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±

and denote Z the standard ¬ber (G1 )0 C± .

Lemma. For every compact set K ‚ Z there is an order r ∈ N and a neigh-

∞ ∞ ∞

borhood V ‚ E± ‚ Tn Q± of j0± s in the C r -topology such that χ± ¤ r on

V — K.

Proof. Let us apply theorem 19.7 to the translation invariant section s and a

compact set K — K ‚ C± — Z = G1 C± , where K is a compact neighborhood

of 0± ∈ Rn . We get an order r and a smooth function µ > 0 except for ¬nitely

many points y ∈ K where µ(y) = 0. Let us ¬x x in the interior of K with

µ(x) > 0. Hence there is a neighborhood V of s in the C r -topology on EC±

∞

and a neighborhood U ‚ C± of x in K such that χU (jy q, (y, z)) ¤ r whenever

∞

(y, z) ∈ U — K and q ∈ V . Now, let W be a neighborhood of j0 s in C r -topology

on E± such that tx — W is contained in the set of all in¬nite jets of sections from

∞

V . Since we might assume that tx acts on G1 C± = Rn — Z by G1 tx = tx — idZ

and we have assumed tx — s = s, the lemma follows from lemma 23.2.

Under the assumptions of 23.3 we get

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

23. The order of natural operators 207

23.4. Corollary. Let s ∈ EC± be a translation invariant section and K ‚ Z a

r