‚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

C ∞ (U, D0 E) since X — : L(D0 E, D0 E) ’ C ∞ (U, D0 E) is continuous linear. It

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

X constant (but arbitrary) we have to consider only the following expression:

’ ω(x), (dY (x)— —¦ ιE )— .X(x)

X(x), ω(x), dY (x) =

E E E

’ ω(x), ι— .dY (x)—— .X(x)

= X(x), ω(x) —¦ dY (x) E E

E

= X(x), dY (x)— .ω(x) ’ ι—— .ω(x), dY (x)—— .X(x)

E E

E

= ιE .ω(x), dY (x)—— .X(x) ’ ι—— .ω(x), dY (x) .X(x)

——

E E

E

’ ι—— ).ω(x), dY (x) .X(x)

——

= (ιE ,

E

E

which is not 0 in general since ker(ιE ’ ι—— ) = ιE (E ) at least for Banach spaces,

E

see [Cigler, Losert, Michor, 1979, 1.15], applied to ιE . So we may assume that

(ιE ’ ι—— ).ω(x) = 0 ∈ E . We choose a non-re¬‚exive Banach space which is

E

isomorphic to its bidual ([James, 1951]) and we choose as dY (x) this isomorphism,

then dY (x)—— is also an isomorphism, and a suitable X(x) makes the expression

nonzero.

Note that this also shows that for general convenient vector spaces E the exterior

(1)

derivative dω is in C ∞ (U, L2 (D0 E, R)) only if ω ∈ C ∞ (M ← T M ). Note that

alt

even for ω : U ’ E a constant 1-form of order 1 we need not have dω = 0.

33.14. Example. There exist c∞ -open subsets U and V in a Banach space E, a

di¬eomorphism f : U ’ V , and a 1-form ω ∈ C ∞ (U, L(E , R)) such that dloc f — ω =

f — dloc ω.

Proof. We start in a more general situation. Let f : U ’ V ‚ F be a smooth

(1)

mapping, and let Xx , Yx ∈ Dx U = E . Then we have

dloc (f — ω)x (Xx , Yx ) = Dx (f — ω( ).Yx ).Xx ’ Dx (f — ω( ).Xx ).Yx

’ ...

= Dx (ω(f ( )).D( ) f.Yx ).Xx

= Xx ω —¦ f, D( ) f.Yx F ’ . . .

ω —¦ f, df ( )—— .Yx F (x)—— .Xx ’ . . . by (32.6)

=d

ω(f ( )), df (x)—— .Yx F (x)—— .Xx +

=d

d ω(f (x)), df ( )—— .Yx F (x)—— .Xx ’ . . . by

+ (32.6)

f — (dloc ω)x (Xx , Yx ) = (dloc ω)f (x) (Dx f.Xx , Dx f.Yx )

) .Dx f.Yx ).Dx f.Xx ’ Df (x) (ω( ) .Dx f.Xx ).Dx f.Yx

= Df (x) (ω(

), df (x)—— .Yx F (f (x))—— .df (x)—— .Xx ’ . . .

= d ω(

——

Recall that for ∈ H = L(H, R) the bidual mapping satis¬es L(H , R) =

ιH ( ) ∈ H . Then for the di¬erence we get

dloc (f — ω)x (Xx , Yx ) ’ f — (dloc ω)x (Xx , Yx )

)—— .Yx (x)—— .Xx ’ d ω(f (x)), df ( )—— .Xx (x)—— .Yx

= d ω(f (x)), df ( F F

) .Yx )(x)—— .Xx ’ d(df (

——

)—— .Xx )(x)—— .Yx

= iF ω(f (x)), d(df ( .

F

33.14

346 Chapter VII. Calculus on in¬nite dimensional manifolds 33.15

This expression does not vanish in general, e.g., when the following choices are

made: We put ω(f (x)) = ιF . = —— for ∈ F , and we have

)—— Yx )(x)—— .Xx = d(d , f )—— Yx )(x)—— .Xx

d(d( —¦ f )( F(

)—— Yx (x)—— .Xx

= d ιF , df ( F

——

) Yx )(x)—— .Xx

——

= ιF , d(df ( ,

F

which is not symmetric in general for —¦ f = ev : G — G ’ R (for a non re¬‚ex-

ive Banach space G) by the argument in (32.7). It remains to show that such a

factorization of ev over a di¬eomorphism f and ∈ (G — G) is possible. Choose

(±, x) ∈ G — G such that ±, x = 1, and consider

f

G — G = G — ker ± — R.x ’ G — ker ± — R.x ’ R

’ ’

(β, y, tx) ’ (β, y, β, y + tx ’ β, y + tx

G .x) G

(β, y, t’β,x .x) ← (β, y, tx).

