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

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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.

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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

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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)

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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.

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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

restriction to a local pointed skeleton, there must be a bijective correspondence

between bundle functors on categories with a common local pointed skeleton.

Hence, although the category FMm,0 does not coincide with Mfm (in the former

category, there are coverings of m-dimensional manifolds), the bundle functors

on Mfm and FMm,0 are in fact the same ones. Analogously, the usual local

skeleton of FM0 coincides with that of Mf . So corollary 20.6 reproves the clas-

sical result on natural bundles due to [Epstein, Thurston, 79] while 20.7 implies

that every bundle functor de¬ned on the whole category of manifolds is regular.

For the same reason our results also apply to the category of (m+n)-dimensional

manifolds with a foliation of codimension m and morphisms transforming leafs

into leafs.

The rest of this section is devoted to the proof of proposition 20.2. Let us ¬x

a bundle functor F on an admissible locally ¬‚at category C over manifolds with

almost W-extendible sets of morphisms and an object (Rm , 0) in a local pointed

skeleton. We shall brie¬‚y write p instead of pRm , „ for the action of Rm on F Rm

and we denote by B(x, µ) the open ball {y ∈ Rm ; |y ’ x| < µ} ‚ Rm .

First the technique used in section 19 will help us to get a lemma that seems

to be near to the continuity of „ claimed in proposition 20.2. However, the

complete proof of 20.2 will require a lot of other analytical considerations.

20.9. Lemma. Let zi ∈ F Rm , i = 1, 2,. . . , be a sequence of points converging

to z ∈ F Rm such that p(zi ) = p(z). Then there is a sequence of real constants

µi > 0 such that for any point a ∈ Rm and any neighborhood W of „a (z) the

inclusion „ (B(a, µi ) — {zi }) ‚ W holds for all large i™s.

Proof. Let us assume that the lemma is not true for some sequence zi ’ z.

Then for any sequence µi of positive real numbers there are a point a ∈ Rm , a

neighborhood W of „a (z) and a sequence ai ∈ B(a, µi ) such that „ (ai , zi ) ∈ W

/

for an in¬nite set of indices i ∈ I0 ‚ N. Let us denote xi := p(zi ), x := p(z).

Passing to a further subset of indices we can arrange that 2|xi ’ xj | > |xi ’ x|

for all i, j ∈ I0 , i = j. If we construct a smooth map f : Rm ’ Rm such that

(1) germ f (xi ) = germ tai (xi )

for an in¬nite subset of indices i ∈ I ‚ I0 and

(2) germ f (xj ) = germ ta (xj )

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20. The regularity of bundle functors 189

for an in¬nite subset of indices j ∈ J ‚ I0 , then using the almost W-extendibility

of C-morphisms we ¬nd some g ∈ C(Rm , Rm ) satisfying F g(zi ) = F tai (zi ) =

„ (ai , zi ) for large i ∈ I and F g(zj ) = „ (a, zj ) for large j ∈ J. Hence F (t’a ) —¦

F g(zi ) = „ (ai ’a, zi ) for large i ∈ I while F (t’a )—¦F g(zj ) = zj for large j ∈ J and

this implies F g(z) = „a (z) which is in contradiction with F g(zi ) = „ (ai , zi ) ∈ W

/

for large i ∈ I.

The existence of a smooth map f : Rm ’ Rm satisfying (1), (2) is ensured

by the Whitney extension theorem (see 19.4) if we choose the numbers µi small

enough. To see this, let us view (1) and (2) as a prescription of all derivatives

of f on some small neighborhoods of the points xi , i ∈ I0 . Then the condition

19.4.(1) reads

|ai ’ aj | |a ’ ai | |a ’ ai |

’ 0, ’0 ’0

lim lim lim

j,i’∞ |xi ’ xj |k i’∞ |xi ’ x|k |xi ’ xj |k

j,i’∞

i∈I

j,i∈I i∈I,j∈J

for all k ∈ N.

Let us choose 0 < µi < e’1/(|xi ’x|) . Now, if i < j then |ai ’ aj | < 2µi and

|xi ’ xj | > 1 |xi ’ x| and the ¬rst estimate follows. Analogously we get the

2

remaining ones.

The next lemma is necessary to overcome di¬culties with constant sequences

in F Rm .

20.10. Lemma. Let zj ∈ F Rm , j = 1, 2, . . . , be a sequence of points converg-

ing to z ∈ F Rm . Then there is a sequence of points ai ∈ Rm , ai = 0, i = 1,

2,. . . , converging to 0 ∈ Rm and a subsequence zji such that F tai (zji ) ’ z if

i ’ ∞.

Proof. Let us recall that F Rm has a countable basis of open sets and let Uj ,

j ∈ N, form a basis of open neighborhoods of the point z satisfying Uj+1 ‚ Uj .

For each number j ∈ N, there is a sequence of points a(j, k) ∈ Rm , k ∈ N, such

that

F (ta )(Uj ) = F (ta(j,k) )(Uj ).

a∈Rm k∈N

Let bj ∈ Rm be such a sequence that for all k ∈ N, bj = a(j, k). Passing

to subsequences, we may assume zj ∈ Uj for all j and consequently we get

F (tbj )(zj ) ∈ k∈N F (ta(j,k) )(Uj ) for all j ∈ N. Let us choose a sequence kj ∈ N,

such that

F (tbj )(zj ) ∈ F (ta(j,kj ) )(Uj )

for all j ∈ N, and denote aj := bj ’ a(j, kj ). Then aj = 0 and F (taj )(zj ) ∈ Uj

for all j ∈ N. Therefore F (taj )(zj ) ’ z and since aj = p(F (taj )(zj )) ’ p(zj ), we

also have aj ’ 0

A further step we need is to exclude the dependence of the balls B(a, µi ) on

the indices i in the formulation of 20.9.

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190 Chapter V. Finite order theorems

20.11. Lemma. Let zi ’ z be a convergent sequence in F Rm , p(zi ) = p(z),

and let W be an open neighborhood of z. Then there exist b ∈ Rm and µ > 0

such that

„ B(b, µ) — {z} ‚ W, „ B(b, µ) — {zi } ‚ W

for large i™s.

Proof. We ¬rst deduce that there is some open ball B(y, ·) ‚ Rm satisfying

B(y, ·) ‚ {a ∈ Rm ; F (ta )(z) ∈ W }.

(1)

Let us apply lemma 20.10 to a constant sequence yj := z. So there is a sequence

ai ∈ Rm , ai = 0, ai ’ 0 such that „ai (z) ’ z. Now we apply lemma 20.9 to the

sequence wi := „ai (z). Since for a = 0 we have „a (z) ∈ W , there is a sequence

of positive constants ·i such that „ B(0, ·i ) — {wi } ‚ W for large i™s. Let us

choose one of these indices, say i0 , and put y := ai0 , · := ·i0 . Now for any

b ∈ B(y, ·) we have „b (z) = „b’y —¦ „y (z) = „b’y (wi ) ‚ W , so that (1) holds.

