¬berwise split injective and linear in each

canonical ¬ber chart, so it is a splitting embedding.

If • is split surjective, let N1 := ker • ⊆ NA , and let V ‚ NA be a topological

linear complement to N1 . Then for m = dim M and for the canonical charts we

have the commutative diagram:

w T uM

u

T (•, M )

TA M B

wT

T (•, U± )

TA (U± ) B (U± )

u u

TA (u± ) TB (u± )

w u (U ) — N

m

Id —•|NA

m m

u± (U± ) — NA ± ± B

w u (U ) — 0 — N

Id —0 — iso

u± (U± ) — N1 — V m

m m

± ± B

So T (•, M ) is a ¬ber bundle projection with standard ¬ber E± — ker •.

31.19. Theorem. Let A and B be augmented convenient C ∞ -algebras. Then we

have:

(1) We get the convenient C ∞ -algebra A back from the functor TA by restricting

to the subcategory of Rn ™s.

(2) The natural transformations TA ’ TB correspond exactly to the bounded

C ∞ -algebra homomorphisms A ’ B.

31.19

31.20 31. Product preserving functors on manifolds 319

Proof. (1) is obvious. (2) For a natural transformation • : TA ’ TB (which is

smooth) its value •R : TA (R) = A ’ TB (R) = B is a C ∞ -algebra homomorphism

which is smooth and thus bounded. The inverse of this mapping is already described

in theorem (31.18.3).

31.20. Proposition. Let A = R · 1 • N be a local augmented convenient C ∞ -

algebra and let M be a smooth ¬nite dimensional manifold.

Then there exists a bijection

µ : TA (M ) ’ Hom(C ∞ (M, R), A)

onto the space of bounded algebra homomorphisms, which is natural in A and M .

Via µ the expression Hom(C ∞ ( , R), A) describes the functor TA in a coordinate

free manner.

Proof. Step 1. Let M = Rn , so TA (Rn ) = An . Then for a = (a1 , . . . , an ) ∈ An

we have µ(a)(f ) = A(f )(a1 , . . . , an ), which gives a bounded algebra homomor-

phism C ∞ (Rn , R) ’ A. Conversely, for • ∈ Hom(C ∞ (Rn , R), A) consider a =

(•(pr1 ), . . . , •(prn )) ∈ An . Since polynomials are dense in C ∞ (Rn , R), • is boun-

ded, and A is Hausdor¬, • is uniquely determined by its values on the coordinate

functions pri (compare [Kainz, Kriegl, Michor, 1987, 2.4.(3)], thus •(f ) = A(f )(a)

and µ is bijective. Obviously µ is natural in A and Rn .

Step 2. Now let i : U ‚ Rn be an embedding of an open subset. Then the image

of the mapping

µ’1,A

(i— )— Rn

∞ ∞

Hom(C (U, R), A) ’ ’ Hom(C (R , R), A) ’ ’ An

n

’’ ’’

’1

is the set πA,Rn (U ) = TA (U ) ‚ An , and (i— )— is injective.

To see this let • ∈ Hom(C ∞ (U, R), A). Then •’1 (N ) is the maximal ideal in

C ∞ (U, R) consisting of all smooth functions vanishing at a point x ∈ U , and

x = π(µ’1 (• —¦ i— )) = π(•(pr1 —¦i), . . . , •(prn —¦i)), so that µ’1 ((i— )— (•)) ∈ TA (U ) =

π ’1 (U ) ‚ An .

Conversely for a ∈ TA (U ) the homomorphism µa : C ∞ (Rn , R) ’ A factors over

i— : C ∞ (Rn , R) ’ C ∞ (U, R), by steps 2 and 3 of (31.17).

Step 3. The two functors Hom(C ∞ ( , R), A) and TA : Mf ’ Set coincide on

all open subsets of Rn ™s, so they have to coincide on all manifolds, since smooth

manifolds are exactly the retracts of open subsets of Rn ™s see e.g. [Federer, 1969] or

[Kol´ˇ, Michor, Slov´k, 1993, 1.14.1]. Alternatively one may check that the gluing

ar a

process described in (31.17), step 4, works also for the functor Hom(C ∞ ( , R), A)

and gives a unique manifold structure on it, which is compatible to TA M .

31.20

320

31.20

321

Chapter VII

Calculus on In¬nite Dimensional Manifolds

32. Vector Fields . . . . . . . . . . . . ............. . 321

33. Di¬erential Forms . . . . . . . . . . ............. . 336

34. De Rham Cohomology . . . . . . . . ............. . 353

35. Derivations on Di¬erential Forms and the Fr¨licher-Nijenhuis Bracket

o . 358

In chapter VI we have found that some of the classically equivalent de¬nitions of

tangent vectors di¬er in in¬nite dimensions, and accordingly we have di¬erent kinds

of tangent bundles and vector ¬elds. Since this is the central topic of any treatment

of calculus on manifolds we investigate in detail Lie brackets for all these notions

of vector ¬elds. Only kinematic vector ¬elds can have local ¬‚ows, and we show

that the latter are unique if they exist (32.16). Note also theorem (32.18) that

any bracket expression of length k of kinematic vector ¬elds is given as the k-th

derivative of the corresponding commutator expression of the ¬‚ows, which is not

well known even in ¬nite dimensions.

We also have di¬erent kinds of di¬erential forms, which we treat in a systematic

way, and we investigate how far the usual natural operations of di¬erential forms

generalize. In the end (33.21) the most common type of kinematic di¬erential forms

turns out to be the right ones for calculus on manifolds; for them the theorem of

De Rham is proved.

We also include a version of the Fr¨licher-Nijenhuis bracket in in¬nite dimensions.

o

The Fr¨licher-Nijenhuis bracket is a natural extension of the Lie bracket for vector

o

¬elds to a natural graded Lie bracket for tangent bundle valued di¬erential forms

(later called vector valued). Every treatment of curvature later in (37.3) and (37.20)

is initially based on the Fr¨licher-Nijenhuis bracket.

o

32. Vector Fields

32.1. Vector ¬elds. Let M be a smooth manifold. A kinematic vector ¬eld X on

M is just a smooth section of the kinematic tangent bundle T M ’ M . The space

of all kinematic vector ¬elds will be denoted by X(M ) = C ∞ (M ← T M ).

By an operational vector ¬eld X on M we mean a bounded derivation of the

sheaf C ∞ ( , R), i.e. for the open U ‚ M we are given bounded derivations

XU : C ∞ (U, R) ’ C ∞ (U, R) commuting with the restriction mappings.

32.1

322 Chapter VII. Calculus on in¬nite dimensional manifolds 32.2

We shall denote by Der(C ∞ (M, R)) the space of all operational vector ¬elds on M .

We shall equip Der(C ∞ (M, R)) with the convenient vector space structure induced

by the closed linear embedding

Der(C ∞ (M, R)) ’ L(C ∞ (U, R), C ∞ (U, R)).

U

Convention. In (32.4) below we will show that for a smoothly regular manifold the