β,y

’

33.15. Proposition. Let f : M ’ N be a smooth mapping between smooth

manifolds. Then we have

f — —¦ d = d —¦ f — : C ∞ (Lk (T N, N — R)) ’ C ∞ (Lk+1 (T M, M — R)).

alt alt

Proof. Since by (33.12) the local and global formula for the exterior derivative

coincide on spaces C ∞ (Lk (D± M, M — R)) we shall prove the result with help of

alt

the local formula. So we may assume that f : U ’ V is smooth between c∞ -open

sets in convenient vector spaces E and F , respectively. Note that we may use the

global formula only if f is a local di¬eomorphism, see (33.9).

For ω ∈ C ∞ (V, Lk (F, R)), x ∈ U , and Xi ∈ E we have

alt

(f — ω)(x)(X1 , . . . , Xk ) = ω(f (x))(df (x).X1 , . . . , df (x).Xk ),

so by (33.12.1) we may compute

k

(df — ω)(x)(X0 , . . . , Xk ) = (’1)i d(f — ω)(x)(Xi )(X0 , . . . , Xi , . . . , Xk )

i=0

k

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

=

i=0

k

(’1)i ω(f (x))(df (x).X0 , . . . , d2 f (x).(Xi , Xj ), . . . , i , . . . , df (x).Xk )

+

i=0 j<i

k

(’1)i ω(f (x))(df (x).X0 , . . . , i , . . . , d2 f (x).(Xi , Xj ), . . . , df (x).Xk )

+

i=0 j>i

33.15

33.18 33. Di¬erential forms 347

k

(’1)i dω(f (x))(df (x).X0 , . . . , df (x).Xk )

=

i=0

(’1)i+j ω(f (x))(d2 f (x).(Xi , Xj ) ’ d2 f (x).(Xj , Xi ),

+

j<i

df (x).X0 , . . . , j , . . . , i , . . . , df (x).Xk )

= (f — dω)(x)(X0 , . . . , Xk ) + 0.

33.16. Example. There exists a smooth function

f ∈ C ∞ (E, R) = C ∞ (E, L0 (D(1) E, R))

alt

such that

0 = dloc dloc f ∈ C ∞ (E, L2 (D(1) E, R)).

alt

(1)

Proof. Let f ∈ C ∞ (E, R), Xx , Yx ∈ Dx E = E . Then we have

(dloc f )x (Xx ) = df (x)—— .Xx = ιF .df (x), Xx E

= Xx , df (x) E

(dloc dloc f )x (Xx , Yx ) =

(x)—— .Xx ’ d Xx , df ( (x)—— .Yx

= d Yx , df ( ) )

E E

= ιE .Yx , d(df )(x)—— .Xx ’ ιE .Xx , d(df )(x)—— .Yx

E E

= d(df )(x)—— .Xx , Yx ’ d(df )(x)—— .Yx , Xx ,

E E

which does not vanish in general by the argument in (32.7).

33.17. Lie derivatives. Let D± denote one of T , D, or D[1,∞) . For a vector

¬eld X ∈ C ∞ (M ← D± M ) and ω ∈ Homk,∞ (M,R) (C ∞ (M ← D± M ), C ∞ (M, R))

alt

C

we de¬ne the Lie derivative LX ω of ω along X by

k

(LX ω)|U (Y1 , . . . , Yk ) = X(ω(Y1 , . . . , Yk )) ’ ω|U (Y1 , . . . , [X, Yi ], . . . , Yk ),

i=1

for Y1 , . . . , Yk ∈ C ∞ (U ← D± U ). From (32.5) it follows that

LX ω ∈ Homk,∞ (M,R) (C ∞ (M ← D± M ), C ∞ (M, R)).

alt

C

33.18. Theorem. The following formulas hold for C ∞ (Lk (T M, M — R)) and

alt

k, alt ∞ ∞

for the spaces HomC ∞ (M,R) (C (M ← D M ), C (M, R)) where D± is any of D,

±

D[1,∞) , or T .