Further, let us apply lemma 20.9 to the sequence zi ’ z and let us ¬x a

neighborhood W of z. Then the conclusion of 20.9 reads as follows. There is

a sequence of positive real constants µi such that for any a ∈ Rm the condition

„a (z) ∈ W implies „ B(a, µi ) — {zi } ‚ W for all large i™s. Therefore

{a ∈ Rm ; „ B(a, µi ) — {zi } ‚ W }.

B(y, ·) ‚

k∈N i≥k

For any natural number k we de¬ne

{a ∈ B(y, ·); „ B(a, µi ) — {zi } ‚ W }.

Bk :=

i≥k

Since ∪k∈N Bk = B(y, ·), the Baire category theorem implies that there is a

¯

natural number k0 such that int(Bk0 ) © B(y, ·) = ….

¯

Now, let us choose b ∈ Rm and µ > 0 such that B(b, µ) ‚ int(Bk0 ) © B(y, ·).

If x ∈ B(b, µ) and i ≥ k0 , then there is x ∈ Bk0 with x ∈ B(¯, µi ) so that we

¯ x

have „x (zi ) ∈ W and (1) implies „x (z) ∈ W .

20.12. Proof of proposition 20.2. Let zi ’ z be a convergent sequence in

F Rm , xi ’ x a convergent sequence in Rm . We have to show

„xi (zi ) = F (txi )(zi ) ’ F (tx )(z) = „x (z).

(1)

Since we can apply the isomorphism F (t’x ), we may assume x = 0. More-

over, it is su¬cient to show that any subsequence of (xi , zi ) contains a further

subsequence satisfying (1). That is why we may assume either p(zi ) = p(z) or

p(zi ) = p(z) for all i ∈ N.

Let us ¬rst deal with the latter case. According to lemma 20.10 there is a

sequence yi ∈ Rm and subsequence zij such that „yj (zij ) ’ z and yj ’ 0,

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20. The regularity of bundle functors 191

yj = 0. But „xij (zij ) ’ z if and only if „xij ’yj —¦ „yj (zij ) ’ z, so that if we

consider zj := „yj (zij ), z := z and xj := xij ’ yj , we transform the problem to

¯ ¯ ¯

the former case.

So we assume p(zi ) = p(z) for all i ∈ N and xi ’ 0. Let us moreover assume

that „xi (zi ) does not converge to z. Then, for each x ∈ Rm , „x+xi (zi ) does not

converge to „x (z) as well. Therefore, if we set

A := {x ∈ Rm ; „x+xi (zi ) does not converge to „x (z)}

we ¬nd A = Rm . Now we use the separability of F Rm . Let Vs , s ∈ N, be a basis

of open sets in F Rm and let

Ls := {x ∈ Rm ; „x+xi (zi ) ∈ Vs for large i™s}

Qs := {x ∈ Rm ; „x (z) ∈ Vs and x ∈ Ls }.

/

We know A ‚ ∪s∈N Qs and consequently ∪s∈N Qs = Rm . By virtue of the Baire

category theorem there is a natural number k such that int(Qk ) = ….

Let us choose a point a ∈ Qk © int(Qk ). Then z ∈ „’a (Vk ) and so

W := p’1 tp(z)’a int(Qk ) „’a (Vk )

is an open neighborhood of z. According to lemma 20.11 there is an open ball

B(b, µ) ‚ Rm such that

„ B(b, µ) — {z} ‚ p’1 tp(z)’a int(Qk )

(2)

„ B(b, µ) — {zi } ‚ „’a (Vk )

(3)

for all large i™s. Inclusion (2) implies p(z) + B(b, µ) ‚ p(z) ’ a + int(Qk ) or,

equivalently,

B(b + a, µ) ‚ int Qk .

(4)

Formula (3) is equivalent to

„ B(b + a, µ) — {zi } ‚ Vk

for large i™s. Since xi ’ 0, we know that for any x ∈ B(b + a, µ) also (x + xi ) ∈

B(b + a, µ) for large i™s and we get the inclusion B(b + a, µ) ‚ Lk . Finally, (4)

implies

B(b + a, µ) ‚ Lk © int Qk ‚ (Rm \ Qk ) © int Qk .

This is a contradiction.

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192 Chapter V. Finite order theorems

21. Actions of jet groups

Let us recall the jet group Gr m,n of the only type in the category FMm,n

which we mentioned in 18.8. In this section, we derive estimates on the possible

order of this jet group acting on a manifold S depending only on dimS. In view

of lemma 20.4, these estimates will imply the ¬niteness of the order of bundle

functors on FMm,n .

21.1. The whole procedure leading to our estimates is rather technical but the

main idea is very simple and can be applied to other categories as well. Consider

a jet group Gr of an admissible category C over manifolds acting on a manifold S

±

and write Bk for the kernel of the jet projection πk : Gr ’ Gk . For every point

r r

± ±

y ∈ S, let Hy be the isotropy subgroup at the point y. The action factorizes to

an action of a group Gk on S if and only if Bk ‚ Hy for all points y ∈ S. So

r

±

if we assume that the order r is essential, i.e. the action does not factorize to

r’1 r

G± , then there is a point y ∈ S such that Hy does not contain Bk’1 . If the

action is continuous, then Hy is closed and the homogeneous space Gr /Hy is

±

mapped injectively and continuously into S. Hence we have

dim S ≥ dim(Gr /Hy )

(1) ±

and we see that dim S is bounded from below by the smallest possible codimen-

sion of Lie subgroups in Gr which do not contain Bk .

r

±

A proof of such a bound in the special case C = FMm,n will occupy the rest

of this section.

21.2. Theorem. Let a jet group Gr , m ≥ 1, n ≥ 0, act continuously on a

m,n

manifold S, dim S = s, s ≥ 0, and assume that r is essential, i.e. the action does

not factorize to an action of Gk , k < r. Then

m,n

r ¤ 2s + 1.

Moreover, if m, n > 1, then

s s s s

r ¤ max{ , + 1, , + 1}

m’1 m n’1 n

and if m > 1, n = 0, then

s s

r ¤ max{ , + 1}.

m’1 m

All these estimates are sharp for all m ≥ 1, n ≥ 0, s ≥ 0.

