(1)

j k fk (xk ) = j k gk (xk )

(2)

(3) Dfk (zk ) = Dgk (zk ).

Since all xk are di¬erent from x0 , by passing to subsequences we can assume

1

|xk+1 ’ x0 | < |xk ’ x0 |.

(4)

4

Let us ¬x Riemannian metrics ρZ or ρW on Z or W , respectively, and choose

further points zk ∈ Z, zk ’ z0 , xk := π(¯k ) and neighborhoods Uk or Vk of xk

¯ ¯ ¯ z

or xk , respectively, in such a way that for all k ∈ N the following six conditions

¯

hold

|xk ’ x0 | ¤ 2|a ’ b| for all a ∈ Uk ∪ Vk , b ∈ Uj ∪ Vj , k = j

(5)

|‚ ± (fk ’ f )(a)| ¤ 2µ(xk ) for all a ∈ Uk ∪ Vk , 0 ¤ |±| ¤ k

(6)

|‚ ± (gk ’ f )(a)| ¤ 2µ(xk ) for all a ∈ Uk ∪ Vk , 0 ¤ |±| ¤ k

(7)

ρW (Dgk (zk ), Dfk (¯k )) ≥ kρZ (zk , zk )

(8) z ¯

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

182 Chapter V. Finite order theorems

and for all m, k ∈ N, and multi-indices ± with |±| + 2m ¤ k, a ∈ Uk , and b ∈ Vk

we require

1 1 ±+β 1

fk (b)(a ’ b)β ’ ‚ ± gk (a) ¤

(9) ‚

|b ’ a|m β! k

|β|¤m

1 1 ±+β 1

gk (a)(b ’ a)β ’ ‚ ± fk (b) ¤ .

(10) ‚

|b ’ a|m β! k

|β|¤m

All these requirements can be satis¬ed. Indeed, the equalities (5), (6), (7) are

valid for all points a, b from some suitable neighborhoods Wk of the points xk .

By the Taylor formula, for any ¬xed k and |±| + m ¤ k, (2) implies ‚ ± gk (a) =

‚ ± fk (a) + o(|a ’ xk |m ). Therefore, if we consider only points a, b ∈ Wk such

that

|b ’ xk | ¤ 2|b ’ a|, |a ’ xk | ¤ 2|b ’ a|,

(11)

then under the condition |±| + 2m ¤ k we get ( note that o(|a ’ xk |m ) or

o(|b ’ xk |m ) now implies o(|a ’ b|m ))

1 ±+β 1 ±+β

fk (b)(a ’ b)β = gk (b)(a ’ b)β + o(|a ’ b|m )

‚ ‚

β! β!

|β|¤m |β|¤m

‚ ± gk (a) + o(|b ’ a|m )

=

1 ±+β 1 ±+β

gk (a)(b ’ a)β = fk (a)(b ’ a)β + o(|a ’ b|m )

‚ ‚

β! β!

|β|¤m |β|¤m

‚ ± fk (b) + o(|b ’ a|m ).

=

Hence also conditions (9), (10) are realizable if we take Uk , Vk in su¬ciently small

neighborhoods Wk of xk in such a way that (11) holds for all a ∈ Uk , b ∈ Vk . By

virtue of (3), there are also neighborhoods of the points zk in Z ensuring (8).

Finally, we are able to choose appropriate points zk and neighborhoods Uk , Vk

¯

using the fact that π is continuous and locally non-constant.

The aim of conditions (1), (4)“(7), (9), (10) is to guarantee the existence of

a map h ∈ C ∞ (Rm , Rn ) satisfying

(12) germ h(xk ) = germ gk (xk ) and germ h(¯k ) = germ fk (¯k ).

x x

Then, by virtue of our requirements on E, we may assume h ∈ E, provided we

use (12) for large indices k, only. But applying D to h, the π-locality and (8)

imply

ρW (Dh(zk ), Dh(¯k )) ≥ kρZ (zk , zk )

z ¯

for large k™s, and this is a contradiction with Dh ∈ C ∞ (Z, W ) and (zk , zk ) ’

¯

(z0 , z0 ).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

19. Peetre-like theorems 183

So it remains to verify condition 19.4.(1) in the Whitney extension theorem

¯ ¯ ¯

with K = k (Uk ∪ Vk ) ∪ {x0 } and f± (x) = ‚ ± gk (x) if x ∈ Uk , f± (x) = ‚ ± fk (x)