(1) iX (• § ψ) = iX • § ψ + (’1)deg • • § iX ψ.

(2) LX (• § ψ) = LX • § ψ + • § LX ψ.

(3) d(• § ψ) = d• § ψ + (’1)deg • • § dψ.

33.18

348 Chapter VII. Calculus on in¬nite dimensional manifolds 33.18

1

d2 = d —¦ d = 2 [d, d] = 0.

(4)

[LX , d] = LX —¦ d ’ d —¦ LX = 0.

(5)

[iX , d] = iX —¦ d + d —¦ iX = LX .

(6)

[LX , LY ] = LX —¦ LY ’ LY —¦ LX = L[X,Y ] .

(7)

[LX , iY ] = LX iY ’ iY LX = i[X,Y ] .

(8)

(9) [iX , iY ] = iX iY + iY iX = 0.

Lf.X • = f.LX • + df § iX •.

(10)

Remark. In this theorem we used the graded commutator for graded derivations

[D1 , D2 ] := D1 —¦ D2 ’ (’1)deg(D1 ) deg(D2 ) D2 —¦ D1 . We will elaborate this notion in

(35.1) below.

The left hand side of (6) maps the subspace C ∞ (Lk (T M, M — R)) of the space of

alt

k, alt ∞

modular di¬erential forms HomC ∞ (M,R) (X(M ), C (M, R)) into itself, thus the Lie

derivative LX also does. We do not know whether this is true for the other spaces

on the left hand side of the diagram in (33.12).

Proof. All results will be proved in Homk,∞ (M,R) (C ∞ (M ← D± M ), C ∞ (M, R)),

alt

C

∞

so they also hold in the subspace C (M ← Lk (T M, M — R)).

alt

(9) is obvious and (1) was shown in (33.11).

(8) Take the di¬erence of the following two expressions:

k

(LX iY ω)(Z1 , . . . , Zk ) = X((iY ω)(Z1 , . . . , Zk )) ’ (iY ω)(Z1 , . . . , [X, Zi ], . . . , Zk )

i=1

k

= X(ω(Y, Z1 , . . . , Zk )) ’ ω(Y, Z1 , . . . , [X, Zi ], . . . , Zk )

i=1

(iY LX ω)(Z1 , . . . , Zk ) = LX ω(Y, Z1 , . . . , Zk )

= X(ω(Y, Z1 , . . . , Zk )) ’ ω([X, Y ], Z1 , . . . , Zk )’

k

’ ω(Y, Z1 , . . . , [X, Zi ], . . . , Zk ).

i=1

(2) Let • be of degree p and ψ of degree q. We prove the result by induction on

p + q. Suppose that (2) is true for p + q < k. Then for X we have by part (8), by

1, and by induction

(iY LX )(• § ψ) = (LX iY )(• § ψ) ’ i[X,Y ] (• § ψ)

= LX (iY • § ψ + (’1)p • § iY ψ) ’ i[X,Y ] • § ψ ’ (’1)p • § i[X,Y ] ψ

= LX iY • § ψ + iY • § LX ψ + (’1)p LX • § iY ψ+

+ (’1)p • § LX iY ψ ’ i[X,Y ] • § ψ ’ (’1)p • § i[X,Y ] ψ

iY (LX • § ψ + • § LX ψ) = iY LX • § ψ + (’1)p LX • § iY ψ+

+ iY • § LX ψ + (’1)p • § iY LX ψ.

33.18

33.18 33. Di¬erential forms 349

Using again (8), we get the result since the iY for all local vector ¬elds Y together act

point separating on each space of di¬erential forms, in both cases of the convention

(33.2).

(6) follows by summing up the following parts.

(LX0 •)(X1 , . . . , Xk ) = X0 (•(X1 , . . . , Xk ))+

k

(’1)0+j •([X0 , Xj ], X1 , . . . , Xj , . . . , Xk )

+

j=1

(iX0 d•)(X1 , . . . , Xk ) = d•(X0 , . . . , Xk )

k

(’1)i Xi (•(X0 , . . . , Xi , . . . , Xk )) +

=

i=0

(’1)i+j •([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xk ).

