¯ ¯ ¯ ¯

X0 (f g) = X0 (f )g(x) + f (x)X0 (g) + s(X(f )).s(X(g)) ’ s(X(f ).X(g)).

¯ ¯

Note that (2) is an a¬ne equation in X0 for ¬xed X. By induction, the X ∈

¯

DA/N0 (U )x form a smooth manifold, and the ¬ber over a ¬xed X consists of all

¯

X = X0 + s —¦ X with X0 in the closed a¬ne subspace described by (2), whose

model vector space is the space of all derivations at x. If we were able to ¬nd

a (local) section DA/N0 (U ) ’ DA (U ) and if these sections ¬tted together nicely

we could then conclude that DA (U ) was the total space of a smooth a¬ne bundle

over DA/N0 (U ), so it would be smooth. But this translates to a lifting problem as

follows: A homomorphism C ∞ (U, R) ’ A/N0 has to be lifted in a ˜natural way™ to

C ∞ (U, R) ’ A. But we know that in general C ∞ (U, R) is not a free C ∞ -algebra,

see (31.16) for comparison.

31.10. The basic facts from the theory of Weil functors are completed by the

following assertion.

31.10

312 Chapter VI. In¬nite dimensional manifolds 31.12

Proposition. Given two Weil algebras A and B, the composed functor TA —¦ TB is

a Weil functor generated by the tensor product A — B.

Proof. For a convenient vector space E we have TA (TB E) = A — B — E, and this

is compatible with the action of smooth mappings, by (31.5).

Corollary. There is a canonical natural equivalence TA —¦ TB ∼ TB —¦ TA generated

=

by the exchange algebra isomorphism A — B ∼ B — A.

=

31.11. Examples. Let A be the algebra R.1+R.δ with δ 2 = 0. Then TA M = T M ,

the tangent bundle, and consequently we get TA—A M = T 2 M , the second tangent

bundle.

31.12. Weil functors and Lie groups. We have (compare (38.10)) that the

tangent bundle T G of a Lie group G is again a Lie group, the semidirect product

g G of G with its Lie algebra g.

Now let A be a Weil algebra, and let TA be its Weil functor. Then in the notation of

(36.1) the space TA (G) is also a Lie group with multiplication TA (µ) and inversion

TA (ν). By the properties (31.7), of the Weil functor TA we have a surjective homo-

morphism πA : TA G ’ G of Lie groups. Following the analogy with the tangent

bundle, for a ∈ G we will denote its ¬ber over a by (TA )a G ‚ TA G, likewise for

mappings. With this notation we have the following commutative diagram, where

we assume that G is a regular Lie group (38.4):

w g—A

g—N

w (T wT wg w0

0 A )0 g Ag

u u u

TA expG expG

(TA )0 expG

w (T wT wG we

πA

e A )e G AG

The structural mappings (Lie bracket, exponential mapping, evolution operator,

adjoint action) are determined by multiplication and inversion. Thus, their images

under the Weil functor TA are the same structural mappings. But note that the

canonical ¬‚ip mappings have to be inserted like follows. So for example

κ

g — A ∼ TA g = TA (Te G) ’ Te (TA G)

’

=

is the Lie algebra of TA G, and the Lie bracket is just TA ([ , ]). Since the bracket

is bilinear, the description of (31.5) implies that [X — a, Y — b]TA g = [X, Y ]g —

ab. Also TA expG = expTA G . If expG is a di¬eomorphism near 0, (TA )0 (expG ) :

(TA )0 g ’ (TA )e G is also a di¬eomorphism near 0, since TA is local. The natural

transformation 0G : G ’ TA G is a homomorphism which splits the bottom row

of the diagram, so TA G is the semidirect product (TA )0 g G via the mapping

TA ρ : (u, g) ’ TA (ρg )(u). So from (38.9) we may conclude that TA G is also a

31.12

31.14 31. Product preserving functors on manifolds 313

regular Lie group, if G is regular. If ω G : T G ’ Te G is the Maurer Cartan form of

G (i.e., the left logarithmic derivative of IdG ) then

κ0 —¦ T A ω G —¦ κ : T T A G ∼ T A T G ’ T A T e G ∼ T e T A G

= =

is the Maurer Cartan form of TA G.

Product preserving functors from ¬nite

dimensional manifolds to in¬nite dimensional ones

31.13. Product preserving functors. Let Mf¬n denote the category of all

¬nite dimensional separable Hausdor¬ smooth manifolds, with smooth mappings

as morphisms. Let F : Mf¬n ’ Mf be a functor which preserves products in the

following sense: The diagram

F (pr ) F (pr )

F (M1 ) ← ’1’ F (M1 — M2 ) ’ ’ 2 F (M2 )

’’ ’’ ’

is always a product diagram.

Then F (point) = point, by the following argument:

u (pr ) F (point u— point) F (pr )T (point)

∼ wF

F

£

∼

1

=R

2

F (point)

RRR

f

=

RRR f

f 1 2

point

Each of f1 , f , and f2 determines each other uniquely, thus there is only one mapping

f1 : point ’ F (point), so the space F (point) is a single point.

We also require that F has the following two properties:

(1) The map on morphisms F : C ∞ (Rn , R) ’ C ∞ (F (Rn ), F (R)) is smooth,

where we regard C ∞ (F (Rn ), F (R)) as Fr¨licher space, see section (23).

o

Equivalently, the associated map C ∞ (Rn , R) — F (Rn ) ’ F (R) is smooth.

(2) There is a natural transformation π : F ’ Id such that for each M the

mapping πM : F (M ) ’ M is a ¬ber bundle, and for an open submanifold

U ‚ M the mapping F (incl) : F (U ) ’ F (M ) is a pullback.

