remains to show that ωU (X± )(x) = 0 for each ±.

n

i=1 •i — ‚i ∈ (D0 E) — D0 E, hence X± = (•i —¦ X).‚i and

We have ± =

ωU (X± )(x) = •i (X(x)).ωU (‚i )(x) = 0 since X(x) = 0.

So we get a ¬ber linear mapping ω : DM ’ M — R which is given by ω(Xx ) =

(x, ωU (X)(x)) for any X ∈ C ∞ (U ← DU ) with X(x) = Xx . Obviously, ω : DM ’

M — R is smooth and gives rise to a smooth section of D M .

If E has the bornological approximation property, then by (28.7) we have D0 E =

E . If ± is a net of ¬nite dimensional bounded operators which converges to IdE

in L(E, E), then the ¬nite dimensional operators —— converge to IdE = IdE in

±

L(E , E ), in the bornological topology. The rest follows from theorem (28.7)

33.6. Queer 1-forms. Let E be a convenient vector space without the borno-

logical approximation property, for example an in¬nite dimensional Hilbert space.

Then there exists a bounded linear functional ± ∈ L(E, E) which vanishes on

33.6

340 Chapter VII. Calculus on in¬nite dimensional manifolds 33.8

E — E such that ±(IdE ) = 1. Then ωU : C ∞ (U, E) ’ C ∞ (U, R), given by

ωU (X)(x) := ±(dX(x)), is a bounded sheaf homomorphism which is a module ho-

momorphism, since ωU (f.X)(x) = ±(df (x) — X(x) + f (x).dX(x)) = f (x)ωU (X)(x).

Note that ωU (X)(x) does not depend only on X(x). So there are many ˜kinematic

modular 1-forms™ which are not kinematic 1-forms.

This process can be iterated to involve higher derivatives like for derivations, see

(28.2), but we resist the temptation to pursue this task. It would be more interesting

to produce queer modular 1-forms which are not operational 1-forms.

33.7. k-forms. For a smooth manifold M there are at least eight interesting

spaces of k-forms, see the diagram below where A := C ∞ (M, R), and where C ∞ (E)

denotes the space of smooth sections of the vector bundle E ’ M :

wC

C ∞ (Λk (D M )) ∞

(Lk (DM, M — R))

alt

wC

C ∞ (Λk (T M )) ∞

(Lk (T M, M — R))

alt

u u

w Hom

k, alt

Λk HomA (C ∞ (DM ), A) (C ∞ (DM ), A)

A A

u u

w Hom k, alt

Λk HomA (C ∞ (T M ), A) (C ∞ (T M ), A)

A A

Here Λk is the bornological exterior product which was treated in (5.9). One could

also start from other tensor products. By Λk = Λk ∞ (M,R) we mean the convenient

A C

module exterior product, the subspace of all skew symmetric elements in the k-fold

bornological tensor product over A, see (5.21). By Homk ∞ (M,R),alt = Homk,∞ (M,R)

alt

C C

we mean the convenient space of all bounded homomorphism between the respective

sheaves of convenient modules over the sheaf of smooth functions.

33.8. Wedge product. For di¬erential forms • of degree k and ψ of degree

and for (local) vector ¬elds Xi (or tangent vectors) we put

(• § ψ)(X1 , . . . , Xk+ ) =

1

sign σ · •(Xσ1 , . . . , Xσk ).ψ(Xσ(k+1) , . . . , Xσ(k+ ) ).

= k! !

σ∈Sk+

This is well de¬ned for di¬erential forms in each of the spaces in (33.7) and others

(see (33.12) below) and gives a di¬erential form of the same type of degree k+ . The

wedge product is associative, i.e (• § ψ) § „ = • § (ψ § „ ), and graded commutative,

i. e. • § ψ = (’1)k ψ § •. These properties are proved in multilinear algebra. There

arise several kinds of algebras of di¬erential forms.