21.3. Proof of the estimate r ¤ 2s + 1. Let us ¬rst assume s > 0. By the

general arguments discussed in 21.1, there is a point y ∈ S such that its isotropy

r

group Hy does not contain the normal closed subgroup Br’1 . We shall denote

gr , br r r

r’1 and h the Lie algebras of Gm,n , Br’1 and Hy , respectively. Since

m,n

r

Br’1 is a connected and simply connected nilpotent Lie group, its exponential

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21. Actions of jet groups 193

map is a global di¬eomorphism of br onto Br’1 , cf. 13.16 and 13.4. Therefore

r

r’1

h does not contain br . In this way, our problem reduces to the determination

r’1

of a lower bound of the codimensions of subalgebras of gr that do not contain

m,n

the whole br .

r’1

Since gm,n is a Lie subalgebra in gr , there is the induced grading

r

m+n

gr = g0 • · · · • gr’1

m,n

where homogeneous components gp are formed by jets of homogeneous pro-

jectable vector ¬elds of degrees p + 1, cf. 13.16.

If we consider the intersections of h with the ¬ltration de¬ning the grading

gm,n = •p gp , then we get the ¬ltration

r

h = h0 ⊃ h1 ⊃ . . . ⊃ hr’1 ⊃ 0

and the quotient spaces hp = hp /hp+1 are subalgebras in gp . Therefore we can

˜

construct a new algebra h = h0 • · · · • hr’1 with grading and since

˜

dim h = dim h/h1 + dim h1 /h2 + · · · + dim hr’1 = dim h,

˜

both the algebras h and h have the same codimension. By the construction,

˜

br ‚ h if and only if hr’1 = gr’1 , so that h does not contain br as well. That

r’1 r’1

is why in the proof of theorem 21.2 we may restrict ourselves to Lie subalgebras

h ‚ gr with grading h = h0 • · · · • hr’1 satisfying hi ‚ gi for all 0 ¤ i ¤ r ’ 1,

m,n

and hr’1 = gr’1 .

Now the proof of the estimate r ¤ 2s + 1 becomes rather easy. To see this,

let us ¬x two degrees p = q with p + q = r ’ 1 and recall [gp , gq ] = gr’1 ,

see 13.16. Hence there is either a ∈ gp or a ∈ gq with a ∈ h, for if not then

/

[gp , gq ] = gr’1 ‚ hr’1 . It follows

1

codim h ≥ (r ’ 1).

2

According to 21.1.(1) we get s ≥ 1 (r ’ 1) and consequently r ¤ 2s + 1.

2

The remaining case s = 0 follows immediately from the fact that given an

action ρ : Gr

m,n ’ Di¬(S) on a zero-dimensional manifold S, then its kernel

ker ρ contains the whole connected component of the unit. Since Gr has two

m,n

components and these can be distinguished by the ¬rst order jet projection, we

see that the order can be at most one.

Let us notice, that the only special property of gr

m,n among the general jet

groups which we used in 21.3 was the equality [gp , gq ] = gp+q . Hence the ¬rst

estimate from theorem 21.2 can be easily generalized to some other categories.

The proof of the better estimates for higher dimensions is based on the same

ideas but supported by some considerations from linear algebra. We choose some

non-zero linear form C on gr’1 with ker C ⊃ hr’1 . Then given p, q, p+q = r ’1,

we de¬ne a bilinear form f : gp — gq ’ R by f (a, b) = C([a, b]) and we study

the dimensions of the annihilators.

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194 Chapter V. Finite order theorems

21.4. Lemma. Let V , W , be ¬nite dimensional real vector spaces and let

f : V — W ’ R be a bilinear form. Denote by V 0 or W 0 the annihilators of V

or W related to f , respectively. Let M ‚ V , N ‚ W be subspaces satisfying

f |(M — N ) = 0. Then

codim M + codim N ≥ codim V 0 .

Proof. Consider the associated form f — : V /W 0 — W/V 0 ’ R and let [M ], [N ]

be the images of M , N in the projections onto quotient spaces. Since f — is not

degenerated, we have

dim[M ] + dim[M ]0 = codim W 0 .

(1)

Note that codim V 0 = codim W 0 . We know dim[M ] = dim(M/M © W 0 ) =

dim(M + W 0 ) ’ dim W 0 and similarly for N . Therefore

dim[M ] + dim[N ] =

= dim(M + W 0 ) ’ dim W 0 + dim(N + V 0 ) ’ dim V 0

(2)

= codim W 0 ’ codim(M + W 0 ) + codim V 0 ’ codim(N + V 0 )

≥ codim W 0 + (codim V 0 ’ codim M ’ codim N ).

According to our assumptions N ‚ M 0 , so that dim[N ] ¤ dim[M ]0 . But then

(1) implies

dim[M ] + dim[N ] ¤ codim W 0

and therefore the term in the last bracket in (2) must be less then zero.

If we ¬x a basis of the vector space Rm then there is the induced basis on

the vector space gr’1 and the induced coordinate expressions of linear forms

C on gr’1 . By naturality of the Lie bracket, using arbitrary coordinates on

Rm the coordinate formula for the Lie bracket does not change. Since ¬ber

respecting linear transformations of Rm+n ’ Rm preserve the projectability of

vector ¬elds, we can use arbitrary a¬ne coordinates on the ¬bration Rm+n ’ Rm

in our discussion on possible codimensions of the subalgebras, which is based on

formula 13.2.(5).

±

The coordinate expression of C will be written like C = (Ci ), i = 1, . . . , m +

i ±‚

±i

n, |±| = r. This means C(X) = ±,i a± x ‚xi ∈ gr’1 ,

±,i Ci a± , if X =

±

where we sum also over repeated indices. For technical reasons we set Ci = 0

whenever i ¤ m and ±j > 0 for some j > m.

If suitable, we also write ± = (±1 , . . . ±m+n ) in the form ± = i1 · · · ir , where

r = |±|, 1 ¤ ij ¤ m + n, so that ±j is the number of indices ik that equal j.

Further we shall use the symbol (j) for a multiindex ± with ±i = 0 for all i = j,

and its length will be clear from the context. As before, the symbol 1j denotes

i

a multiindex ± with ±i = δj .

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21. Actions of jet groups 195

21.5. Lemma. Let C be a non-zero form on gr’1 , m ≥ 1, n ≥ 0. Then in

suitable a¬ne coordinates on the ¬bration Rm+n ’ Rm , the induced coordinate

expression of C satis¬es one of the following conditions:

(1)

(i) Cm = 0 and m > 1.

±+1

(1) ±+1

±

(ii) C1 = 0; Cj = 0 whenever ±j = 0 and 1 ¤ j ¤ m; C1 1 = Cj j (no

summation) for all |±| = r ’ 1, 1 ¤ j ¤ m.

(m+1) ±

(iii) Cm+n = 0, n > 1, and Cj = 0 whenever j ¤ m.

±+1j

±+1

(m+1) ±

(iv) Cm+1 = 0; Cj = 0 if j ¤ m or ±j = 0; and Cm+1m+1 = Cj (no

summation) for all |±| = r ’ 1, j ≥ m + 1.