31.14. C ∞ -algebras. An R-algebra is a commutative ring A with unit together

with a ring homomorphism R ’ A. Then every map p : Rn ’ Rm which is given

by an m-tuple of real polynomials (p1 , . . . , pm ) can be interpreted as a mapping

A(p) : An ’ Am in such a way that projections, composition, and identity are

preserved, by just evaluating each polynomial pi on an n-tuple (a1 , . . . , an ) ∈ An .

Compare with (17.1).

A C ∞ -algebra A is a real algebra in which we can moreover interpret all smooth

mappings f : Rn ’ Rm . There is a corresponding map A(f ) : An ’ Am , and

again projections, composition, and the identity mapping are preserved.

31.14

314 Chapter VI. In¬nite dimensional manifolds 31.15

More precisely, a C ∞ -algebra A is a product preserving functor from the category

C ∞ to the category of sets, where C ∞ has as objects all spaces Rn , n ≥ 0, and all

smooth mappings between them as arrows. Morphisms between C ∞ -algebras are

then natural transformations: they correspond to those algebra homomorphisms

which preserve the interpretation of smooth mappings.

Let us explain how one gets the algebra structure from this interpretation. Since A

is product preserving, we have A(point) = point. All the laws for a commutative

ring with unit can be formulated by commutative diagrams of mappings between

products of the ring and the point. We do this for the ring R and apply the product

preserving functor A to all these diagrams, so we get the laws for the commutative

ring A(R) with unit A(1) with the exception of A(0) = A(1) which we will check

later for the case A(R) = point. Addition is given by A(+) and multiplication by

A(m). For » ∈ R the mapping A(m» ) : A(R) ’ A(R) equals multiplication with

the element A(») ∈ A(R), since the following diagram commutes:

eeeeee

eeeeg )

A(R)

e

A(m

u

»

∼

=

w A(R) — A(R) Aw A(R)

Id —A(»)

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

space of derivations on the algebra C ∞ (M, R) of globally de¬ned smooth functions

coincides with the derivations of the sheaf. Thus we shall follow the convention,

that either the manifolds in question are smoothly regular, or that (as de¬ned

above) Der means the space of derivations of the corresponding sheaf also denoted

by C ∞ (M, R).

32.2. Lemma. On any manifold M the operational vector ¬elds correspond exactly

to the smooth sections of the operational tangent bundle. Moreover we have an

isomorphism of convenient vector spaces Der(C ∞ (M, R)) ∼ C ∞ (M ← DM ).

=

Proof. Every smooth section X ∈ C ∞ (M ← DM ) de¬nes an operational vector

¬eld by ‚U (f )(x) := X(x)(germx f ) = pr2 (Df (X(x))) for f ∈ C ∞ (U, R) and x ∈ U .

We have that ‚U (f ) = pr2 —¦Df —¦ X = df —¦ X ∈ C ∞ (U, R) by (28.15). Then ‚U

is obviously a derivation, since df (Xx ) = Xx (f ) by (28.15). The linear mapping

‚U : C ∞ (U, R) ’ C ∞ (U, R) is bounded if and only if evx —¦‚U : C ∞ (U, R) ’ R is

bounded, by the smooth uniform boundedness principle (5.26), and this is true by

(28.15), since (evx —¦X)(f ) = df (Xx ).

Moreover, the mapping

C ∞ (M ← DM ) ’ Der(C ∞ (M, R)) ’ L(C ∞ (U, R), C ∞ (U, R))

U

given by X ’ (‚U )U is linear and bounded, since by the uniform boundedness

principle (5.26) this is equivalent to the boundedness of X ’ ‚U (f )(x) = df (Xx )

for all open U ⊆ M , f ∈ C ∞ (U, R) and x ∈ X.

Now let conversely ‚ be an operational vector ¬eld on M . Then the family evx —¦‚U :

C ∞ (U, R) ’ R, where U runs through all open neighborhoods of x, de¬nes a unique

bounded derivation Xx : C ∞ (M ⊇ {x}, R) ’ R, i.e. an element of Dx M . We have

to show that x ’ Xx is smooth, which is a local question, so we assume that M is

a c∞ -open subset of a convenient vector space E. The mapping

X

M ’ DM ∼ M — D0 E ⊆ M — L(C ∞ (U, R), R)

’ =

U

is smooth if and only if for every neighborhood U of 0 in E the component M ’

L(C ∞ (U, R), R), given by ‚ ’ Xx (f ( ’x)) = ‚Ux (f ( ’x))(x) is smooth, where

Ux := U + x. By the smooth uniform boundedness principle (5.18) this is the case

if and only if its composition with evf is smooth for all f ∈ C ∞ (U, R). If t ’ x(t)

32.2

32.5 32. Vector ¬elds 323

is a smooth curve in M ⊆ E, then there is a δ > 0 and an open neighborhood W

of x(0) in M such that W ⊆ U + x(t) for all |t| < δ and hence Xx(t) (f ( ’x(t))) =

‚W (f ( ’x(t)))(x(t)), which is by the exponential law smooth in t.

Moreover, the mapping Der(C ∞ (M, R)) ’ C ∞ (M ← DM ) given by ‚ ’ X is

linear and bounded, since by the uniform boundedness principle in proposition

(30.1) this is equivalent to the boundedness of ‚ ’ Xx ∈ Dx M ’ U C ∞ (U, R)

for all x ∈ M , i.e. to that of ‚ ’ Xx (f ) = ‚U (f )(x) for all open neighborhoods U

of x and f ∈ C ∞ (U, R), which is obviously true.

32.3. Lemma. There is a natural embedding of convenient vector spaces

X(M ) = C ∞ (M ← T M ) ’ C ∞ (M ← DM ) ∼ Der(C ∞ (M, R)).

=

Proof. Since T M is a closed subbundle of DM this is obviously true.

32.4. Lemma. Let M be a smoothly regular manifold.

Then each bounded derivation X : C ∞ (M, R) ’ C ∞ (M, R) is already an opera-

tional vector ¬eld. Moreover, we have an isomorphism

C ∞ (M ← DM ) ∼ Der(C ∞ (M, R), C ∞ (M, R))

=

of convenient vector spaces.