33.8

33.11 33. Di¬erential forms 341

33.9. Pullback of di¬erential forms. Let f : N ’ M be a smooth mapping

between smooth manifolds, and let • be a di¬erential form on M of degree k in

any of the following spaces: C ∞ (Lk (D± M, M — R)) for D± = D, D(k) , D[1,∞) , T .

alt

— ±

In this situation the pullback f • is de¬ned for tangent vectors Xi ∈ Dx N by

(f — •)x (X1 , . . . , Xk ) := •f (x) (Dx f.X1 , . . . , Dx f.Xk ).

± ±

(1)

Then we have f — (• § ψ) = f — • § f — ψ, so the linear mapping f — is an algebra

homomorphism. Moreover, we have (g—¦f )— = f — —¦g — if g : M ’ P , and (IdM )— = Id,

and (f, •) ’ f — • is smooth in all these cases.

If f : N ’ M is a local di¬eomorphism, then we may de¬ne the pullback f — • also

for a modular di¬erential form • ∈ Homk,∞ (M,R) (C ∞ (M ← D± M ), C ∞ (M, R)), by

alt

C

(2) (f — •)|U (X1 , . . . , Xk ) := •|f (U ) (D± f —¦X1 —¦(f |U )’1 , . . . , D± f —¦Xk —¦(f |U )’1 )—¦f.

These two de¬nitions are intertwined by the canonical mappings between di¬erent

spaces of di¬erential forms.

33.10. Insertion operator. For a vector ¬eld X ∈ C ∞ (M ← D± M ) where

D± = D, D(k) , D[1,∞) , T we de¬ne the insertion operator

iX = i(X) : Homk,∞ (M,R) (C ∞ (M ← D± M ), C ∞ (M, R)) ’

alt

C

’ Homk’1, alt (C ∞ (M ← D± M ), C ∞ (M, R))

C ∞ (M,R)

(iX •)(Y1 , . . . , Yk’1 ) := •(X, Y1 , . . . , Yk’1 ).

It restricts to operators

iX = i(X) : C ∞ (Lk (D± M, M — R)) ’ C ∞ (Lk’1 (D± M, M — R)).

alt alt

33.11. Lemma. iX is a graded derivation of degree ’1, so we have iX (• § ψ) =

iX • § ψ + (’1)deg • • § iX ψ.

Proof. We have

(iX1 (• § ψ))(X2 , . . . , Xk+ ) = (• § ψ)(X1 , . . . , Xk+ )

1

= sign(σ) •(Xσ1 , . . . , Xσk )ψ(Xσ(k+1) , . . . , Xσ(k+ ) ).

k! !

σ

k

(iX1 • § ψ + (’1) • § iX1 ψ)(X2 , . . . , Xk+ )

1

= sign(σ) •(X1 , Xσ2 , . . . , Xσk )ψ(Xσ(k+1) , . . . , Xσ(k+ ) )

(k’1)! !

σ

k

(’1)

+ sign(σ) •(Xσ2 , . . . , Xσ(k+1) )ψ(X1 , Xσ(k+2) , . . . ).

k! ( ’ 1)! σ

Using the skew symmetry of • and ψ we may distribute X1 to each position by

adding an appropriate sign. These are k + summands. Since (k’1)! ! + k! ( 1

1

’1)! =

k+

k! ! , and since we can generate each permutation in Sk+ in this way, the result

follows.

33.11

342 Chapter VII. Calculus on in¬nite dimensional manifolds 33.12

33.12. Exterior derivative. Let U ‚ E be c∞ -open in a convenient vector

space E, and let ω ∈ C ∞ (U, Lk (E; R)) be a kinematic k-form on U . We de¬ne

alt

the exterior derivative dω ∈ C (U, Lk+1 (E; R)) as the skew symmetrization of the

∞

alt

k

derivative dω(x) : E ’ Lalt (E; R) (sorry for the two notions of d, it™s only local);

i.e.

k

(’1)i dω(x)(Xi )(X0 , . . . , Xi , . . . , Xk )

