given by the derivation f ’ (f —¦ c)(k) (0) at x. Namely, we have

k

k

(k) (k)

(f —¦ c)(j) (0)(g —¦ c)(k’j) (0)

((f.g) —¦ c) (0) = ((f —¦ c).(g —¦ c)) (0) = j

j=0

= (f —¦ c)(k) (0)g(x) + f (x)(g —¦ c)(k) (0),

since all other summands vanish: (f —¦ c)(j) (0) = 0 for 1 ¤ j < k. That c(k) (0) is a

kinematic tangent vector follows from the chain rule in a local chart.

Step 2. Let (pr1 , •) : R — M ⊃ U• ’ V• ‚ R — M be a di¬eomorphism between

open neighborhoods of {0} — M in R — M , such that •0 = IdM . We say that •t is

a curve of local di¬eomorphisms though IdM . Note that a local ¬‚ow of a kinematic

vector ¬eld is always such a curve of local di¬eomorphisms.

j k

‚ 1‚

From step 1 we see that if ‚tj |0 •t = 0 for all 1 ¤ j < k, then X := k! ‚tk |0 •t

is a well de¬ned vector ¬eld on M . We say that X is the ¬rst non-vanishing

derivative at 0 of the curve •t of local di¬eomorphisms. We may paraphrase this

as (‚t |0 •— )f = k!LX f .

k

t

Claim 3. Let •t , ψt be curves of local di¬eomorphisms through IdM , and let

f ∈ C ∞ (M, R). Then we have

k

j k’j

— —

•— )f (‚t |0 ψt )(‚t |0 •— )f.

—

k

k k

‚t |0 (•t —¦ ψt ) f = ‚t |0 (ψt —¦ =

t t

j

j=0

The multinomial version of this formula holds also:

k! j j

‚t |0 (•1 —¦ . . . —¦ •t )— f = (‚t 1 |0 (•t )— ) . . . (‚t 1 |0 (•1 )— )f.

k

t t

j1 ! . . . j !

j1 +···+j =k

32.18

32.18 32. Vector ¬elds 335

We only show the binomial version. For a function h(t, s) of two variables we have

k

j k’j

k

k

‚t h(t, t) = ‚t ‚s h(t, s)|s=t ,

j

j=0

since for h(t, s) = f (t)g(s) this is just a consequence of the Leibniz rule, and linear

combinations of such decomposable tensors are dense in the space of all functions of

two variables in the compact C ∞ -topology (41.9), so that by continuity the formula

holds for all functions. In the following form it implies the claim:

k

j k’j

k

k

‚t |0 f (•(t, ψ(t, x))) = ‚t ‚s f (•(t, ψ(s, x)))|t=s=0 .

j

j=0

Claim 4. Let •t be a curve of local di¬eomorphisms through IdM with ¬rst non-

k

vanishing derivative k!X = ‚t |0 •t . Then the inverse curve of local di¬eomorphisms

•’1 has ¬rst non-vanishing derivative ’k!X = ‚t |0 •’1 .

k

t t

Since we have •’1 —¦ •t = Id, by claim 3 we get for 1 ¤ j ¤ k

t

j

j j’i

‚t |0 (•’1 (‚t |0 •— )(‚t (•’1 )— )f =

—¦ •t )— f = j i

0= t t

t

i

i=0

j j

‚t |0 •— (•’1 )— f + •— ‚t |0 (•’1 )— f,

= which says

t

t 0

0

j j

‚t |0 •— f = ’‚t |0 (•’1 )— f, as required.

t

t

Claim 5. Let •t be a curve of local di¬eomorphisms through IdM with ¬rst non-

m

vanishing derivative m!X = ‚t |0 •t , and let ψt be a curve of local di¬eomorphisms

n

through IdM with ¬rst non-vanishing derivative n!Y = ‚t |0 ψt . Then the curve of

local di¬eomorphisms [•t , ψt ] = ψt —¦•’1 —¦ψt —¦•t has ¬rst non-vanishing derivative

’1

t

m+n

|0 [•t , ψt ].