Proof. Let ‚ be a bounded derivation of the algebra C ∞ (M, R). If f ∈ C ∞ (M, R)

vanishes on an open subset U ‚ M then also ‚(f ): For x ∈ U we take a bump

function gx,U ∈ C ∞ (M, R) at x, i.e. gx,U = 1 near x and supp(gx,U ) ‚ U . Then

‚(f ) = ‚((1 ’ gx,U )f ) = ‚(1 ’ gx,U )f + (1 ’ gx,U )‚(f ), and both summands are

zero near x. So ‚(f )|U = 0.

Now let f ∈ C ∞ (U, R) for a c∞ -open subset U of M . We have to show that we can

de¬ne ‚U (f ) ∈ C ∞ (U, R) in a unique manner. For x ∈ U let gx,U ∈ C ∞ (M, R) be

a bump function as before. Then gx,U f ∈ C ∞ (M, R), and ‚(gx,U f ) makes sense.

By the argument above, ‚(gf ) near x is independent of the choice of g. So let

‚U (f )(x) := ‚(gx,U f )(x). It has all the required properties since the topology on

C ∞ (U, R) is initial with respect to all mappings f ’ gx,U f for x ∈ U .

This mapping ‚ ’ ‚U is bounded, since by the uniform boundedness principles

(5.18) and (5.26) this is equivalent with the boundedness of ‚ ’ ‚U (f )(x) :=

‚(gx,U f )(x) for all f ∈ C ∞ (U, R) and all x ∈ U

32.5. The operational Lie bracket. Recall that operational vector ¬elds are

the bounded derivations of the sheaf C ∞ ( , R), see (32.1). This is a convenient

vector space by (32.2) and (30.1).

If X, Y are two operational vector ¬elds on M , then the mapping f ’ X(Y (f )) ’

Y (X(f )) is also a bounded derivation of the sheaf C ∞ ( , R), as a simple compu-

tation shows. We denote it by [X, Y ] ∈ Der(C ∞ ( , R)) ∼ C ∞ (M ← DM ).

=

32.5

324 Chapter VII. Calculus on in¬nite dimensional manifolds 32.6

The R-bilinear mapping

] : C ∞ (M ← DM ) — C ∞ (M ← DM ) ’ C ∞ (M ← DM )

[,

is called the Lie bracket. Note also that C ∞ (M ← DM ) is a module over the

algebra C ∞ (M, R) by pointwise multiplication (f, X) ’ f X, which is bounded.

Theorem. The Lie bracket [ , ] : C ∞ (M ← DM ) — C ∞ (M ← DM ) ’

C ∞ (M ← DM ) has the following properties:

[X, Y ] = ’[Y, X],

[X, [Y, Z]] = [[X, Y ], Z] + [Y, [X, Z]], the Jacobi identity,

[f X, Y ] = f [X, Y ] ’ (Y f )X,

[X, f Y ] = f [X, Y ] + (Xf )Y.

The form of the Jacobi identity we have chosen says that ad(X) = [X, ] is a

derivation for the Lie algebra (C ∞ (M ← DM ), [ , ]).

Proof. All these properties can be checked easily for the commutator [X, Y ] =

X —¦ Y ’ Y —¦ X in the space of bounded derivations of the algebra C ∞ (U, R).

32.6. Lemma. Let b : E1 — . . . — Ek ’ R be a bounded multilinear mapping on a

product of convenient vector spaces. Let f ∈ C ∞ (E, R), let fi : E ’ Ei be smooth

(1)

mappings, and let Xx ∈ E = Dx E.

Then we have

= df (x)—— .Xx

Xx (f ) = Xx , df (x) E

Xx (b—¦(f1 , . . . , fk )) = d(b —¦ (f1 , . . . , fk ))(x)—— .Xx

, fi+1 (x), . . . , fk (x))—— .dfi (x)—— .Xx

= b(f1 (x), . . . , fi’1 (x),

1¤i¤k

dfi (x)—— .Xx , b(f1 (x), . . . , fi’1 (x),

= , fi+1 (x), . . . , fk (x)) Ei .

1¤i¤k

If B : E1 — . . . — Ek ’ F is a vector valued bounded multilinear mapping, and if

g : E ’ F is a smooth mapping, then we have

Dx g.Xx = dg(x)—— .Xx ∈ F

(1)

(1)

Dx (B —¦ (f1 , . . . , fk )).Xx =

(1)

, fi+1 (x), . . . )—— .dfi (x)—— .Xx ∈ DB(f1 (x),...,fk (x)) F.

= B(. . . , fi’1 (x),

1¤i¤k

H : H — H ’ R is the duality pairing for any convenient vector space

Here ,

H. We will further denote by ιH : H ’ H the canonical embedding into the

bidual space.

32.6

32.6 32. Vector ¬elds 325

Proof. The ¬rst equation is immediate.

We have

k

d(b —¦ (f1 , . . . , fk ))(x) = , fj+1 (x), . . . , fk (x)) —¦ dfj (x)

b(f1 (x), . . . , fj’1 (x),

j=1

k

dfj (x)— b(f1 (x), . . . , fj’1 (x),

= , fj+1 (x), . . . , fk (x)) .

j=1

(1)

Thus for Xx ∈ Dx E we have

Xx (b—¦(f1 , . . . , fk )) = Xx d(b —¦ (f1 , . . . , fk ))(x)

k

dfj (x)— b(f1 (x), . . . , fj’1 (x),

= Xx , fj+1 (x), . . . , fk (x))

j=1

k

Xx dfj (x)— b(f1 (x), . . . , fj’1 (x),

= , fj+1 (x), . . . , fk (x))

j=1

k

dfj (x)—— (Xx ) b(f1 (x), . . . , fj’1 (x),

= , fj+1 (x), . . . , fk (x)) .

j=1

For the second assertion we choose a test germ

h ∈ C ∞ (F ⊇ {B(f1 (x), . . . , fk (x))}, R)

and proceed as follows:

(1)

(Dx g.Xx )(h) = Xx (h —¦ g) = Xx , d(h —¦ g)(x) E

—

= Xx , dh(g(x)) —¦ dg(x) = Xx , dg(x) .dh(g(x))

E E

= dg(x)—— .Xx , dh(g(x)) = (dg(x)—— .Xx )(h).