¯

if x ∈ Vk and f± (x0 ) = ‚ ± f (x0 ). This follows by our construction for all couples

¯ ¯

(a, b) ∈ k (Uk — Vk ), see (9), (10). In all other cases and for all m ∈ N we have

to use (6) and (7), (5), the Taylor formula, (6) and (7), and (5) to get

1 1

’ a)β = ±+β

f (a) + o(|xk(a) ’ x0 |m ) (b ’ a)β

|β|¤m β! f±+β (a)(b |β|¤m β! ‚

1 ±+β

f (a)(b ’ a)β + o(|b ’ a|m )

= |β|¤m β! ‚

± m

= ‚ f (b) + o(|b ’ a| )

= f± (b) + o(|xk(b) ’ x0 |m ) + o(|b ’ a|m )

= f± (b) + o(|b ’ a|m ).

19.12. Lemma. Let z0 ∈ Z be a point, x = π(z0 ) and f ∈ E. Then there is

a neighborhood V of z0 in π ’1 (x) and a natural number r such that for every

z ∈ V and all maps g ∈ E the condition j r g(x) = j r f (x) implies Dg(z) = Df (z).

Proof. The proof is quite similar to that of 19.11, but we ¬rst have to prove the

dependence on in¬nite jets. Consider g1 , g2 ∈ E with j ∞ g1 (x) = j ∞ g2 (x) and

a point y ∈ π ’1 (x). Let us choose a sequence yk ’ y in Z, π(yk ) = : xk = x

and neighborhoods Uk of xk satisfying |a ’ x| ≥ 2|a ’ b| for all a ∈ Uk , b ∈ Uj ,

k = j. Using the Whitney extension theorem 19.4, the Taylor formula, and our

assumptions on E we ¬nd a map h ∈ E satisfying for all large k™s

germ h(x2k ) = germ g1 (x2k ) and germ h(x2k+1 ) = germ g2 (x2k+1 ).

This implies Dh(y2k ) = Dg1 (y2k ), Dh(y2k+1 ) = Dg2 (y2k+1 ) and consequently

Dg1 (y) = Dg2 (y).

Now, we assume the assertion of the lemma is not true. So we can construct

a sequence zk ’ z0 , π(zk ) = x and maps gk ∈ E satisfying for all k ∈ N

j k f (x) = j k gk (x)

(1)

(2) Dgk (zk ) = Df (zk ).

We choose further points zk ’ z0 in Z, xk := π(¯k ), xk = x, and neighborhoods

¯ ¯ z ¯

Vk of xk in such a way that

¯

ρW (Dgk (¯k ), Df (zk )) ≥ kρZ (¯k , zk ) for all k ∈ N

(3) z z

|a ’ x| ≥ 2|a ’ b| for all a ∈ Vk , b ∈ Vj , k = j

(4)

|‚ ± (gk ’ f )(a)| 1

¤ for all a ∈ Vk , |±| + m ¤ k.

(5)

|a ’ x|m k

This is possible by virtue of (1), (2) and the Taylor formula analogously to 19.11.

Finally, using (4), (5), the Whitney extension theorem and our assumptions, we

get a map h ∈ E satisfying

germ h(¯k ) = germ gk (¯k ) and j ∞ h(x) = j ∞ f (x)

x x

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

184 Chapter V. Finite order theorems

for large k™s. Hence (3) and the ¬rst part of this proof imply

ρW (Dh(¯k ), Dh(zk )) = ρW (Dgk (¯k ), Df (zk )) ≥ kρZ (¯k , zk )

z z z

which is a contradiction with Dh ∈ C ∞ (Z, W ).

Proof of theorem 19.7. According to lemmas 19.11 and 19.12, for every point

z ∈ K we ¬nd a neighborhood Vz of z, an order rz and a smooth function

µz : π(Vz ) ’ R which is strictly positive with a possible exception of the point

π(z), such that the conclusion of 19.7 is true for these data. The proof is then

completed by the standard compactness argument.