(m + n)![X, Y ] = ‚t

From this claim the theorem follows.

By the multinomial version of claim 3, we have

AN f := ‚t |0 (ψt —¦ •’1 —¦ ψt —¦ •t )— f

’1

N

t

N! j

(‚t |0 •— )(‚t |0 ψt )(‚t |0 (•’1 )— )(‚t |0 (ψt )— )f.

’1

—

i k

= t

t

i!j!k! !

i+j+k+ =N

Let us suppose that 1 ¤ n ¤ m; the case m ¤ n is similar. If N < n all summands

are 0. If N = n we have by claim 4

AN f = (‚t |0 •— )f + (‚t |0 ψt )f + (‚t |0 (•’1 )— )f + (‚t |0 (ψt )— )f = 0.

’1

—

n n n n

t

t

If n < N ¤ m we have, using again claim 4:

N! j

(‚t |0 ψt )(‚t |0 (ψt )— )f + δN (‚t |0 •— )f + (‚t |0 (•’1 )— )f

’1

— m m m

AN f = t

t

j! !

j+ =N

’1

= (‚t |0 (ψt —¦ ψt )— )f + 0 = 0.

N

32.18

336 Chapter VII. Calculus on in¬nite dimensional manifolds 33

Now we come to the di¬cult case m, n < N ¤ m + n.

AN f = ‚t |0 (ψt —¦ •’1 —¦ ψt )— f +

’1

(‚t |0 •— )(‚t ’m |0 (ψt —¦ •’1 —¦ ψt )— )f

’1

N N

N m

t t

t

m

+ (‚t |0 •— )f,

N

(6) t

by claim 3, since all other terms vanish, see (8) below. Again by claim 3 we get:

N! j

‚t |0 (ψt —¦ •’1 —¦ ψt )— f =

’1

(‚t |0 ψt )(‚t |0 (•’1 )— )(‚t |0 (ψt )— )f

’1

—

N k

t t

j!k! !

j+k+ =N

j ’1

(‚t ’m |0 ψt )(‚t |0 (•’1 )— )f

(‚t |0 ψt )(‚t |0 (ψt )— )f +

— —

N N N m

(7) = t

j m

j+ =N

(‚t |0 (•’1 )— )(‚t ’m |0 (ψt )— )f + (‚t |0 (•’1 )— )f

’1

N N

m N

+ t t

m

N ’m

|0 ψt )m!L’X f + m m!L’X (‚t ’m |0 (ψt )— )f

’1

—

N N N

=0+ m (‚t

+ (‚t |0 (•’1 )— )f

N

t

= δm+n (m + n)!(LX LY ’ LY LX )f + (‚t |0 (•’1 )— )f

N N

t

= δm+n (m + n)!L[X,Y ] f + (‚t |0 (•’1 )— )f

N N

t

From the second expression in (7) one can also read o¬ that

‚t ’m |0 (ψt —¦ •’1 —¦ ψt )— f = ‚t ’m |0 (•’1 )— f.

’1

N N

(8) t t

If we put (7) and (8) into (6) we get, using claims 3 and 4 again, the ¬nal result

which proves claim 5 and the theorem:

AN f = δm+n (m + n)!L[X,Y ] f + (‚t |0 (•’1 )— )f

N N

t

(‚t |0 •— )(‚t ’m |0 (•’1 )— )f + (‚t |0 •— )f

N N

m N

+ t

t t

m

= δm+n (m + n)!L[X,Y ] f + ‚t |0 (•’1 —¦ •t )— f

N N

t

N

= δm+n (m + n)!L[X,Y ] f + 0.

33. Di¬erential Forms

This section is devoted to the search for the right notion of di¬erential forms which

are stable under Lie derivatives LX , exterior derivative d, and pullback f — . Here

chaos breaks out (as one referee has put it) since the classically equivalent descrip-

tions of di¬erential forms give rise to many di¬erent classes; in the table (33.21) we

shall have 12 classes. But fortunately it will turn out in (33.22) that there is only

one suitable class satisfying all requirements, namely

„¦k (M ) := C ∞ (Lk (T M, M — R)).

alt

33

33.2 33. Di¬erential forms 337

33.1. Cotangent bundles. We consider the contravariant smooth functor which

associates to each convenient vector space E its dual E of bounded linear func-

tionals, and we apply it to the kinematic tangent bundle T M described in (28.12)

of a smooth manifold M (see (29.5)) to get the kinematic cotangent bundle T M .