E

(1)

(Dx (B —¦ (f1 , . . . , fk ))Xx )(h) = Xx (h —¦ B —¦ (f1 , . . . , fk ))

= d(h —¦ B —¦ (f1 , . . . , fk ))(x)—— .Xx

——

k

dh(B(f1 (x), . . . )) —¦ , fi+1 (x), . . . ) —¦ dfi (x)

= B(. . . , fi’1 (x), .Xx

i=1

k

= dh(B(f1 (x), . . . ))—— . , fi+1 (x), . . . )—— .dfi (x)—— .Xx

B(. . . , fi’1 (x),

i=1

k

, fi+1 (x), . . . )—— .dfi (x)—— .Xx

= B(. . . , fi’1 (x), (h).

i=1 B(f1 (x),... )

32.6

326 Chapter VII. Calculus on in¬nite dimensional manifolds 32.7

32.7. The Lie bracket of operational vector ¬elds of order 1. One could

hope that the Lie bracket restricts to a Lie bracket on C ∞ (D(1) M ). But this is

not the case. We will see that for a c∞ -open set U in a convenient vector space E

and for X, Y ∈ C ∞ (U, E ) the bracket [X, Y ] has also components of order 2, in

general.

For a bounded linear mapping : F ’ G the transposed mapping t : G ’ F

is given by t := — —¦ ιG , where ιG : G ’ G is the canonical embedding into the

H : H — H ’ R is the duality pairing, then this may also be

bidual. If ,

described by (x), y G = t (y), x F .

For X, Y ∈ C ∞ (U, E ), for f ∈ C ∞ (U, E) and for x ∈ U we get:

X(f )(x) = Xx (f ) = Xx (df (x))

X(f ) = ev —¦(X, df )

Y (X(f ))(x) = Yx (X(f )) = Yx (ev —¦(X, df ))

= Yx dX(x)— ev( , df (x)) + d(df )(x)— ev(X(x), )

= Yx dX(x)— ι(df (x)) + d(df )(x)— (Xx )

= Yx ι(df (x)) —¦ dX(x) + Xx —¦ d(df )(x)

= Yx dX(x)t df (x) + Xx —¦ d(df )(x)

= Yx —¦ dX(x)t df (x) + Yx Xx —¦ d(df )(x) .

Here we used the equation:

ι(y) —¦ T = T t (y) for y ∈ F, T ∈ L(E, F ),

which is true since

ι(y) —¦ T (x) = ι(y)(T (x)) = T (x)(y) = T t (y)(x).

Note that for the symmetric bilinear form b := d(df )(x)§ : E — E ’ R a canonical

extension to a bilinear form ˜ on E is given by

b

˜ x , Yx ) := Xx (Yx —¦ b∨ )

b(X

However, this extension is not symmetric as the following remark shows: Let b :=

ev : E — E ’ R. Then ˜ : E — E ’ R is given by

b

˜

b(X, Y ) := X(Y —¦ b∨ ) = X(Y —¦ Id) = X(Y ) = ιE (Y )(X)

For b := ev —¦ ¬‚ip : E — E ’ R we have that ˜ : E — E ’ R is given by

b

˜ X) := Y (X —¦ b∨ ) = Y (X —¦ ιE ) = Y (ι— (X)) = (Y —¦ ι— )(X) = (ιE )—— (Y )(X).

b(Y, E E

Thus, ˜ is not symmetric in general, since ker(ι—— ’ ιE ) = ιE (E), at least for

b E

Banach spaces, see [Cigler, Losert, Michor, 1979, 1.15], applied to ιE .

32.7

32.8 32. Vector ¬elds 327

Lemma. For X ∈ C ∞ (T M ) and Y ∈ C ∞ (D(1) M ) we have [X, Y ] ∈ C ∞ (D(1) M ),

and the bracket is given by the following local formula for M = U , a c∞ -open subset

in a convenient vector space E:

[X, Y ](x) = Y (x) —¦ dX(x)— ’ dY (x).X(x) ∈ E .

Proof. From the computation above we get:

Y (X(f ))(x) = (d(ιE —¦ X)(x)t )— .Y (x), df (x) + d(df )(x)—— .Y (x), ιE .X(x)

E E

+ Y (x), d(df )(x)— .ιE .X(x)

= Y (x), (ιE —¦ dX(x))t .df (x) E E

= Y (x), dX(x)— .df (x) + Y (x), d(df )(x)t .X(x)

E E

= Y (x) —¦ dX(x)— , df (x) + Y (x), d(df )(x)t .X(x)

E E

X(Y (f ))(x) = (dY (x)t )— .ιE .X(x), df (x) + d(df )(x)—— .ιE .X(x), Y (x)

E E

+ ιE .X(x), d(df )(x)— .Y (x)

= ιE .X(x), dY (x)t .df (x) E E

+ d(df )(x)— .Y (x), X(x)

= dY (x)t .df (x), X(x) E E

= dY (x).X(x), df (x) + Y (x), d(df )(x).X(x)

E E

Since d(df )(x) : E ’ E is symmetric in the sense that d(df )(x)t = d(df )(x), the

result follows.

32.8. Theorem. The Lie bracket restricts to the following mappings between split-

ting subspaces

] : C ∞ (M ← D(k) M ) — C ∞ (M ← D( ) M ) ’ C ∞ (M ← D(k+ ) M ).