(1) (dω)(x)(X0 , . . . , Xk ) =

i=0

k

(’1)i d(ω(

= )(X0 , . . . , Xi , . . . , Xk ))(x)(Xi )

i=0

where Xi ∈ E. Next we view the Xi as ˜constant vector ¬elds™ on U and try to

replace them by kinematic vector ¬elds. Let us compute ¬rst for Xj ∈ C ∞ (U, E),

where we suppress obvious evaluations at x ∈ U :

(’1)i Xi (ω —¦ (X0 , . . . , Xi , . . . , Xk ))(x) =

i

(’1)i (dω(x).Xi )(X0 , . . . , Xi , . . . , Xk )+

=

i

(’1)i ω —¦ (X0 , . . . , dXj (x).Xi , . . . , Xi , . . . , Xk )+

+

j<i

(’1)i ω —¦ (X0 , . . . , Xi , . . . , dXj (x).Xi , . . . , Xk ) =

(2) +

i<j

(’1)i (dω(x).Xi )(X0 , . . . , Xi , . . . , Xk )+

=

i

(’1)i+j ω —¦ (dXj (x).Xi ’ dXi (x).Xj , X0 , . . . , Xj , . . . , Xi , . . . , Xk )

+

j<i

(’1)i (dω(x).Xi )(X0 , . . . , Xi , . . . , Xk )+

=

i

(’1)i+j ω —¦ ([Xi , Xj ], X0 , . . . , Xj , . . . , Xi , . . . , Xk ).

+

j<i

Combining (2) and (1) gives the global formula for the exterior derivative

k

(’1)i Xi (ω —¦ (X0 , . . . , Xi , . . . , Xk ))+

(3) (dω)(x)(X0 , . . . , Xk ) =

i=0

(’1)i+j ω —¦ ([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xk ).

+

i<j

Formula (3) de¬nes the exterior derivative for modular forms on X(M ), C ∞ (M ←

DM ), and C ∞ (M ← D[1,∞) M ), since it gives multilinear module homomorphisms

by the Lie module properties of the Lie bracket, see (32.5) and (32.8).

The local formula (1) gives the exterior derivative on C ∞ (Lk (T M, M — R)): Local

alt

expressions (1) for two di¬erent charts describe the same di¬erential form since both

33.12

33.12 33. Di¬erential forms 343

can be written in the global form (3), and the canonical mapping C ∞ (Lk (T M, M —

alt

k, alt ∞

R)) ’ HomC ∞ (M,R) (X(M ), C (M, R)) is injective, since we use sheaves on the

right hand side.

The ¬rst line of the local formula (1) gives an exterior derivative dloc also on the

space C ∞ (Lk (DU, R)), where U is an open subset in a convenient vector space

alt

E, if we replace dω(x) by Dx ω : D0 E ’ D0 (Lk (D0 E, R)) composed with the

alt

canonical mapping

(‚ [1] )’1

( )[1]

k k

Lk (D0 E, R) =

D0 (Lalt (D0 E, R)) ’ ’ ’

’’ D0 (Lalt (D0 E, R)) ’ ’ ’

’ ’’ alt

ι—

= (Λk (D0 E)) ’ (Λk (D0 E)) = Lk (D0 E, R).

’ alt

Here ι : Λk D0 E ’ (Λk D0 E) is the canonical embedding into the bidual. If we

replace d by D in the second expression of the local formula (1) we get the same

expression. For ω ∈ C ∞ (U, Lk (D0 E, R)) we have

alt

k

(dloc ω)(x)(X0 , . . . , Xk ) = (’1)i Dx (ω( )(X0 , . . . , Xi , . . . , Xk ))(Xi )

i=0

k

(’1)i Dx (ev(X0 ,...,Xi ,...,Xk ) —¦ω)(Xi )

=

i=0

k

(’1)i Dω(x) (ev(X0 ,...,Xi ,...,Xk ) ).Dx ω.Xi

=

i=0

k

(1)

(’1)i (Dω(x) ev(X0 ,...,Xi ,...,Xk ) .(Dx ω.Xi )[1]