A smooth atlas (U± , u± : U± ’ E± ) of M gives the cocycle of transition functions

x ’ d(uβ —¦ u’1 )(u± (x))— ∈ GL(Eβ , E± ).

U±β ±

If we apply the same duality functor to the operational tangent bundle DM de-

scribed in (28.12) we get the operational cotangent bundle D M . A smooth atlas

(U± , u± : U± ’ E± ) of M now gives rise to the following cocycle of transition

functions

x ’ D(uβ —¦ u’1 )(u± (x))— ∈ GL((D0 Eβ ) , (D0 E± ) ),

U±β ±

see (28.9) and (28.12).

For each k ∈ N we get the operational cotangent bundle (D(k) ) M of order ¤ k,

which is described by the same cocycle of transition functions but now restricted

(k) (k)

to have values in GL((D0 Eβ ) , (D0 E± ) ), see (28.10).

33.2. 1-forms. Let M be a smooth manifold. A kinematic 1-form is just a smooth

section of the kinematic cotangent bundle T M . So C ∞ (M ← T M ) denotes the

convenient vector space (with the structure from (30.1)) of all kinematic 1-forms

on M .

An operational 1-form is just a smooth section of the operational cotangent bundle

D M . So C ∞ (M ← D M ) denotes the convenient vector space (with the structure

from (30.1)) of all operational 1-forms on M .

For each k ∈ N we get the convenient vector space C ∞ (M ← (D(k) ) (M )) of all

operational 1-forms of order ¤ k, a closed linear subspace of C ∞ (M ← D M ).

A modular 1-form is a bounded linear sheaf homomorphism ω : Der(C ∞ ( , R)) ’

C ∞ ( , R) which satis¬es ωU (f.X) = f.ωU (X) for X ∈ Der(C ∞ (U, R)) = C ∞ (U ←

DU ) and f ∈ C ∞ (U, R) for each open U ‚ M . We denote the space of all modular

1-forms by

HomC ∞ (M,R) (C ∞ (M ← DM ), C ∞ (M, R))

and we equip it with the initial structure of a convenient vector space induced by

the closed linear embedding

HomC ∞ (M,R) (C ∞ (M ← DM ), C ∞ (M, R)) ’ L(C ∞ (U ← DU ), C ∞ (U, R)).

U

Convention. Similarly as in (32.1), we shall follow the convention that either the

manifolds in question are smoothly regular or that Hom means the space of sheaf

homomorphisms (as de¬ned above) between the sheafs of sections like C ∞ (M ←

DM ) of the respective vector bundles. This is justi¬ed by (33.3) below.

33.2

338 Chapter VII. Calculus on in¬nite dimensional manifolds 33.4

33.3. Lemma. If M is smoothly regular, the bounded C ∞ (M, R)-module homo-

morphisms ω : C ∞ (M ← DM ) ’ C ∞ (M, R) are exactly the modular 1-forms and

this identi¬cation is an isomorphism of the convenient vector spaces.

Proof. If X ∈ C ∞ (M ← DM ) vanishes on an open subset U ‚ M then also

ω(X): For x ∈ U we take a bump function g ∈ C ∞ (M, R) at x, i.e. g = 1 near x

and supp(g) ‚ U . Then ω(X) = ω((1 ’ g)X) = (1 ’ g)ω(X) which is zero near x.

So ω(X)|U = 0.