±

Proof. Let C be a non-zero form on gr’1 with coordinates Cj in the canonical

basis of Rm+n ’ Rm . Let us consider a matrix A ∈ GL(m + n) whose ¬rst row

consists of arbitrary real parameters a1 = t1 = 0, a1 = t2 , . . . , a1 = tm , a1 = 0

m

1 2 j

i

for j > m, and let all the other elements be like in the unit matrix. Let aj be the

˜

’1

elements of the inverse matrix A . If we perform this linear transformation,

we get a new coordinate expression of C, in particular

¯ (1)

Cj = a11 · · · a1r Cs1 ...ir as .

i

(1) ˜j

i i

Hence we get

1

¯ (1)

C1 = ti1 · · · tir C11 ...ir

i

(2)

t1

tj i1 ...ir

¯ (1) Cj1 ...ir ’

i

Cj = ti1 · · · tir for 1 < j ¤ m.

(3) C

t1 1

¯ (1) ±

Formula (2) implies that either we can obtain C1 = 0 or C1 = 0 for all multi

indices ±, |±| = r. Let us assume m > 1 and try to get condition (i). According

to (3), if (i) does not hold after performing any of our transformations, then

the expression on the right hand side of (3) has to be identically zero for all

±

values of the parameters and this implies Cj = 0 whenever ±j = 0, |±| = r,

±+1

±+1

and C1 1 = Cj j for all |±| = r ’ 1, 1 ¤ j ¤ m. Hence we can summarize:

¯±

either (i) can be obtained, or (ii) holds, or Cj = 0 for all 1 ¤ j ¤ m, |±| = r, in

suitable a¬ne coordinates.

Analogously, let us take a matrix A ∈ GL(m + n) whose (m + 1)-st row

consists of real parameters t1 , . . . , tm+n , tm+1 = 0 and let the other elements be

like in the unit matrix. The new coordinates of C are obtained as above

1

¯ (m+1) i1 ...i

Cm+1 =ti1 · · · tir Cm+1r

(4)

tm+1

tj

¯ (m+1) =ti · · · ti Cj1 ...ir ’

i i1 ...i

(5) Cj Cm+1r .

1 r

tm+1

±

Now we may assume Cj = 0 whenever 1 ¤ j ¤ m, for if not then (i) or (ii) could

(m+1)

be obtained. As before, either there is a basis relative to which Cm+1 = 0 or

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196 Chapter V. Finite order theorems

±

Cm+1 = 0 for all |±| = r. Further, according to (5) either we can get (iii) or

±+1

±+1

±

Cj = 0 whenever ±j = 0, and Cm+1m+1 = Cj j , for all |±| = r ’ 1, j ≥ m + 1.

Therefore if both (iii) and (iv) do not hold after arbitrary transformations, then

±

all Cj have to be zero, but this is contradictory to the fact that C is non-zero.

21.6. Lemma. Let p, q be two degrees with p + q = r ’ 1 > 0 and p ≥ q ≥ 0.

Let m > 1 and n > 1 or n = 0, and let hp , hq be subspaces of gp , gq . Let C be a

±

non-zero linear form on gr’1 and suppose [hp , hq ] ‚ ker C. If Ci , 1 ¤ i ¤ m + n,

|±| = r, is a coordinate expression of C satisfying one of the conditions 21.5.(i)-

(iv), then

2m ’ 2, if 21.5.(i) holds

±

2m, if 21.5.(ii) holds and q > 0

m, if 21.5.(ii) holds and q = 0

codim hp + codim hq ≥

2n ’ 2, if 21.5.(iii) holds

2n, if 21.5.(iv) holds and q > 0

n, if 21.5.(iv) holds and q = 0.

Proof. De¬ne a bilinear form

f : gp — gq ’ R f (a, b) = C([a, b]) .

By our assumptions f (hp , hq ) = {0}. Hence by lemma 21.4 it su¬ces to prove

that the codimension of the f -annihilator of gq in gp has the above lower bounds.

Let h0 be this annihilator and consider elements a ∈ h0 , b ∈ gq . We get

Ci ([a, b])i = 0.

±

C([a, b]) = ±

1¤i¤m+n

|±|=r

Using formula for the bracket 13.2.(5) we obtain

µ+»’1j

»j bj ai ’ µj aj bi

0= Ci µ» »µ

1¤i,j¤m+n

|µ|=q+1

|»|=p+1

µ+»’1j µ+»’1i

µi ai bj .

»j ’ Cj

= Ci »µ

1¤i,j¤m+n

|µ|=q+1

|»|=p+1

Since b ∈ gq is arbitrary, we have got a system of linear equations for the

annihilator h0 containing one equation for each couple (j, µ), where 1 ¤ j ¤

m + n, |µ| = q + 1 and µi = 0 whenever i > m and j ¤ m. The (j, µ)-equation

reads

µ+»’1j µ+»’1i

µi ai = 0.

»j ’ Cj

(1) Ci »

1¤i¤m+n

|»|=p+1

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21. Actions of jet groups 197

A lower bound of the codimension of h0 is given by any number of linearly

independent (j, µ)-equations and we have to discuss this separately for the cases

21.5.(i)“(iv).

(1)

Let us ¬rst assume that 21.5.(i) holds, i.e. Cm = 0, m > 1. We denote by Es

the (s, (1))-equation, 1 ¤ s < m and by Fk the (m, (1)+1k )-equation, 1 ¤ k < m

(note that if q = 0 then (1) + 1k = 1k ). We claim that this subsystem is of full

rank. In order to verify this, consider a linear combination

m’1 m’1

s

b k Fk = 0 as , bk ∈ R.

a Es +

s=1 k=1

From (1) we get

m’1

(1)+»’1s (1)+»’1i 1

δi (q + 1) as +

»s ’ Cs

(2) Ci

s=1

1¤i¤m+n

|»|=p+1

m’1

(1)+1k +»’1m

»m ’ Cm k +»’1i (δi q + δi ) bk ai = 0.

(1)+1 1 k

+ Ci »

k=1

Hence all the coe¬cients at the variables ai with 1 ¤ i ¤ m + n, » = p + 1, and

»

»j = 0 whenever j > m and i ¤ m, have to vanish. Therefore, we get equations

on reals as , bk , whenever we choose i and ». We have to show that all these

reals are zero.

(1)

First, let us substitute » = (1) and i = m. Then (2) implies Cm (p + 1)a1 = 0

and consequently a1 = 0. Now we choose » = (1) + 1v , i = m, with 1 < v < m,

(1)

and we get Cm av = 0 so that as = 0 for 1 ¤ s ¤ m ’ 1. Further, take » = (1)

(1)

and 1 < i < m to obtain ’Cm bi = 0. Finally, the choice i = 1 and » = (1)

(1)

leads to ’Cm (q + 1)b1 = 0. In this way, we have proved that the chosen 2m ’ 2

equations Es and Fk are independent and this implies the ¬rst lower bound in

21.6.