19.13. Let us note that our de¬nition of Whitney-extendibility was not fully

exploited in the proof of lemma 19.12. Namely, we dealt with ˜fast converging™

sequences only. However, we might be unable to verify the W-extendibility for

certain domains E ‚ C ∞ (X, Y ) while the proof of lemma 19.12 might still go

through. So we ¬nd it pro¬table to present explicit formulations. For technical

reasons, we consider the case X = Rm .

De¬nition. A subset E ‚ C ∞ (Rm , Y ) is said to be almost Whitney-extendible

if for every map f ∈ C ∞ (Rm , Y ), sequence fk ∈ E, f0 ∈ E and every convergent

sequence xk ’ x satisfying for all k ∈ N, |xk ’ x| ≥ 2|xk+1 ’ x|, germ f (xk ) =

germ fk (xk ), j ∞ f (x) = j ∞ f0 (x), there is a map g ∈ E and a natural number k0

satisfying germ g(xk ) = germ fk (xk ) for all k ≥ k0 .

19.14. Proposition. Let π : Z ’ Rm be a locally non-constant continuous

map, E ‚ C ∞ (Rm , Y ) be an almost Whitney-extendible subset and let D : E ’

C ∞ (Z, W ) be a π-local operator. Then for every ¬xed map f ∈ E, point x ∈ Rm ,

and for every compact subset K ‚ π ’1 (x), there exists a natural number r such

that for all maps g ∈ E the condition j r g(x) = j r f (x) implies Dg|K = Df |K.

Proof. The proposition is implied by lemma 19.12 and by the standard com-

pactness argument.

At the end of this section, we present an example showing that the results in

19.7 are the best possible ones in our general setting.

19.15. Example. We shall construct a simple idR -local operator

D : C ∞ (R, R) ’ C ∞ (R, R)

such that if we take f = idR , then for any order r and any compact neighborhood

K of 0 ∈ R, every function µ : R ’ R from 19.7 satis¬es µ(0) = 0.

Let g : R2 ’ R be a function with the following three properties

(1) g is smooth in all points x ∈ R2 \ {(0, 1)}

(2) lim supx’1 g(0, x) = ∞

(3) g is identically zero on the closed unit discs centered in (’1, 1) and (1, 1).

Further, let a : R2 ’ R be a smooth function satisfying a(t, x) = 0 if and only if

|x| > t > 0.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

20. The regularity of bundle functors 185

Given f ∈ C ∞ (R, R), x ∈ R, we de¬ne

∞

dk f

df

(a(k, ’) —¦ g —¦ (f —

Df (x) = )(x) (x) .

dxk

dx

k=0

df

The sum is locally ¬nite if g —¦ (f — dx ) is locally bounded. Hence Df is well

df

de¬ned and smooth if g —¦ (f — dx ) is smooth. The only di¬culty may happen if

df

we deal with some f ∈ C ∞ (R, R) and x ∈ R with f (x) = 0, dx (x) = 1. However,

in this case it holds

df

d2 f

dx (y) ’ 1

lim = (x)

dx2

f (y)

y’x

df

and the property (3) of g implies g —¦ (f — dx ) = 0 on some neighborhood of x.

On the other hand, for f = idR , arbitrary µ > 0 and order r ∈ N, there are

dk

functions h1 , h2 ∈ C ∞ (R, R) such that j r h1 (0) = j r h2 (0), | dxk (h1 ’ idR )(0)| < µ

for all 0 ¤ k ¤ r, and Dh1 (0) = Dh2 (0). This is caused by property (2) of g.

20. The regularity of bundle functors

20.1. De¬nition. A category C over manifolds is called locally ¬‚at if C admits

a local pointed skeleton (C± , 0± ) where each C-object C± is over some Rm(±) and

if all translations tx on Rm(±) are C-morphisms.

Each local pointed skeleton of a locally ¬‚at category will be assumed to have

this property.

Every bundle functor F : C ’ Mf on a locally ¬‚at category C determines the

induced action „ of the abelian subgroup Rm(±) ‚ C(C± , C± ) on the manifold

F C± , „x = F (tx ). In section 14 we used this action and the regularity of the

natural bundles to ¬nd canonical di¬eomorphisms F Rm ∼ Rm — p’1 (0). The

= Rm

same consideration applies also in our general case, but we have ¬rst to prove

the smoothness of „ . The most di¬cult and rather technical job is to prove that

„ is continuous. Therefore we ¬rst formulate this result, then we deduce some of

its consequences including the regularity of bundle functors and only at the very

end of this section we present the proof consisting of several analytical lemmas.