[ ,

The spaces X(M ) = C ∞ (M ← T M ) and C ∞ (D[1,∞) M ) := C ∞ (M ←

1¤i<∞

D(i) M ) are sub Lie algebras of C ∞ (M ← DM ).

] maps C ∞ (M ← D( ) M ) into

If X ∈ X(M ) is a kinematic vector ¬eld, then [X,

itself.

This suggests to introduce the notation D(0) := T , but here it does not indicate

the order of di¬erentiation present in the tangent vector.

Proof. All assertions can be checked locally, so we may assume that M = U is

open in a convenient vector space E.

We prove ¬rst that the kinematic vector ¬elds form a Lie subalgebra. For X,

Y ∈ C ∞ (U, E) we have then for the vector ¬eld ‚X |x (f ) = df (x)(X(x)), compare

the notation set up in (28.2)

[‚X , ‚Y ](f ) = ‚X (‚Y (f )) ’ ‚Y (‚X (f ))

= d(df.Y ).X ’ d(df.X).Y

= d2 f.(X, Y ) + df.(dY.X) ’ d2 f.(Y, X) ’ df.(dX.Y )

= ‚dY.X’dX.Y f.

32.8

328 Chapter VII. Calculus on in¬nite dimensional manifolds 32.9

k

Let ‚X ∈ C ∞ (U ← D(k) U ) for X = i=1 X [i] , where X [i] ∈ C ∞ (U, Li (E; R) )

sym

∞

vanishes on decomposable forms. Similarly, let ‚Y ∈ C (U ← D( ) U ), and

suppose that f : (U, x) ’ R is a (k + )-¬‚at germ at x. Since ‚Y (f )(y) =

1i

[i]

i=1 Y (y)( i! d f (y)) the germ ‚Y (f ) is still k-¬‚at at x, so ‚X (‚Y (f ))(x) = 0.

Thus, [‚X , ‚Y ](f )(x) = ‚X (‚Y (f ))(x) ’ ‚Y (‚X (f ))(x) = 0, and we conclude that

[‚X , ‚Y ] ∈ C ∞ (U ← D(k+ ) U ).

Now we suppose that X ∈ C ∞ (U, E) and Y ∈ C ∞ (U, Lsym (E; R) ). Let f :

(U, x) ’ R be an -¬‚at germ at x. Then we have

1

‚Y (‚X (f ))(x) = Y (x) !d df, X E (x)

’k

sym dk (df )(x), d

1

= Y (x) X(x) E

! k

k=0

1

= Y (x) ! d (df )(x), X(x) E + 0

(x), 1! d1+ f (x)( , X(x)) Lsym (E;R)

=Y

‚X (‚Y (f ))(x) = ‚X(x) Y, 1! d f Lsym (E;R)

= d Y, 1! d f Lsym (E;R) (x).X(x)

= dY (x).X(x), 1! d f (x) + Y (x), 1! d(d f )(x).X(x)

Lsym (E;R) Lsym (E;R)

+1

= 0 + Y (x), 1! d f (x)(X(x), ) Lsym (E;R)

So [‚X , ‚Y ](f )(x) = 0.

Remark. In the notation of (28.2) we have shown that on a convenient vector

space we have

k+

] : C ∞ (E ← D[k] E) — C ∞ (E ← D[ ] E) ’ C ∞ (E ← D[i] E).

[,

i=min(k, )

Thus, the space C ∞ (E ← D[k,∞) E) := k¤i<∞ C ∞ (E ← D[i] E) for k ≥ 1 is a sub

Lie algebra. The (possibly larger) space C ∞ (D[k,∞] E) of all operational tangent

¬elds which vanish on all polynomials of degree less than k is obviously a sub Lie

algebra. But beware, none of these spaces of vector ¬elds is invariant under the

action of di¬eomorphisms.

32.9. f -related vector ¬elds. Let D± be one of the following functors D, D(k) ,

T . If f : M ’ M is a di¬eomorphism, then for any vector ¬eld X ∈ C ∞ (M ←

D± M ) the mapping D± f ’1 —¦ X —¦ f is also a vector ¬eld, which we will denote by

f — X. Analogously, we put f— X := D± f —¦ X —¦ f ’1 = (f ’1 )— X.

But if f : M ’ N is a smooth mapping and Y ∈ C ∞ (N ← D± N ) is a vector

¬eld there may or may not exist a vector ¬eld X ∈ C ∞ (M ← D± M ) such that the

following diagram commutes:

wu

u

D± f

±

D± N

DM

(1) X Y

w N.

f

M

32.9

32.12 32. Vector ¬elds 329

De¬nition. Let f : M ’ N be a smooth mapping. Two vector ¬elds X ∈

C ∞ (M ← D± M ) and Y ∈ C ∞ (N ← D± N ) are called f -related, if D± f —¦ X = Y —¦ f

holds, i.e. if diagram (1) commutes.

32.10. Lemma. Let Xi ∈ C ∞ (M ← DM ) and Yi ∈ C ∞ (N ← DN ) be vector

¬elds for i = 1, 2, and let f : M ’ N be smooth. If Xi and Yi are f -related for

i = 1, 2, then also »1 X1 + »2 X2 and »1 Y1 + »2 Y2 are f -related, and also [X1 , X2 ]

and [Y1 , Y2 ] are f -related.

Proof. The ¬rst assertion is immediate. To prove the second we choose h ∈

C ∞ (N, R), and we view each vector ¬eld as operational. Then by assumption

we have Df —¦ Xi = Yi —¦ f , thus:

(Xi (h —¦ f ))(x) = Xi (x)(h —¦ f ) = (Dx f.Xi (x))(h) =

= (Df —¦ Xi )(x)(h) = (Yi —¦ f )(x)(h) = Yi (f (x))(h) = (Yi (h))(f (x)),

so Xi (h —¦ f ) = (Yi (h)) —¦ f , and we may continue:

[X1 , X2 ](h —¦ f ) = X1 (X2 (h —¦ f )) ’ X2 (X1 (h —¦ f )) =

= X1 (Y2 (h) —¦ f ) ’ X2 (Y1 (h) —¦ f ) =

= Y1 (Y2 (h)) —¦ f ’ Y2 (Y1 (h)) —¦ f = [Y1 , Y2 ](h) —¦ f.