Now suppose 21.5.(ii) takes place and let us denote Es the (s, (1))-equation,

1 ¤ s ¤ m, and if q > 0, then Fk will be the (m, (1) + 1m + 1k )-equation,

m m

1 ¤ k ¤ m. As before, we assume s=1 as Es + k=1 bk Fk = 0 for some reals

as and bk and we compare the coe¬cients at ai to show that all these reals are

»

zero. But before doing this, we can simplify all (j, µ)-equations with 1 ¤ j ¤ m

using the relations from 21.5.(ii). Indeed, (1) reduces to

µ+»’1i

(»j ’ µi )ai + R = 0.

Cj »

1¤i¤m

|»|=p+1

where R involves all terms with indices i > m. Consequently Es and Fk have

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

198 Chapter V. Finite order theorems

the forms

(1)+»’1i

(»s ’ δi (q + 1))ai + R = 0

1

Cs »

1¤i¤m

|»|=p+1

Cm k +1m +»’1i (»m ’ δi (q ’ 1) ’ δi ’ δi )ai + R = 0.

(1)+1 1 k m

»

1¤i¤m

|»|=p+1

Assume ¬rst q > 0. If we choose 1 < i ¤ m, » = (1), then the variables ai do

»

not appear in the equations Es at all. Hence the choice i = m, » = (1) gives

(1)+1 (1)

(see 21.5.(ii)) 0 = ’2Cm m bm = ’2C1 bm ; and 1 < i < m, » = (1) now

(1)+1

yields ’Cm m bi = 0. Hence bi = 0 for all 1 < i ¤ m. Further, we take i = m,

» = (1) + 1v + 1m , v = m (note p ≥ q > 0), so that all the coe¬cients in F1 are

(1)

zero. In particular, v = 1 implies C1 a1 = 0 so that a1 = 0. Now, if 1 < v < m,

(1)+1v v (1)

a = C1 av = 0 and what remains are am and b1 , only. Taken

then Cv

(1) (1)+1 (1)

» = (1), i = 1, we see 0 = ’(q + 1)Cm am ’ qCm m b1 = C1 b1 and, ¬nally,

(1)+1

the choice i = m and » = (1) + 1m + 1m gives Cm m 2am = 0. This completes

the proof of the second lower bound in 21.6.

But if q = 0 and 21.5.(ii) holds, we can perform the above procedure after

forgetting all the equations Fk which are not de¬ned. We have only to notice

p + q = r ’ 1 > 0, so that |»| = p + 1 = r ≥ 2.

If n > 1, then the remaining three parts of the proof are complete recapitu-

lations of the above ones. This becomes clear if we notice, that we have used

± ±

neither any information on Cj , j > m, nor the fact that Cj = 0 if j ¤ m and

±i = 0 for some i > m. That is why we can go step by step through the above

proof on replacing 1 or m by m + 1 or m + n, respectively.

If n = 0, then neither 21.5.(iii) nor 21.5.(iv) can hold.

21.7. Proposition. Let h be a subalgebra of gr , m ≥ 1, n ≥ 0, r ≥ 2, which

m,n

does not contain br . Then

r’1

1

codim h ≥ (r ’ 1).

(1)

2

Moreover, if m > 1, n > 1, then

codim h ≥ min{r(m ’ 1), (r ’ 1)m, r(n ’ 1), (r ’ 1)n}

(2)

and if m > 1, n = 0, then

codim h ≥ min{r(m ’ 1), (r ’ 1)m}.

(3)

Proof. In 21.3 we deduced that we may suppose h is a subalgebra with grading

h = h0 • · · · • hr’1 , hi ‚ gi , hr’1 = gr’1 , and we proved the lower bound

(1). Let us assume m > 1, n = 0 and choose a non-zero form C on gr’1 with

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21. Actions of jet groups 199

ker C ⊃ hr’1 . Then we know [hj , hr’j’1 ] ‚ hr’1 ‚ ker C and by lemma 21.5

there is a suitable coordinate expression of C satisfying one of the conditions

21.5.(i), 21.5.(ii). Therefore we can apply lemma 21.6.

±

Assume ¬rst Ci satis¬es 21.5.(i). Then for all j

codim hj + codim hr’j’1 ≥ 2m ’ 2

and consequently

r’1

codim hj ≥ r(m ’ 1).

codim h =

j=0

If 21.5.(ii) holds, then

codim h0 + codim hr’1 ≥ m

codim hj + codim hr’j’1 ≥ 2m

for 1 ¤ j ¤ r ’ 2, so that codim h ≥ m + (r ’ 2)m = (r ’ 1)m. This completes

the proof of (3) and analogous considerations lead to the estimate (2) if n > 1

and the coordinate expression of C satis¬es 21.5.(iii) or 21.5.(iv).

21.8. Examples.

1. Let h1 ‚ gr , m > 1, be de¬ned by

m

‚

; aj = 0 for j = 2, . . . , m, 1 ¤ |(1)| ¤ r}.

h1 = {ai x»

»

‚xi (1)

One sees immediately that the linear subspace h1 consists just of polynomial

vector ¬elds of degree r tangent to the line x2 = x3 = · · · = xm = 0, so that

h1 clearly is a Lie subalgebra in gr of codimension r(m ’ 1). Consider now the

m

subalgebra h ‚ gr consisting of projectable polynomial vector ¬elds of degree

m,n

r over polynomial vector ¬elds from h1 . This is a subalgebra of codimension

r(m ’ 1) in gr .

m,n

2. Consider the algebra h2 ‚ gr , n > 1, de¬ned analogously to the subal-

m+n

gebra h1

‚

; aj

h2 = {ai x» = 0 for 1 ¤ j ¤ m + n, j = m + 1, 1 ¤ |(m + 1)| ¤ r}

»

‚xi (m+1)

and de¬ne h = h2 © gr . Since every polynomial vector ¬eld in gr is tangent

m,n m,n

to the ¬ber over zero, this clearly is a Lie subalgebra with coordinate description

‚

; aj

h = {ai x» = 0 for m + 1 < j ¤ m + n, 1 ¤ |(m + 1)| ¤ r}

»

‚xi (m+1)

and codimension r(n ’ 1).

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200 Chapter V. Finite order theorems

3. Let us recall that the divergence div X of a polynomial vector ¬eld X ∈ grm

r’1 ‚

can be viewed as the jet j0 (div X), see 13.6. So for an element a = ai x» ‚xi

»

we have

»i ai x»’1i .

div a = »

1¤i¤m

1¤|»|¤r

Let M be the line in Rm , m > 1, de¬ned by x2 = x3 = · · · = xm = 0 and

denote by h3 the linear subspace in g1 • · · · • gr’1 (note g0 is missing!)