20.2. Proposition. Let C be an admissible locally ¬‚at category over manifolds

with almost Whitney-extendible sets of morphisms and with the faithful functor

m : C ’ Mf . Let (C± , 0± ) be its local pointed skeleton. Let F : C ’ Mf be a

functor endowed with a natural transformation p : F ’ m such that the locality

condition 18.3.(i) holds. Then the induced actions of the abelian groups Rm(±)

on F C± are continuous.

The proof will be given in 20.9“20.12.

20.3. Theorem. Let C be an admissible locally ¬‚at category over manifolds

with almost Whitney-extendible sets of morphisms, (C± , 0± ) its local pointed

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

186 Chapter V. Finite order theorems

skeleton, and m : C ’ Mf the faithful functor. Let F : C ’ Mf be a func-

tor endowed with a natural transformation p : F ’ m such that the locality

condition 18.3.(i) holds. Then there are canonical di¬eomorphisms

mC± — p’1 (0± ) ∼ F C± , (x, z) ’ F tx (z)

(1) =

C±

and for every A ∈ ObC of type ± the map pA : F A ’ A is a locally trivial ¬ber

bundle with standard ¬ber p’1 (0± ). In particular F is a bundle functor on C.

C±

Proof. Let us ¬x a type ± and write Rm for mC± . By proposition 20.2, the action

„ : Rm — F C± ’ F C± is a continuous action and each map „x : F C± ’ F C± is

a di¬eomorphism. But then a general theorem, see 5.10, implies that this action

is smooth. It follows that for every z ∈ p’1 (0± ) the map s : Rm ’ F C± , s(x) =

C±

„x (z) is smooth and pC± —¦ s = idRm . Therefore pC± is a submersion and p’1 (0± )

C±

is a manifold. Since both the maps (x, z) ’ „ (x, z) and y ’ „ (’pC± (y), y) are

smooth, (1) is a di¬eomorphism. The rest of the theorem follows now from the

locality of functor F .

20.4. Consider a bundle functor F on an admissible category C. Since for every

C-object A the action of C(A, A) on F A determined by F can be viewed as a

pA -local operator, a simple application of our results from section 19 will enable

us to get near to the ¬niteness of the order of bundle functors.

Consider a point x ∈ A and a compact set K ‚ p’1 (x) ‚ F A. We de¬ne

A

QK := ∪f ∈invC(A,A) F f (K).

Lemma. If C(A, A) ‚ C ∞ (mA, mA) is almost Whitney-extendible, then for

every compact K as above there is an order r ∈ N such that for all invertible

r r

C-morphisms f , g and for every point y ∈ A the equality jy f = jy g implies

F f |(QK © p’1 (y)) = F g|(QK © p’1 (y)).

A A

Proof. Let us ¬x the map idA ∈ C(A, A) and let us apply proposition 19.14 to

F : C(A, A) ’ C ∞ (F A, F A), π = pA and K. We denote by r the resulting order.

For every z ∈ QK there are y ∈ K and g ∈ invC(A, A) with F g(y) = z. Consider

f1 , f2 ∈ invC(A, A) such that j r f1 (π(z)) = j r f2 (π(z)). Then j r (f1 —¦ g)(π(y)) =

’1

j r (f2 —¦ g)(π(y)) and therefore j r (g ’1 —¦ f1 —¦ f2 —¦ g)(π(y)) = j r idA (π(y)). Hence

F f1 (z) = F f1 —¦ F g(y) = F f2 —¦ F g(y) = F f2 (z).

20.5. Theorem. Let C be an admissible locally ¬‚at category over manifolds

with almost Whitney-extendible sets of morphisms. If all C-morphisms are lo-

cally invertible, then every bundle functor F on C is regular.

