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9
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

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