But this means Df —¦ [X1 , X2 ] = [Y1 , Y2 ] —¦ f .

32.11. Corollary. Let D± be one of the following functors D, D(k) , T . Let

f : M ’ N be a local di¬eomorphism so that (Tx f )’1 makes sense for each x ∈ M .

Then for Y ∈ C ∞ (N ← D± N ) a vector ¬eld f — Y ∈ C ∞ (M ← D± M ) is de¬ned

by (f — Y )(x) = (Tx f )’1 .Y (f (x)), and the linear mapping f — : C ∞ (N ← Dβ N ) ’

C ∞ (M ← Dβ M ) is a Lie algebra homomorphism, i.e. f — [Y1 , Y2 ] = [f — Y1 , f — Y2 ],

where Dβ is one of D, T , D[1,∞) .

32.12. Integral curves. Let c : J ’ M be a smooth curve in a manifold M

de¬ned on an interval J. It will be called an integral curve or ¬‚ow line of a kinematic

vector ¬eld X ∈ X(M ) if c (t) = X(c(t)) holds for all t ∈ J.

For a given kinematic vector ¬eld integral curves need not exist locally, and if they

exist they need not be unique for a given initial value. This is due to the fact that

the classical results on existence and uniqueness of solutions of equations like the

inverse function theorem, the implicit function theorem, and the Picard-Lindel¨f o

theorem on ordinary di¬erential equations can be deduced essentially from one

another, and all depend on Banach™s ¬xed point theorem. Beyond Banach spaces

these proofs do not work any more, since the reduction does no longer lead to a

contraction on a metrizable space. We are now going to give examples, which show

that almost everything that might fail indeed fails.

32.12

330 Chapter VII. Calculus on in¬nite dimensional manifolds 32.12

Example 1. Let E := s be the Fr´chet space of rapidly decreasing sequences.

e

Note that s = C ∞ (S 1 , R) by the theory of Fourier series. Consider the continuous

linear operator T : E ’ E given by T (x0 , x1 , x2 , . . . ) := (0, 12 x1 , 22 x2 , 32 x3 , . . . ).

The ordinary linear di¬erential equation x (t) = T (x(t)) with constant coe¬cients

and initial value x(0) := (1, 0, 0, . . . ) has no solution, since the coordinates would

have to satisfy the recursive relation xn (t) = n2 xn’1 (t) with x1 (t) = 0, and hence

we must have xn (t) = n!tn . But the so de¬ned curve t ’ x(t) has only for t = 0

values in E. Thus, no local solution exists. By recursion one sees that the solution

for an arbitrary initial value x(0) should be given by

n

tn’i

n! 2

xn (t) = xi (0) .

i! (n ’ i)!

i=0

If the initial value is a ¬nite sequence, say xn (0) = 0 for n > N and xN (0) = 0,

then

N

tn’i

n! 2

xn (t) = xi (0)

i! (n ’ i)!

i=0

N

(n!)2 12

xi (0) (n’N )! tN ’i

tn’N

= i! (n’i)!

(n ’ N )! i=0

N ’1

(n!)2 12 12

|xi (0)| (n’N )! |t|N ’i

|t|n’N

|xn (t)| ≥ |xN (0)| ’

N! i! (n’i)!

(n ’ N )! i=0

N ’1

(n!)2 2 12

|xi (0)||t|N ’i

|t|n’N 1

≥ |xN (0)| ’ ,

N! i!

(n ’ N )! i=0

where the ¬rst factor does not lie in the space s of rapidly decreasing sequences,

and where the second factor is larger than µ > 0 for t small enough. So at least for

a dense set of initial values this di¬erential equation has no local solution.

This also shows that the theorem of Frobenius is wrong in the following sense: The

vector ¬eld x ’ T (x) generates a 1-dimensional subbundle E of the tangent bundle

on the open subset s \ {0}. It is involutive since it is 1-dimensional. But through

points representing ¬nite sequences there exist no local integral submanifolds (M

with T M = E|M ). Namely, if c were a smooth non-constant curve with c (t) =

f (t).T (c(t)) for some smooth function f , then x(t) := c(h(t)) would satisfy x (t) =

T (x(t)), where h is a solution of h (t) = 1/f (h(t)).

Example 2. Next consider E := RN and the continuous linear operator T : E ’ E

given by T (x0 , x1 , . . . ) := (x1 , x2 , . . . ). The corresponding di¬erential equation

has solutions for every initial value x(0), since the coordinates must satisfy the

recursive relation xk+1 (t) = xk (t), and hence any smooth function x0 : R ’ R

(k) (k)

gives rise to a solution x(t) := (x0 (t))k with initial value x(0) = (x0 (0))k . So

by Borel™s theorem there exist solutions to this equation for all initial values and

the di¬erence of any two functions with same initial value is an arbitrary in¬nite

32.12

32.13 32. Vector ¬elds 331

¬‚at function. Thus, the solutions are far from being unique. Note that RN is

a topological direct summand in C ∞ (R, R) via the projection f ’ (f (n))n , and

hence the same situation occurs in C ∞ (R, R).

Note that it is not possible to choose the solution depending smoothly on the initial

value: suppose that x is a local smooth mapping R—E ⊃ I—U ’ E with x(0, y) = y

and ‚t x(t, y) = T (x(t, y)), where I is an open interval containing 0 and U is open

in E. Then x0 : I — U ’ R induces a smooth local mapping x0 ∨ : U ’ C ∞ (I, R),

which is a right inverse to the linear in¬nite jet mapping j0 : C ∞ (I, R) ’ RN = E.