‚ ‚

h3 = {ai x» ∈ g1 • · · · • gr’1 ; div(ai x» ) = 0}.

» » M

‚xi ‚xi

Of course, h3 is not a Lie subalgebra in gr . Let us further consider the Lie

m

subalgebra h4 ‚ gr’1 consisting of all polynomial vector ¬elds without absolute

m

’1

r

h4 ‚ gr and let us

terms and tangent to M , cf. example 1. Let h5 = πr’1 m

de¬ne a linear subspace

h6 = (h5 © g0 ) • (h3 © h5 ).

First we claim that h3 © h5 is a subalgebra. Indeed, if X, Y ∈ h3 © h5 , then

either the degree of both of them is less then r or their bracket is zero. But in

the ¬rst case, X and Y are tangent to M and their divergences are zero on M ,

so that 13.6.(1) implies div([X, Y ])|M = 0.

Now, consider a polynomial vector ¬eld X from the subalgebra h5 © g0 and a

¬eld Y ∈ h3 © h5 . Since every ¬eld from g0 has constant divergence everywhere

and X is tangent to M , 13.6.(1) implies div([X, Y ])|M = 0. So we have proved

that h6 is a subalgebra. In coordinates, we have

m

‚

∈ gr ; aj = 0,

{ai x» ai 1

(1)+1i (1 + δi |(1)|) = 0

h6 = » m (1)

‚xi i=1

for j = 2, . . . , m, 1 ¤ |(1)| ¤ r ’ 1}.

Now, we take the subalgebra h in gr consisting of polynomial vector ¬elds

m,n

over the ¬elds from h6 . The codimension of h is (r ’ 1)m.

4. Analogously to example 2, let us consider the subalgebra h7 in gr ,

m+n

n > 1,

m+n

‚

∈ gr ; aj m+1

{ai x» ai |(m + 1)|) = 0

h7 = m+n (m+1) = 0, (m+1)+1i (1 + δi

»

‚xi i=1

for j = 1, . . . , m + n, j = m + 1, 1 ¤ |(m + 1)| ¤ r ’ 1}

and let us de¬ne h = h7 © gr . Then

m,n

m+n

‚

∈ gr ; aj m+1

{ai x» ai |(m + 1)|) = 0

h= (m+1) = 0, (m+1)+1i (1 + δi

» m,n

‚xi i=m+1

for j = m + 2, . . . , m + n, 1 ¤ |(m + 1)| ¤ r ’ 1}

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

21. Actions of jet groups 201

and we have found a Lie subalgebra in gr of codimension (r ’ 1)n.

m,n

Let us look at the subgroups corresponding to the above subalgebras. In

the ¬rst and the second examples, the groups consist of polynomial ¬bered iso-

morphisms keeping invariant the given lines. These are closed subgroups. In

the remaining two examples, we have to consider analogous subgroups in Gr’1 ,

m,n

r

then to take their preimages in the group homomorphism πr’1 . Further we con-

sider the subgroups of polynomial local isomorphisms at the origin identical in

linear terms and without the absolute ones. Their subsets consisting of maps

keeping the volume form along the given lines are subgroups. Finally, we take

the intersections of the above constructed subgroups. All these subgroups are

closed.

21.9. Proof of 21.2. The idea of the proof was explained in 21.1 and 21.3. In

particular, we deduced that the dimension of every manifold with an action of

Gr , r ≥ 2, which does not factorize to an action of Gr’1 , is bounded from

m,n m,n

below by the smallest possible codimension of Lie subalgebras h = h0 •· · ·•hr’1 ,

hi ‚ gi , hr’1 = gr’1 , with grading. We also got the lower bound 1 (r ’ 1) for

2

the codimensions and this implied the estimate r ¤ 2 dim S + 1. But now, we

can use proposition 21.7 to get a better lower bound for every m > 1 and n > 1.

Indeed,

s = dim S ≥ min{r(m ’ 1), (r ’ 1)m, r(n ’ 1), (r ’ 1)n}

and consequently

s s s s

r ¤ max{ , + 1, , + 1}.

m’1 m n’1 n

If n = 0 we get

s ≥ min{r(m ’ 1), (r ’ 1)m},

so that

s s

r ¤ max{ , + 1}.

m’1 m

Since all the groups determined by the subalgebras we have constructed in 21.8

are closed, the corresponding homogeneous spaces are examples of manifolds

with actions of Gr with the extreme values of r.

m,n

If m = 1, let us consider h = g0 • gs • gs+1 • · · · • g2s’1 ‚ g2s+1 . Since

1

[gs , gs ] = 0 in dimension one, this is a Lie subalgebra and one can see that the

corresponding subgroup H in G2s+1 is closed (in general, every connected Lie

1

subgroup in a simply connected Lie group is closed, see e.g. [Hochschild, 68, p.

137]). The homogeneous space G2s+1 /H has dimension s and G2s+1 acts non

1 1

trivially. Since there are group homomorphisms Gr ’ Gr and Gr ’ Gr

m,n m m,n n

(the latter one is the restriction of the polynomial maps to the ¬ber over zero),

we have found the two remaining examples.

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

202 Chapter V. Finite order theorems

22. The order of bundle functors

Now we will collect the results from the previous sections to get a description

of bundle functors on ¬bered manifolds. Let us remark that the bundle functors

on categories with the same local skeletons in fact coincide. So we also describe

bundle functors on Mfm and Mf in this way, cf. remark 20.8. In view of

the general description of ¬nite order regular bundle functors on admissible

categories and natural transformations between them deduced in theorems 18.14

and 18.15, the next theorem presents a rather detailed information. As usual

m : FMm,n ’ Mf is the faithful functor forgetting the ¬brations.

22.1. Theorem. Let F : FMm,n ’ Mf , m ≥ 1, n ≥ 0, be a functor endowed

with a natural transformation p : F ’ m and satisfying the localization property

18.3.(i). Then S := p’1 (0) is a manifold of dimension s ≥ 0 and for every

Rm+n

(Y ’ M ) in ObFMm,n the mapping pY : F Y ’ Y is a locally trivial ¬ber

bundle with standard ¬ber S, i.e. F : FMm,n ’ FM. The functor F is a

regular bundle functor of a ¬nite order r ¤ 2s + 1. If moreover m > 1, n = 0,

then

s s

r ¤ max{ , + 1},

m’1 m

and if m > 1, n > 1, then

s s s s

r ¤ max{ , + 1, , + 1}.

m’1 m n’1 n

All these estimates are sharp.

