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.

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

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: