<<

. 14
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>>

(2) X is an expansion at x with values in N/N0 , and X0 satis¬es

¯ ¯ ¯ ¯
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

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