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0, . . . , c(k’1) (0) = 0, then c(k) (0) is a well de¬ned tangent vector in Tx M , which is
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

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