+

0¤i<j

k

(’1)i’1 Xi ((iX0 •)(X1 , . . . , Xi , . . . , Xk )) +

(diX0 •)(X1 , . . . , Xk ) =

i=1

(’1)i+j’2 (iX0 •)([Xi , Xj ], X1 , . . . , Xi , . . . , Xj , . . . , Xk )

+

1¤i<j

k

(’1)i Xi (•(X0 , X1 , . . . , Xi , . . . , Xk )) ’

=’

i=1

(’1)i+j •([Xi , Xj ], X0 , X1 , . . . , Xi , . . . , Xj , . . . , Xk ).

’

1¤i<j

(3) We prove the result again by induction on p + q. Suppose that (3) is true for

p+q < k. Then for each local vector ¬eld X we have by (6), (2), 1, and by induction

iX d(• § ψ) = LX (• § ψ) ’ d iX (• § ψ)

= LX • § ψ + • § LX ψ ’ d(iX • § ψ + (’1)p • § iX ψ)

= iX d• § ψ + diX • § ψ + • § iX dψ + • § diX ψ ’ diX • § ψ

’ (’1)p’1 iX • § dψ ’ (’1)p d• § iX ψ ’ • § diX ψ

= iX (d• § ψ + (’1)p • § dψ).

Since X is arbitrary, the result follows.

(4) This follows by a long but straightforward computation directly from the the

global formula (33.12.3), using only the de¬nition of the Lie bracket as a commu-

tator, the Jacobi identity, and cancellation.

(5) dLX = d iX d + ddiX = diX d + iX dd = LX d.

(7) By the (graded) Jacobi identity, by (5), by (6), and by (8) we have [LX , LY ] =

[LX , [iY , d]] = [[LX , iY ], d] + [iY , [LX , d]] = [i[X,Y ] , d] + 0 = L[X,Y ] .

(10) Lf.X • = [if.X , d]• = [f.iX , d]• = f iX d• + d(f iX •) = f iX d• + df § iX • +

f diX • = f LX • + df § iX •.

33.18

350 Chapter VII. Calculus on in¬nite dimensional manifolds 33.20

33.19. Lemma. Let X ∈ X(M ) be a kinematic vector ¬eld which has a local ¬‚ow

FlX . Or more generally, let us suppose that • : R — M ⊃ U ’ M is a smooth

t

mapping such that (t, x) ’ (t, •(t, x) = •t (x)) is a di¬eomorphism U ’ V , where

U and V are open neighborhoods of {0} — M in R — M , and such that •0 = IdM

‚

and ‚t 0 •t = X ∈ X(M ).

k, alt

Then for ω any k-form in HomC ∞ (M,R) (C ∞ (M ← DM ), C ∞ (M, R)) we have

|0 (•t )— ω

‚

= LX ω,

‚t

|0 (FlX )— ω

‚

= LX ω,

t

‚t

X—

= (FlX )— LX ω = LX (FlX )— ω.

‚

‚t (Flt ) ω t t

In particular, for a vector ¬eld X with a local ¬‚ow the Lie derivative LX maps the

spaces C ∞ (Lk (D± M, M — R)) into themselves, for D± = T , D, and D(p) .

alt

Proof. For Yi ∈ C ∞ (M ← D± M ) we have

|0 (ω((•’1 )— Y1 , . . . , (•’1 )— Yk )

( ‚t |0 (•t )— ω)(Y1 , . . . , Yk ) =

‚ ‚

—¦ •t )

t t

‚t

k

ω(Y1 , . . . , ‚t |0 (•’1 )— Yi , . . . , Yk ) + —

‚ ‚

‚t |0 (•t ) (ω(Y1 , . . . , Yp ))

= t

i=1

k

= X(ω(Y1 , . . . , Yk )) ’ ω|U (Y1 , . . . , [X, Yi ], . . . , Yk ),

i=1

where at the end we used (32.15). This proves the ¬rst two assertions.

For the third assertion we proceed as follows

(FlX )— ω |0 (FlX )— (FlX )— ω

d d

=

t t s

dt ds

(FlX )— ds |0 (FlX )— ω

d

= t s

(FlX )— LX ω

= t

X— X— X—

d d

ds |0 (Fls ) (Flt ) ω

dt (Flt ) ω =

LX (FlX )— ω.

= t

We may commute ds |0 with the bounded linear mapping (FlX )— from the space

d

t

of di¬erential forms on U to that of forms on V , where V is open in U such that

FlX (V ) ‚ U for all r ∈ [0, t]. We may ¬nd such open U and V because the

r

c -topology on R — M is the product of the c∞ -topologies, by corollary (4.15).