Proof. Since all morphisms are locally invertible and the functors are local, we

may restrict ourselves to objects of one ¬xed type, say ±. We shall write (C, 0) for

(C± , 0± ), mC = Rm , p = pC . Let us consider a smoothly parameterized family

gs ∈ C(C, C) with parameters in a manifold P . For any z ∈ F C, x = p(z),

f ∈ C(C, C) we have

F f (z) = „f (x) —¦ F (t’f (x) —¦ f —¦ tx ) —¦ „’x (z)

(1)

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

20. The regularity of bundle functors 187

and the mapping in the brackets transforms 0 into 0. Since „ is a smooth action

by theorem 20.3, the regularity will follow from (1) if we show that for families

with gs (0) = 0 the restrictions of F gs to the standard ¬ber S = p’1 (0) are

smoothly parameterized. Since the case m = 0 is trivial, we may assume m > 0.

By lemma 20.4 F is of order ∞. We ¬rst show that the induced action of the

group of in¬nite jets G∞ = invJ0 (C, C)0 on S is continuous with respect to the

∞

±

inverse limit topology.

Consider converging sequences zn ’ z in S and j0 fn ’ j0 f0 in G∞ . We

∞ ∞

±

shall show that any subsequence of F fn (zn ) contains a further subsequence con-

’1

verging to the point F f0 (z). On replacing fn by fn —¦ f0 , we may assume

f0 = idC . By passing to subsequences, we may assume that all absolute values

of the derivatives of (fn ’ idC ) at 0 up to order 2n are less then e’n . Let us

choose positive reals µn < e’n in such a way that on the open balls B(0, µn )

centered at 0 with diameters µn all the derivatives in question vary at most by

e’n . Let xn := (2’n , 0, . . . , 0) ∈ Rm . By the Whitney extension theorem there

is a local di¬eomorphism f : Rm ’ Rm such that

f |B(x2n+1 , µ2n+1 ) = idC and f |B(x2n , µ2n ) = tx2n —¦ f2n —¦ t’x2n

for large n™s. Since the sets of C-morphisms are almost Whitney extendible,

there is a C-morphism h satisfying the same equalities for large n™s. Now

„’xn —¦ F h —¦ „xn (zn ) = F fn (zn ) if n is even

„’xn —¦ F h —¦ „xn (zn ) = zn if n is odd.

Hence, by virtue of proposition 20.2, F f2n (z2n ) converges to z and we have

proved the continuity of the action of G∞ on S as required.

±

Now, let us choose a relatively compact open neighborhood V of z and de¬ne

QV := (∪f ∈invC(C,C) F f (V )) © S. This is an open submanifold in S and the

functor F de¬nes an action of the group G∞ on QV . According to lemma 20.4

±

this action factorizes to an action of a jet group Gr on QV which is continuous

±

by the above part of the proof. Hence this action has to be smooth for the reason

discussed in the proof of theorem 20.3 and since smoothness is a local property

and all C-morphisms are locally invertible this concludes the proof.

20.6. Corollary. Every bundle functor on FMm,n is regular.

We can also deduce the regularity for bundle functors on FMm using theo-

rems 20.3 and 20.5.

20.7. Corollary. Every bundle functor on FMm is regular.

Proof. The system (Rm+n ’ Rm , 0), n ∈ N0 , is a local pointed skeleton of

FMm . Every morphism f : Rm+n ’ Rm+k is locally of the form f = h —¦ g

where g = g0 — idRn : Rm+n ’ Rm+n and h is a morphism over identity on Rm

’1

(g0 = f0 , h1 (x, y) = f1 (f0 (x), y)). So we can deal separately with this two

special types of morphisms.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

188 Chapter V. Finite order theorems

The restriction Fn of functor F to subcategory FMm,n is a regular bundle

functor according to 20.6 and the morphisms of the type g0 — idRn are FMm,n -

morphisms.

Hence it remains to discuss the latter type of morphisms. We may restrict

ourselves to families hp : Rm+n ’ Rm+k parameterized by p ∈ Rq , for some

q ∈ N. Let us consider i : Rm+n ’ Rm+n — Rq , (x, y) ’ (x, y, 0), h : Rm+n+q ’

Rm+k , h(’, ’, p) = hp . Since all the maps hp are over the identity, h is a ¬bered

morphism. We have hp = h —¦ t(0,0,p) —¦ i, so that F hp = F h —¦ F t(0,0,p) —¦ F i.

According to theorem 20.3 F hp is smoothly parameterized.

20.8. Remarks. Since every bundle functor is completely determined by its