∞

Then the derivative of x0 ∨ at any point in U would be a continuous linear right

∞

inverse to j0 , which does not exist (since RN does not admit a continuous norm,

whereas C ∞ (I, R) does for compact I, see also [Tougeron, 1972, IV.3.9]).

Also in this example the theorem of Frobenius is wrong, now in the following

sense: On the complement of T ’1 (0) = R — 0 we consider again the 1-dimensional

subbundle generated by the vector ¬eld T . For every smooth function f ∈ C ∞ (R, R)

∞

the in¬nite jet t ’ jt (f ) is an integral curve of T . We show that integral curves

through a ¬xed point sweep out arbitrarily high dimensional submanifolds of RN :

Let • : R ’ [0, 1] be smooth, •(t) = 0 near t = 0, and •(t) = 1 near t = 1. For

each (s2 , . . . , sN ) we get an integral curve

s2 s3 sN

•(t)(t ’ 1)2 + •(t)(t ’ 1)3 + · · · + •(t)(t ’ 1)N

t ’ jt t +

2! 3! N!

connecting (0, 1, 0, . . . ) with (1, 1, s2 , s3 , . . . , sN , 0, . . . ), and for small s this integral

curve lies in RN \ 0.

Problem: Can any two points be joined by an integral curve in RN \ 0: One has to

¬nd a smooth function on [0, 1] with prescribed jets at 0 and 1 which is nowhere

¬‚at in between.

Example 3. Let now E := C ∞ (R, R), and consider the continuous linear operator

T : E ’ E given by T (x) := x . Let x : R ’ C ∞ (R, R) be a solution of the

‚ ‚

equation x (t) = T (x(t)). In terms of x : R2 ’ R this says ‚t x(t, s) = ‚s x(t, s).

ˆ ˆ ˆ

Hence, r ’ x(t ’ r, s + r) has vanishing derivative everywhere, and so this function

ˆ

is constant, and in particular x(t)(s) = x(t, s) = x(0, s + t) = x(0)(s + t). Thus, we

ˆ ˆ

have a smooth solution x uniquely determined by the initial value x(0) ∈ C ∞ (R, R),

which even describes a ¬‚ow for the vector ¬eld T in the sense of (32.13) below. In

general however, this solution is not real-analytic, since for any x(0) ∈ C ∞ (R, R)

which is not real-analytic in a neighborhood of a point s the composite evs —¦x =

x(s+ ) is not real-analytic around 0.

32.13. The ¬‚ow of a vector ¬eld. Let X ∈ X(M ) be a kinematic vector ¬eld.

FlX

A local ¬‚ow FlX for X is a smooth mapping M — R ⊃ U ’ ’ M de¬ned on a

’

c∞ -open neighborhood U of M — 0 such that

(1) U © ({x} — R) is a connected open interval.

(2) If FlX (x) exists then FlX (x) exists if and only if FlX (FlX (x)) exists, and

s t+s t s

we have equality.

32.13

332 Chapter VII. Calculus on in¬nite dimensional manifolds 32.15

(3) FlX (x) = x for all x ∈ M .

0

(4) dt FlX (x) = X(FlX (x)).

d

t t

In formulas similar to (4) we will often omit the point x for sake of brevity, without

signalizing some di¬erentiation in a space of mappings. The latter will be done

whenever possible in section (42).

32.14. Lemma. Let X ∈ X(M ) be a kinematic vector ¬eld which admits a local

¬‚ow FlX . Then each for each integral curve c of X we have c(t) = FlX (c(0)), thus

t t

X

there exists a unique maximal ¬‚ow. Furthermore, X is Flt -related to itself, i.e.,

T (FlX ) —¦ X = X —¦ FlX .

t t

Proof. We compute

FlX (’t, c(t)) = ’ ds |s=’t FlX (s, c(t)) + FlX (’t, c(s))

d d d

ds |s=t

dt

= ’ ds |s=0 FlX FlX (s, c(t)) + T (FlX ).c (t)

d

’t ’t

= ’T (FlX ).X(c(t)) + T (FlX ).X(c(t)) = 0.

’t ’t

Thus, FlX (c(t)) = c(0) is constant, so c(t) = FlX (c(0)). For the second assertion

’t t

we have X —¦ Flt = dt Flt = ds |0 Flt+s = ds |0 (FlX —¦ FlX ) = T (FlX ) —¦ ds |0 FlX =

X X X

d d d d

t s t s

X

T (Flt ) —¦ X, where we omit the point x ∈ M for the sake of brevity.

32.15. The Lie derivative. For a vector ¬eld X ∈ X(M ) which has a local

¬‚ow FlX and f ∈ C ∞ (M, R) we have dt (FlX )— f = dt f —¦ FlX = df —¦ X —¦ FlX =

d d

t t t t

X X—

X(f ) —¦ Flt = (Flt ) X(f ).

We will meet situations (in (37.19), e.g.) where we do not know that the ¬‚ow of X

exists but where we will be able to produce the following assumption: Suppose that

• : R — M ⊃ U ’ M is a smooth 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 ). Then we have

•’1 = ’X, and still we get dt |0 (•t )— f = dt |0 (f —¦ •t ) = df —¦ X = X(f ) and

‚ d d

t

‚t 0

similarly ‚t 0 (•’1 )— f = ’X(f ).