∞

33.20. Lemma of Poincar´. Let ω ∈ C ∞ (U, Lk+1 (D0 E; F )) be a closed form

±

e alt

i.e., dω = 0, where U is a star-shaped c∞ -open subset of a convenient vector space

E, with values in a convenient vector space F . Here D± may be any of T , D, D(k) ,

etc.

Then ω is exact, i.e. ω = d• where

1

tk ω(tx)(x, v1 , . . . , vk )dt

•(x)(v1 , . . . , vk ) =

0

33.20

33.21 33. Di¬erential forms 351

is a di¬erential form • ∈ C ∞ (U, Lk (D0 E, F )).

±

alt

Proof. We consider µ : R — E ’ E, given by µ(t, x) = µt (x) = tx. Let I ∈ X(E)

be the vector ¬eld I(x) = x, then µ(et , x) = FlI (x). So for (x, t) in a neighborhood

t

k, alt

of U — (0, 1], in HomC ∞ (U,R) (C (U ← D U ), C ∞ (U, R)) we have

∞ ±

d—

(FlI t )— ω = 1 (FlI t )— LI ω by

d

dt µt ω = (33.19)

log log

dt t

1— —

1

= t µt (iI dω + diI ω) = t dµt iI ω.

For X1 , . . . , Xk ∈ D0 E we may compute

( 1 µ— iI ω)x (X1 , . . . , Xk ) = 1 (iI ω)tx (Dx µt .X1 , . . . , Dx µt .Xk )

tt t

= 1 ωtx (tx, Dx µt .X1 , . . . , Dx µt .Xk ) = ωtx (x, Dx µt .X1 , . . . , Dx µt .Xk ).

t

Since Tx (µt ) = t. IdE and D(1) µt = µ—— = t. IdE we can make the last com-

t

putation more explicit if all Xi ∈ E or E . So if k ≥ 0, the k-form 1 µ— iI ω

tt

is de¬ned and smooth in (t, x) for all t ∈ [0, 1] and describes a smooth curve in

C ∞ (U, Lk (D0 E, F )). Clearly, µ— ω = ω and µ— ω = 0, thus

±

1 0

alt

1

µ— ω µ— ω d—

’

ω= = dt µt ωdt

1 0

0

1 1

d( 1 µ— iI ω)dt 1—

= =d t µt iI ωdt = d•.

tt

0 0

Remark. We were unable to prove the Lemma of Poincar´ for modular forms

e

which are given by module homomorphisms, because µ— ω does not make sense in

t

a di¬erentiable way for t = 0. One may ask whether a closed modular di¬eren-

tial form ω ∈ Homk,∞ (M,R) (C ∞ (M ← D± M ), C ∞ (M, R)) already has to be in

alt

C

C ∞ (Lk (D± M, M — R)).

alt

33.21. Review of operations on di¬erential forms.

f—

LX

Space d

C ∞ (M ← Λ— (D M )) “ “ +

C ∞ (M ← Λ— (T M )) “ “ +

Λ— ∞ (M,R) HomC ∞ (M,R) (C ∞ (M ← DM ), C ∞ (M, R)) “ “ di¬

C

Λ— ∞ (M,R) C ∞ (M ← T M ) “ “ +

C

C ∞ (L— (DM, M — R)) ¬‚ow “ +

alt

—, alt

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

∞

+ + di¬

C ∞ (L— (D[1,∞) M, M — R)) ¬‚ow “ +

alt

—, alt

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

∞

+ + di¬

C ∞ (M ← L— (D(1) M, M — R)) ¬‚ow “ +

alt

—, alt

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

∞

¬‚ow ? di¬

C ∞ (M ← L— (T M, M — R)) + + +

alt

—, alt

HomC ∞ (M,R) (X(M ), C ∞ (M, R)) + + di¬

33.21

352 Chapter VII. Calculus on in¬nite dimensional manifolds 33.22

In this table a ˜“™ means that the space is not invariant under the operation on

top of the column, a ˜+™ means that it is invariant, ˜di¬™ means that it is invariant

under f — only for di¬eomorphisms f , and ˜¬‚ow™ means that it is invariant under

LX for all kinematic vector ¬elds X which admit local ¬‚ows.