= by (28.11.4)

i=0

k

(’1)i (ev(X0 §...Xi ···§Xk ) )—— .(‚ [1] )’1 .(Dx ω.Xi )[1]

= by (28.11.3)

i=0

k

(’1)i ev(X0 §...Xi ···§Xk ) .ι— .(‚ [1] )’1 .(Dx ω.Xi )[1]

=

i=0

k

(’1)i ι— —¦ (‚ [1] )’1 —¦ ( )[1] —¦ Dx ω (Xi )(X0 , . . . , Xi , . . . , Xk ),

=

i=0

since the following diagram commutes:

wR

ev(X0 §...Xi ···§Xk )

u

k

(Λ D0 E)

ι—

wR

(ev(X0 §...Xi ···§Xk ) )——

(Λk D0 E)

The local formula (1) describes by a similar procedure the local exterior derivative

dloc also on C ∞ (Lk (D[1,∞) M, R)).

alt

33.12

344 Chapter VII. Calculus on in¬nite dimensional manifolds 33.13

For the forms of tensorial type (involving Λk ) there is no exterior derivative in

general, since the derivative is not tensorial in general.

For a manifold M let us now consider the following diagram of certain spaces of

di¬erential forms.

w Hom k,alt

C ∞ (Lk (DM, M — R)) ∞

← DM ), C ∞ (M, R))

C ∞ (M,R) (C (M

alt

u u

w k,alt

∞

HomC ∞ (M,R) (C ∞ (D[1,∞) M ), C ∞ (M, R))

(Lk (D[1,∞) M, M — R))

C alt

u u

w Hom k,alt

C ∞ (Lk (T M, M — R) ∞

C ∞ (M,R) (X(M ), C (M, R))

alt

If M is a c∞ -open subset in a convenient vector space E, on the two upper left

spaces there exists only the local (from formula (1)) exterior derivative dloc . On all

other spaces the global (from formula (3)) exterior derivative d makes sense. All

canonical mappings in this diagram commute with the exterior derivatives except

the dashed ones. The following example (33.13) shows that

(1) The dashed arrows do not commute with the respective exterior derivatives.

(2) The (global) exterior derivative does not respect the spaces on the left hand

side of the diagram except the bottom one.

(3) The dashed arrows are not surjective.

The example (33.14) shows that the local exterior derivative on the two upper

left spaces does not commute with pullbacks of smooth mappings, not even of

di¬eomorphisms, in general. So it does not even exist on manifolds. Furthermore,

dloc —¦ dloc is more interesting than 0, see example (33.16).

33.13. Example. Let U be c∞ -open in a convenient vector space E. If ω ∈

(1)

C ∞ (U, E ) = C ∞ (U, L(D0 E, R)) then in general the exterior derivative

dω ∈ Hom2,∞ (U,R) (C ∞ (U ← DU ), C ∞ (U, R))

alt

C

is not contained in C ∞ (U ← L2 (DU, U — R)).

alt

Proof. Let X, Y ∈ C ∞ (U, E ). The Lie bracket [X, Y ] is given in (32.7), and ω

depends only on the D(1) -part of the bracket. Thus, we have

dω(X, Y )(x) = X(ω(Y ))(x) ’ Y (ω(X))(x) ’ ω([X, Y ])(x)

’ Y (x), d ω, X

= X(x), d ω, Y (x) (x)

E E E E

’ ω(x), (dY (x)t )— .X(x) ’ (dX(x)t )— .Y (x) E

’

= X(x), dω(x), Y (x) + X(x), ω(x), dY (x)

E E E E

’ Y (x), dω(x), X(x) ’ Y (x), ω(x), dX(x)

E E E E

’ ω(x), (dY (x)— —¦ ιE ) .X(x) —

+ ω(x), (dX(x)— —¦ ιE )— .Y (x) .

E E

33.13

33.14 33. Di¬erential forms 345

Let us treat the terms separately which contain derivatives of X or Y . Choosing