‚

t

Lemma. In this situation we have for Y ∈ C ∞ (M ← DM ):

|0 (•t )— Y

d

= [X, Y ],

dt

|0 (FlX )— Y

d

= [X, Y ],

t

dt

X—

= (FlX )— [X, Y ].

d

dt (Flt ) Y t

Proof. Let f ∈ C ∞ (M, R) be a function, and let ±(t, s) := Y (•(t, x))(f —¦ •’1 ),

s

which is locally de¬ned near 0. It satis¬es

±(t, 0) = Y (•(t, x))(f ),

±(0, s) = Y (x)(f —¦ •’1 ),

s

‚ ‚ ‚

‚t ±(0, 0) = Y (•(t, x))(f ) = ‚t 0 (Y f )(•(t, x)) = X(x)(Y f ),

‚t 0

|0 Y (x)(f —¦ •’1 ) = Y (x) ‚s |0 (f —¦ •’1 ) = ’Y (x)(Xf ).

‚ ‚ ‚

‚s ±(0, 0) = s s

‚s

32.15

32.17 32. Vector ¬elds 333

‚

‚u |0 ±(u, u)

Hence, = [X, Y ]x (f ). But on the other hand we have

’1

‚ ‚

‚u |0 ±(u, u) ‚u |0 Y (•(u, x))(f —¦ •u ) =

=

’1

‚

‚u |0 D(•u ) —¦ Y —¦ •u x (f )

=

( ‚u |0 (•u )— Y )x (f ),

‚

=

so the ¬rst two assertions follow. For the third claim we compute as follows:

X—

D(FlX ) —¦ D(FlX ) —¦ Y —¦ FlX —¦ FlX

‚ ‚

‚s |0

‚t (Flt ) Y = ’t ’s s t

= D(FlX ) —¦ D(FlX ) —¦ Y —¦ FlX —¦ FlX

‚

‚s |0

’t ’s s t

= D(FlX ) —¦ [X, Y ] —¦ FlX = (FlX )— [X, Y ].

’t t t

32.16. Lemma. Let X ∈ X(M ) and Y ∈ X(N ) be f -related vector ¬elds for a

smooth mapping f : M ’ N which have local ¬‚ows FlX and FlY . Then we have

f —¦ FlX = FlY —¦f , whenever both sides are de¬ned.

t t

—

Moreover, if f is a di¬eomorphism we have Flf Y = f ’1 —¦ FlY —¦f in the following

t t

sense: If one side exists then also the other and they are equal.

For f = IdM this again implies that if there exists a ¬‚ow then there exists a unique

maximal ¬‚ow FlX .

t

Proof. We have Y —¦ f = T f —¦ X, and thus for small t we get, using (32.13.1),

Y

—¦f —¦ FlX ) = Y —¦ FlY —¦f —¦ FlX ’T (FlY ) —¦ T f —¦ X —¦ FlX

d

dt (Flt ’t ’t ’t

t t

= T (FlY ) —¦ Y —¦ f —¦ FlX ’T (FlY ) —¦ T f —¦ X —¦ FlX = 0.

’t ’t

t t

So (FlY —¦f —¦FlX )(x) = f (x) or f (FlX (x)) = FlY (f (x)) for small t. By the ¬‚ow prop-

’t

t t t

erties (32.13.2), we get the result by a connectedness argument as follows: In the

common interval of de¬nition we consider the closed subset Jx := {t : f (FlX (x)) =

t

Y X

Flt (f (x))}. This set is open since for t ∈ Jx and small |s| we have f (Flt+s (x)) =

f (FlX (FlX (x))) = FlY (f (FlX (x))) = FlY (FlY (f (x))) = FlY (f (x)).

s t s t s t t+s

The existence of the unique maximal ¬‚ow now follows since two local ¬‚ows have to

agree on their common domain of de¬nition.

32.17. Corollary. Take X ∈ X(M ) be a vector ¬eld with local ¬‚ow, and let

Y ∈ C ∞ (M ← DM ). Then the following assertions are equivalent

(1) [X, Y ] = 0.

(2) (FlX )— Y = Y , wherever de¬ned.

t

If also Y is kinematic and has a local ¬‚ow then these are also equivalent to

(3) FlX —¦ FlY = FlY —¦ FlX , wherever de¬ned.

t s s t

Proof. (1) ” (2) is immediate from lemma (32.15). To see (2) ” (3) we note

X—

that FlX —¦ FlY = FlY —¦ FlX if and only if FlY = FlX —¦ FlY —¦ FlX = Fl(Flt ) Y by

’t

t s s t s s t s

X—

lemma (32.16), and this in turn is equivalent to Y = (Flt ) Y , by the uniqueness

of ¬‚ows.

32.17

334 Chapter VII. Calculus on in¬nite dimensional manifolds 32.18

32.18. Theorem. [Mauhart, Michor, 1992] Let M be a manifold, and let •i :

R — M ⊃ U•i ’ M be smooth mappings for i = 1, . . . , k such that (t, x) ’

(t, •i (t, x) = •i (x)) is a di¬eomorphism U•i ’ V•i . Here the U•i and V•i are open

t

‚

neighborhoods of {0} — M in R — M such that •i = IdM and ‚t 0 •i = Xi ∈ X(M ).

t

0

j j ’1 j

i ’1

i j i i

We put [• , • ]t = [•t , •t ] := (•t ) —¦ (•t ) —¦ •t —¦ •t . Then for each formal bracket

expression P of length k we have

|0 P (•1 , . . . , •k )

‚

for 1 ¤ < k,

0= t t

‚t

1 ‚k 1 k

k! ‚tk |0 P (•t , . . . , •t ) ∈ X(M )

P (X1 , . . . , Xk ) =

as ¬rst non-vanishing derivative in the sense explained in step (2 of the proof. In

particular, we have for vector ¬elds X, Y ∈ X(M ) admitting local ¬‚ows

(FlY —¦ FlX —¦ FlY —¦ FlX ),

‚

0= ’t ’t t t

‚t 0

1 ‚2 Y X Y X

2 ‚t2 |0 (Fl’t —¦ Fl’t —¦ Flt —¦ Flt ).

[X, Y ] =

Proof. Step 1. Let c : R ’ M be a smooth curve. If c(0) = x ∈ M , c (0) =

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