33.22. Remark. From the table (33.21) we see that for many purposes only one

space of di¬erential forms is fully suited. We will denote from now on by

„¦k (M ) := C ∞ (M ← Lk (T M, M — R))

alt

the space of di¬erential forms, for a smooth manifold M . By (30.1) it carries the

structure of a convenient vector space induced by the closed embedding

C ∞ (U± , Lk (E, R))

„¦k (M ) ’ alt

±

s ’ pr2 —¦ ψ± —¦ (s|U± ),

where (U± , u± : U± ’ E) is a smooth atlas for the manifold M , and where ψ± :=

Lk (T u’1 , R)) is the induced vector bundle chart.

±

alt

Similarly, we denote by

„¦k (M, V ) := C ∞ (M ← Lk (T M, M — V ))

alt

the space of di¬erential forms with values in a convenient vector space V , and by

„¦k (M ; E) := C ∞ (M ← Lk (T M, E))

alt

the space of di¬erential forms with values in a vector bundle p : E ’ M .

Lemma. The space „¦k (M ) is isomorphic as convenient vector space to the closed

linear subspace of C ∞ (T M —M . . . —M T M, R) consisting of all ¬berwise k-linear

alternating smooth functions in the vector bundle structure T M • · · · • T M from

(29.5).

Proof. By (27.17), the space C ∞ (T M —M . . . —M T M, R) carries the initial struc-

ture with respect to the closed linear embedding

C ∞ (T M —M . . . —M T M, R) ’ C ∞ (u± (U± ) — E — . . . — E, R),

±

and C ∞ (u± (U± )—E—. . .—E, R) contains an isomorphic copy of C ∞ (U± , Lk (E, R))

alt

as closed linear subspace by cartesian closedness.

Corollary. All the important mappings are smooth:

d : „¦k (M ) ’ „¦k+1 (M )

i : X(M ) — „¦k (M ) ’ „¦k’1 (M )

L : X(M ) — „¦k (M ) ’ „¦k (M )

f — : „¦k (M ) ’ „¦k (N )

33.22

34.1 34. De Rham cohomology 353

where f : N ’ M is a smooth mapping. The last mappings is even smooth consid-

ered as mapping (f, ω) ’ f — ω, C ∞ (N, M ) — „¦k (M ) ’ „¦k (N ).

Recall once more the formulas for ω ∈ „¦k (M ) and Xi ∈ X(M ), from (33.12.3),

(33.10), (33.17) :

k

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

(dω)(x)(X0 , . . . , Xk ) =

i=0

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

+

i<j

(iX •)(X1 , . . . , Xk’1 ) = •(X, X1 , . . . , Xk’1 ),

k

(LX ω)(X1 , . . . , Xk ) = X(ω(X1 , . . . , Xk )) ’ ω(X1 , . . . , [X, Xi ], . . . , Xk ).

i=1

Proof. For d we use the local formula (33.12.1), smoothness of i is obvious, and

for the Lie derivative we may use formula (33.18.6). The pullback mapping f — is

induced from T f — . . . — T f .

34. De Rham Cohomology

Section (33) provides us with several graded commutative di¬erential algebras con-

sisting of various kinds of di¬erential forms for which we can de¬ne De Rham

cohomology, namely all those from the list (33.21) which have + in the d-column.

But among these only C ∞ (L— (T M, M — R)) behaves functorially for all smooth

alt

mappings; the others are only functors over categories of manifolds where the mor-

phisms are just the local di¬eomorphisms. So we treat here cohomology only for

these di¬erential forms.

34.1. De Rham cohomology. Recall that for a smooth manifold M we have

denoted

„¦k (M ) := C ∞ (Lk (T M, M — R)).

alt

We now consider the graded algebra „¦(M ) = k≥0 „¦k (M ) of all di¬erential forms

on M . Then the space Z(M ) := M ) := {ω ∈ „¦(M ) : dω = 0} of closed forms is

a graded subalgebra of „¦ (i. e. it is a subalgebra, and „¦k (M ) © Z(M ) = Z k (M )),

and the space B(M ) := {d• : • ∈ „¦(M )} of exact forms is a graded ideal in

Z(M ). This follows directly from d2 = 0 and the derivation property d(• § ψ) =

d• § ψ + (’1)deg • • § dψ of the exterior derivative.