Proof. Since FMm,n is a locally ¬‚at category with Whitney-extendible sets of

morphisms, we have only to prove the assertion concerning the order. The rest

of the theorem follows from theorems 20.3 and 20.5. By de¬nition of bundle

functors, it su¬ces to prove that the action of the group G of germs of ¬bered

morphisms f : Rm+n ’ Rm+n with f (0) = 0 on the standard ¬ber S factorizes

to an action of Gr with the above bounds of r depending on s, m, n.

m,n

As in the proof of theorem 20.5, let V ‚ S be a relatively compact open set

and QV ‚ S be the open submanifold invariant with respect to the action of G,

as de¬ned in 20.4. By virtue of lemma 20.4 the action of G on QV factorizes to

an action of Gk for some k ∈ N. But then theorem 21.2 yields the necessary

m,n

estimates. Moreover, if we consider the Gr -spaces with the extreme orders

m,n

from theorem 21.2, then the general construction of a bundle functor from an

action of the r-th skeleton yields bundle functors with the extreme orders, cf.

18.14.

22.2. Example. All objects in the category FMm,n are of the same type. Now

we will show that the order of bundle functors may vary on objects of di¬erent

types. We shall construct a bundle functor on Mf of in¬nite order.

Consider the sequence of the r-th order tangent functors T (r) from 12.14.

These are bundle functors of orders r ∈ N with values in the category VB of

vector bundles. Let us denote dk the dimension of the standard ¬ber of T (k) Rk

and de¬ne a functor F : Mf ’ FM as follows.

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

22. The order of bundle functors 203

Consider the functors Λk operating on category VB of vector bundles. For

every manifold M the value F M is de¬ned as the Whitney sum over M

Λdk T (k) M

FM =

1¤k¤∞

and for every smooth map f : M ’ N we set

Λdk T (k) f : F M ’ F N .

Ff =

1¤k¤∞

Since Λdk T (k) M = M — {0} whenever k > dim M , the value F M is a well

de¬ned ¬nite dimensional smooth manifold and F f is a smooth map. The ¬ber

projections on T (k) M yield a ¬bration of F M and all the axioms of bundle

functors are easily veri¬ed. Since the order of Λdk T (k) is at least k the functor

F is of in¬nite order.

22.3. The order of bundle functors on FMm . Consider a bundle functor

F : FMm ’ FM and let Fn be its restriction to the subcategory FMm,n ‚

FMm . Write Sn for the standard ¬bers of functors Fn and sn := dim Sn . We

have proved that functors Fn have ¬nite orders rn bounded by the estimates

given in theorem 22.1.

Theorem. Let F : FMm ’ FM be a bundle functor. Then for all ¬bered

¯

manifolds Y with n-dimensional ¬bers and for all ¬bered maps f , g : Y ’ Y ,

rn+1 rn+1 ¯

the condition jx f = jx g implies F f |Fx Y = F g|Fx Y . If dimY ¤ dimY ,

then even the equality of rn -jets implies that the values on the corresponding

¬bers coincide.

Proof. We may restrict ourselves to the case f , g : Rm+n ’ Rm+k , f (0) = g(0) =

0 ∈ Rm+k .

r r

(a) First we discuss the case n = k. Let us assume j0 f = j0 g, r = rn and

consider families ft = f +tidRm+n , gt = g+tidRm+n , t ∈ R. The Jacobians at zero

are certain polynomials in t, so that the maps ft and gt are local di¬eomorphisms

r r

at zero except a ¬nite number of values of t. Since j0 ft = j0 gt for all t, we have

F ft |Sn = F gt |Sn except a ¬nite number of values of t. Hence the regularity of

F implies F f |Sn = F g|Sn .

Every ¬bered map f ∈ FMm (Rm+n , Rm+k ) over f0 : Rm ’ Rm locally de-

composes as f = h —¦ g where g = f0 — idRn : Rm+n ’ Rm+n and h = f —¦ g ’1 is

over the identity on Rm . Hence in the rest of the proof we will restrict ourselves

to morphisms over the identity.

(b) Next we assume n = k + q, q > 0, f , g : Rm+k+q ’ Rm+k , and let

¯

j0 f = j0 g with r = rn . Consider f = (f, pr2 ), g = (g, pr2 ) : Rm+n ’ Rm+n ,

r r

¯

r¯

m+k+q q r

’ R is the projection onto the last factor. Since j0 f = j0 g ,

where pr2 : R ¯

¯ and g = pr —¦¯, the functoriality and (a) imply F f |Sn = F g|Sn .

f = pr1 —¦f 1g

¯

r r

(c) If k = n + 1 and if j0 f = j0 g with r = rn+1 , then we consider f ,

¯

g : Rm+n+1 ’ Rm+n+1 de¬ned by f = f —¦ pr1 , g = g —¦ pr1 . Let us write

¯ ¯

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

204 Chapter V. Finite order theorems

i : Rm+n ’ Rm+n+1 for the inclusion x ’ (x, 0). For every y ∈ Rm+n+1 with

r¯ ¯

r

pr1 (y) = 0 we have jy f = jy g and since f = f —¦i, g = g —¦i, we get F f |Sn = F g|Sn .

¯ ¯

m+n m+n+q

’R , x ’ (x, 0). Analogously

(d) Let k = n + q, q > 0, and i : R

to (a) we may assume that f and g have maximal rank at 0. Hence according

to the canonical local form of maps of maximal rank we may assume g = i.

(e) Let us write f = (idRm , f 1 , . . . , f k ) : Rm+n ’ Rm+k , k > n, and assume

j0 f = j0 i with r = rn+1 . We de¬ne h : Rm+n+1 ’ Rm+k

r r

h(x, y) = (idRm , f 1 (x), . . . , f n (x), y, f n+2 (x), . . . , f k (x)).

Then we have

h —¦ (idRm , idRn , f n+1 ) = f

h —¦ i = (idRm , f 1 , . . . , f n , 0, f n+2 , . . . , f k ).

Since j0 (idRm , idRn , f n+1 ) = j0 i, part (c) of this proof implies

r r

F (idRm+n , f n+1 )|Sn = F i|Sn

and we get for every z ∈ Sn

F f (z) = F h —¦ F i(z) = F (idRm , f 1 , . . . , f n , 0, f n+2 , . . . , f k )(z).

Now, we shall proceed by induction. Let us assume

F f (z) = F (idRm , f 1 , . . . , f n , 0, . . . , 0, f n+s , . . . , f k )(z), s > 1,

r r

for every z ∈ Sn and j0 n+1 f = j0 n+1 i. Let σ : Rm+n+k ’ Rm+n+k be the map

which exchanges the coordinates xn+1 and xn+s , i.e.