Now let X ∈ C ∞ (U ← DU ) for a c∞ -open subset U of M . We have to show that

we can de¬ne ωU (X) ∈ C ∞ (U, R) in a unique manner. For x ∈ U let g ∈ C ∞ (M, R)

be a bump function at x, i.e. g = 1 near x and supp(g) ‚ U . Then gX ∈ C ∞ (M ←

DM ), and ω(gX) makes sense. By the argument above, ω(gX)(x) is independent

of the choice of g. So let ωU (X)(x) := ω(gX)(x). It has all required properties since

the topology on C ∞ (U ← DU ) is initial with respect to all mappings X ’ gX,

where g runs through all bump functions as above.

That this identi¬cation furnishes an isomorphism of convenient vector spaces can

be seen as in (32.4).

33.4. Lemma. On any manifold M the space C ∞ (M ← D M ) of operational

1-forms is a closed linear subspace of modular 1-forms HomC ∞ (M,R) (C ∞ (M ←

DM ), C ∞ (M, R)).

The closed vector bundle embedding T M ’ DM induces a bounded linear mapping

C ∞ (M ← D M ) ’ C ∞ (M ← T M ).

We do not know whether C ∞ (M ← D M ) ’ C ∞ (M ← T M ) is surjective or even

¬nal.

Proof. A smooth section ω ∈ C ∞ (M ← D M ) de¬nes a modular 1-form which

assigns ωU (X)(x) := ω(x)(X(x)) to X ∈ C ∞ (U ← DU ) and x ∈ U , by (32.2),

since this gives a bounded sheaf homomorphism which is C ∞ ( , R)-linear.

To show that this gives an embedding onto a c∞ -closed linear subspace we consider

the following diagram, where (U± ) runs through an open cover of charts of M . Then

the vertical mappings are closed linear embeddings by (30.1), (33.1), and (32.2).

w Hom

C ∞ (M ← D M ) ∞

(M ← DM ), C ∞ (M, R))

C ∞ (M,R) (C

u u

w

C ∞ (U± , (D0 E± ) ) L(C ∞ (U± ← DU± ), C ∞ (U± , R))

± ±

u u

w

C ∞ (U± — D0 E± , R) C ∞ (C ∞ (U± , D0 E± ) — U± , R)

± ±

33.4

33.6 33. Di¬erential forms 339

The horizontal bottom arrow is the mapping f ’ ((X, x) ’ f (x, X(x))), which is

an embedding since (X, x) ’ (x, X(x)) has (x, Y ) ’ (const(Y ), x) as smooth right

inverse.

33.5. Lemma. Let M be a smooth manifold such that for all model spaces E the

convenient vector space D0 E has the bornological approximation property (28.6).

Then

C ∞ (M ← D M ) ∼ HomC ∞ (M,R) (C ∞ (M ← DM ), C ∞ (M, R)).

=

If all model spaces E have the bornological approximation property then D0 E = E ,

and the space E also has the bornological approximation property. So in this case,

HomC ∞ (M,R) (C ∞ (M ← DM ), C ∞ (M, R)) ∼ C ∞ (M ← T M ).

=

If, moreover, all E are re¬‚exive, we have

HomC ∞ (M,R) (C ∞ (M ← DM ), C ∞ (M, R)) ∼ C ∞ (M ← T M ),

=

as in ¬nite dimensions.

Proof. By lemma (33.4) the space C ∞ (M ← D M ) is a closed linear subspace of

the convenient vector space HomC ∞ (M,R) (C ∞ (M ← DM ), C ∞ (M, R)). We have to

show that any sheaf homomorphism ω ∈ HomC ∞ (M,R) (C ∞ (M ← DM ), C ∞ (M, R))

lies in C ∞ (M ← D M ). This is a local question, hence we may assume that M is

a c∞ -open subset of E.

We have to show that for each X ∈ C ∞ (U, D0 E) the value ωU (X)(x) depends only

on X(x) ∈ D0 E. So let X(x) = 0, and we have to show that ωU (X)(x) = 0.

By assumption, there is a net ± ∈ (D0 E) — D0 E ‚ L(D0 E, D0 E) of bounded

linear operators with ¬nite dimensional images, which converges to IdD0 E in the

bornological topology of L(D0 E, D0 E). Then X± := ± —¦ X converges to X in