De¬nition. The algebra

{ω ∈ „¦(M ) : dω = 0}

Z(M )

H — (M ) := =

{d• : • ∈ „¦(M )}

B(M )

34.1

354 Chapter VII. Calculus on in¬nite dimensional manifolds 34.2

is called the De Rham cohomology algebra of the manifold M . It is graded by

ker(d : „¦k (M ) ’ „¦k+1 (M ))

— k

H (M ) = H (M ) = .

im(d : „¦k’1 (M ) ’ „¦k (M ))

k≥0 k≥0

If f : M ’ N is a smooth mapping between manifolds then f — : „¦(N ) ’ „¦(M )

is a homomorphism of graded algebras by (33.9), which satis¬es d —¦ f — = f — —¦ d

by (33.15). Thus, f — induces an algebra homomorphism which we also call f — :

H — (N ) ’ H — (M ). Obviously, each H k is a contravariant functor from the category

of smooth manifolds and smooth mappings into the category of real vector spaces.

34.2. Lemma. Let f , g : M ’ N be smooth mappings between manifolds which

are C ∞ -homotopic, i.e., there exists h ∈ C ∞ (R — M, N ) with h(0, x) = f (x) and

h(1, x) = g(x). Then f and g induce the same mapping in cohomology f — = g — :

H — (N ) ’ H — (M ).

Remark. f , g ∈ C ∞ (M, N ) are called homotopic if there exists a continuous

mapping h : [0, 1] — M ’ N with h(0, x) = f (x) and h(1, x) = g(x). For ¬nite

dimensional manifolds this apparently looser relation in fact coincides with the

relation of C ∞ -homotopy. We sketch a proof of this statement: let • : R ’ [0, 1]

be a smooth function with •(t) = 0 for t ¤ 1/4, •(t) = 1 for t ≥ 3/4, and •

¯ ¯

monotone in between. Then consider h : R — M ’ N , given by h(t, x) = h(•(t), x).

¯ ˜

Now we may approximate h by smooth functions h : R — M ’ N without changing

it on (’∞, 1/8) — M where it equals f , and on (7/8, ∞) — M , where it equals g.

This is done chartwise by convolution with a smooth function with small support

on Rm . See [Br¨cker, J¨nich, 1973] for a careful presentation of the approximation.

o a

It is an open problem to extend this to some in¬nite dimensional manifolds.

The lemma of Poincar´ (33.20) is an immediate consequence of this result.

e

Proof. For ω ∈ „¦k (M ) we have h— ω ∈ „¦k (R — M ). We consider the insertion

operator inst : M ’ R — M , given by inst (x) = (t, x). For • ∈ „¦k (R — M ) we then

have a smooth curve t ’ ins— • in „¦k (M ).

t

Consider the integral operator I0 : „¦k (R — M ) ’ „¦k (M ) given by I0 (•) :=

1 1

1

ins— • dt. Let T := ‚t ∈ C ∞ (R — M ← T (R — M )) be the unit vector ¬eld

‚

t

0

in direction R.

We have inst+s = FlT —¦ inss for s, t ∈ R, so

t

— —

— T—

T

‚ ‚ ‚

‚t 0 (Flt —¦ inss ) • = ‚t 0 inss (Flt ) •

‚s inss • =

ins— ‚t 0 (FlT )— • = (inss )— LT •

‚

= by (33.19).

s t

We have used that (inss )— : „¦k (R — M ) ’ „¦k (M ) is linear and continuous, and so

34.2

34.4 34. De Rham cohomology 355

one may di¬erentiate through it by the chain rule. Then we have in turn

1 1

ins— • dt d ins— • dt

1

d I0 •=d =

t t

0 0

1

ins— d• dt = I0 d •

1

= by (33.15).

t

0

1 1

(ins— ins— )• —

ins— LT • dt

‚

’ = ‚t inst • dt =

1 0 t

0 0

1 1

I0 LT •

= = I0 (d iT + iT d)• by (33.18.6).

¯

Now we de¬ne the homotopy operator h := I0 —¦ iT —¦ h— : „¦k (M ) ’ „¦k’1 (M ). Then

1

we get

g — ’ f — = (h —¦ ins1 )— ’ (h —¦ ins0 )— = (ins— ’ ins— ) —¦ h—

1 0

¯¯

= (d —¦ I 1 —¦ iT + I 1 —¦ iT —¦ d) —¦ h— = d —¦ h ’ h —¦ d,