σ(x, x1 , . . . , xn , xn+1 , . . . , xn+s , . . . , xk ) =

= (x, x1 , . . . , xn , xn+s , . . . , xn+1 , . . . , xk ).

We get

F (idRm ,f 1 , . . . , f n , 0, . . . , 0, f n+s , . . . , f k )(z) =

= F σ —¦ (idRm , f 1 , . . . , f n , f n+s , 0, . . . , 0, f n+s+1 , . . . , f k ) (z)

= F σ —¦ F (idRm , f 1 , . . . , f n , 0, . . . , 0, f n+s+1 , . . . , f k )(z)

= F (idRm , f 1 , . . . , f n , 0, . . . , 0, f n+s+1 , . . . , f k )(z).

So the induction yields F f (z) = F (idRm , f 1 , . . . , f n , 0, . . . , 0). Since we always

have rn+1 ≥ rn , (a) implies

F (idRm , f 1 , . . . , f n )|Sn = F idRm+n |Sn .

Finally, we get

F f |Sn = F (idRm , f 1 , . . . , f n , 0, . . . , 0)|Sn

= F (i —¦ (idRm , f 1 , . . . , f n ))|Sn = F i|Sn .

Theorem 22.3 reads that every bundle functor on FMm is of locally ¬nite

order and we also have estimates on these ˜local orders™. But there still remains

an open question. Namely, all values on morphisms with an m-dimensional

source manifold depend on rm+1 -jets. It is not clear whether one could get a

better estimate.

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

23. The order of natural operators 205

23. The order of natural operators

In this section, we shall continue the general discussion on natural operators

started in 18.16“18.20. Let us ¬x an admissible category C over manifolds, its

local pointed skeleton (C± , 0± ), ± ∈ I, and consider bundle functors on C.

23.1. The local order. We call a natural domain E of a natural operator

(G1 , G2 ) W-extendible (or Whitney-extendible) if all the domains EA ‚

D: E

∞

CmA (F1 A, F2 A), A ∈ ObC, are W-extendible. We recall that the set of all

sections of any ¬bration is W-extendible, so that the classical natural operators

between natural bundles always have W-extendible domains.

Let us recall that we can apply corollary 19.8 to each q-local operator D : E ‚

C (Y1 , Y2 ) ’ C ∞ (Z1 , Z2 ), where Y1 , Y2 , Z1 , Z2 are smooth manifolds, q : Z1 ’

∞

Y1 is a surjective submersion and E is Whitney-extendible. In particular, D is

of some order k, 0 ¤ k ¤ ∞. Let us consider a mapping s ∈ E, z ∈ Z1

and the compact set K = {z} ‚ Z1 . According to 19.8 applied to K and s,

there is the smallest possible order r =: χ(j ∞ s(q(z)), z) ∈ N such that for all

s ∈ E the condition j r s(q(z)) = j r s(q(z)) implies Ds(z) = D¯(z). Let us write

¯ ¯ s

k k

E ‚ J (Y1 , Y2 ) for the set of all k-jets of mappings from the domain E. The

just de¬ned mapping χ : E ∞ —Y1 Z1 ’ N is called the local order of D.

For every π-local natural operator D : E (G1 , G2 ) with a natural W-

extendible domain E, the operators

∞ ∞

DA : EA ‚ CmA (F1 A, F2 A) ’ CmA (G1 A, G2 A)

are πA -local. The system of local orders (χA )A∈ObC is called the local order of

the natural π-local operator D.

∞

Every locally invertible C-morphism f : A ’ B acts on EA —F1 A G1 A by

f — (jx s, z) = j ∞ (F2 f —¦ s —¦ F1 f ’1 )(F1 f (x)), G1 f (z) .

∞

Lemma. Let D : E (G1 , G2 ) be a natural operator with a natural Whitney-

extendible domain E. For every locally invertible C-morphism f : A ’ B and

∞ ∞

every (jx s, z) ∈ EA —F1 A G1 A we have

χB f — (jx s, z) = χA (jx s, z).

∞ ∞

Proof. Since C is admissible and the domain E is natural, we may restrict

∞

ourselves to A = B = C± , for some ± ∈ I. Assume χA (jx s, z) = r and

j r q(F1 f (x)) = j r (F2 f —¦ s —¦ F1 f ’1 )(F1 f (x)) for some x ∈ F1 C± and s, q ∈ EC± .

Then j r ((F2 f )’1 —¦q—¦F1 f )(x) = j r s(x) and therefore DC± ((F2 f )’1 —¦q—¦F1 f )(z) =

DC± s(z). We have locally for each s ∈ EC±

s —¦ F1 f ’1 = F2 f ’1 —¦ (F2 f —¦ s —¦ F1 f ’1 )

DC± s = G2 f ’1 —¦ DC± (F2 f —¦ s —¦ F1 f ’1 ) —¦ G1 f .

Hence DC± q(G1 f (z)) = G2 f —¦ DC± s(z) and we have proved χB —¦ f — ¤ χA .

Applying the action of the inverse f ’1 we get the converse inequality.

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

206 Chapter V. Finite order theorems

23.2. Consider the associated maps

∞

DC± : EC± —F1 C± G1 C± ’ G2 C±

determined by a natural π-local operator D with a natural W-extendible domain

r r

E. We shall write brie¬‚y E± ‚ EC± for the subset of jets with sources in the

’1

¬ber S± over 0± in F1 C± , and Z± := πC± (S± ) ‚ G1 C± . By naturality, the whole

operator D is determined by the restrictions

∞

D± : E± —S± Z± ’ G2 C±

∞

of the maps DC± . Let us write χ± : E± —S± Z± ’ N for the restrictions of χC± .

Lemma. The maps χ± are G∞ -invariant and if χ± ¤ r, then the operator D is

±

of order r on all objects of type ±.

Proof. The lemma follows immediately from the de¬nition of naturality, the

homogeneity of category C and lemma 23.1.

23.3. The above lemma suggests how to prove ¬niteness of the order in concrete

situations. Namely, theorem 19.7 implies that ˜locally™ χ± is bounded and so it

must be bounded on each orbit under the action of G∞ . Assume now F1 = IdC ,

±

i.e. we deal with a natural pG1 -local operator D : E (G1 , G2 ) with a natural

∞ ∞ ∞

W-extendible domain (EA ‚ C (F A)). Then E± ‚ Tn Q± , where Q± = F0 C±

is the standard ¬ber and n = dim(mC± ). Further assume that the category

C is locally ¬‚at and that the bundle functors F and G1 have the properties

asserted in theorem 20.3 (so this always holds if C has almost W-extendible sets

of morphisms). Consider a section s ∈ EC± ‚ C ∞ (F C± ) invariant with respect

to all translations, i.e. F (tx ) —¦ s —¦ t’x (y) = s(y) for all x ∈ mC± = Rn , y ∈ C±