on an admissible category C. A natural operator D : (F1 , F2 ) (G1 , G2 ) is a

∞ ∞

system of regular operators DA : CmA (F1 A, F2 A) ’ CmA (G1 A, G2 A), A ∈ ObC,

∞ ∞

such that for all s1 ∈ CmA (F1 A, F2 A), s2 ∈ CmB (F1 B, F2 B) and f ∈ C(A, B)

the right-hand diagram commutes whenever the left-hand one does.

u wG A

s1 DA s1

F2 A F1 A G1 A 2

u u u u

F2 f F1 f G1 f G2 f

u wG B

s2 DB s2

F2 B F1 B G1 B 2

This implies the localization property

DL(A,U ) (s|(pF1 )’1 (U )) = (DA s)|(pG1 )’1 (U )

for every A ∈ ObC and every open subset U ‚ mA. If q : G1 ’ F1 is a natural

transformation formed by surjective submersions qA and if all operators DA are

qA -local, then we say that D is q-local.

In the special case F1 = m we have CmA (mA, F2 A) = C ∞ (F2 A), so that DA

∞

transforms sections of F2 A into base-preserving morphisms G1 A ’ G2 A; in this

(G1 , G2 ). Then D is always pG1 -local by de¬nition. If

case we write D : F2

we have a natural surjective submersion qM : G2 M ’ G1 M and we require the

(G2 ’ G1 )

values of operator D to be sections of q, we write D : (F1 , F2 )

(G2 ’ G1 ) in the special case F1 = m. In particular, if G2 is

and D : F2

of the form G2 = H —¦ G1 , where H is a bundle functor on a suitable category,

and q = pH is the bundle projection of H, we write D : (F1 , F2 ) HG1 and

D : F2 HG1 for F1 = m. In the case F1 = m = G1 , we have an operator

G2 transforming sections of F2 A into sections of G2 A for all A ∈ ObC.

D : F2

The classical natural operators from 14.15 correspond to the case C = Mfm .

Example 1. The tangent functor T is de¬ned on the whole category Mf . The

Lie bracket of vector ¬elds is a natural operator [ , ] : T • T T , see 3.10 for

the veri¬cation. Let us remark that the naturality of the bracket with respect to

local di¬eomorphisms follows directly from the fact that its de¬nition does not

depend on any coordinate construction.

Example 2. Let F be a natural bundle over m-manifolds and X be a vector

¬eld on an m-manifold M . If we apply F to the ¬‚ow of X, we obtain the ¬‚ow

of a vector ¬eld FM X on F M . This de¬nes a natural operator F : T TF.

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18. Bundle functors and natural operators 175

18.18. Natural domains. This concept re¬‚ects the situation when the oper-

ators are de¬ned on some morphisms only.

∞

De¬nition. A system of subsets EA ‚ CmA (F1 A, F2 A), A ∈ ObC, is called a

natural domain, if

(i) the restriction of every s ∈ EA to L(A, U ) belongs to EL(A,U ) for every

open subset U ‚ mA,

(ii) for every C-isomorphism f : A ’ B it holds f— (EA ) = EB , where f— (s) =

F2 f —¦ s —¦ (F1 f )’1 , s ∈ EA .

∞

If we replace CmA (F1 A, F2 A) by a natural domain EA in 18.17, we obtain the

de¬nition of a natural operator E (G1 , G2 ).

Example 1. For every admissible category m : C ’ Mf we de¬ne the C-¬elds

on the C-objects as those vector ¬elds on the underlying manifolds, the ¬‚ows

of which are formed by local C-morphisms. For every regular bundle functor

on C there is the ¬‚ow operator F : T T F de¬ned on all C-¬elds. Indeed,

if we apply F to the ¬‚ow of a C-¬eld X ∈ X(mA), we get a ¬‚ow of a vector

¬eld FX on F A. The naturality of F follows from 3.14. In particular, if C is

the category of symplectic 2m-dimensional manifolds, then the C-¬elds are the

locally Hamiltonian vector ¬elds. For the category C of Riemannian manifolds

and isometries, the C-¬elds are the Killing vector ¬elds. If C = FM, we obtain

the projectable vector ¬elds.

Example 2. The Fr¨licher-Nijenhuis bracket is a natural operator [ , ] : T —

o

Λ T • T — Λ T ’ T — Λk+l T — with respect to local di¬eomorphisms by the

k— l—

de¬nition. The functors in question do not act on the whole category Mf .

However, we have proved more than this naturality in section 8. Let us consider

∞

EM = „¦k (M ; T M ) ‚ CM (•k T M, T M ). Then we can view the bracket as

k

an operator [ , ] : (•k T • •l T, T • T ) (•k+l T, T ) with the natural domain

k l

(EM = EM —EM )M ∈ObMf and its naturality follows from 8.15. We remark that

even the Schouten-Nijenhuis bracket satis¬es such a kind of naturality, [Michor,

87b].

18.19. To deduce a result analogous to 14.17 for natural morphism operators,

we shall assume that all C-objects are of the same type and all C-morphisms

are local isomorphisms. Hence the r-th order skeleton of C is one Lie group

Gr ‚ Gr , where m is the dimension of the only object C of a local skeleton of

m

C.

Consider four bundle functors F1 , F2 , G1 , G2 on C and a q-local natural

operator D : (F1 , F2 ) (G1 , G2 ). Then the rule

k

A ’ JmA (F1 A, F2 A) —F1 A G1 A =: HA

with its canonical extension to the C-morphisms de¬nes a bundle functor H on

C. Using 18.16.(1), we deduce quite similarly to 14.15 the following assertion

Proposition. k-th order natural operators D : (F1 , F2 ) (G1 , G2 ) are in bi-

jection with the natural transformations H ’ G2 .

By 18.15, these natural transformations are in bijective correspondence with

the Gs -equivariant maps H0 ’ (G2 )0 between the standard ¬bers, where s is

the maximum of the orders of G2 and H.

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176 Chapter V. Finite order theorems

If we pose some additional natural conditions on such an operator D, they

are re¬‚ected directly in our model. For example, in the case F1 = m assume

we have a natural surjective submersion p : G2 ’ G1 and require every DA s

to be a section of pA . Then the k-order operators of this type are in bijection

with the Gs -maps f : (J k F2 )0 — (G1 )0 ’ (G2 )0 satisfying p0 —¦ f = pr2 , where

p0 : (G2 )0 ’ (G1 )0 is the map induced by p.

18.20. We are going to extend 18.19 to the case of a natural domain E ‚

(F1 , F2 ). Such a domain will be called k-admissible, if

k k

(i) the space EA ‚ JmA (F1 A, F2 A) of all k-jets of the maps from EA is a

k

¬bered submanifold of JmA (F1 A, F2 A) ’ F1 A,

k

(ii) for every smooth curve γ(t) : R ’ EC there is a smoothly parametrized

k

family st ∈ EC such that γ(t) = jc(t) st , where c(t) is the source of γ(t).

The second condition has a similar technical character as the smooth splitting

property in 18.9.

Then the rule

k

A ’ EA —F1 A G1 A =: HA

with its canonical extension to the C-morphisms de¬nes a bundle functor H on

C. Analogously to 18.19 we deduce

Proposition. If E is a k-admissible natural domain, then k-th order natural

(G1 , G2 ) are in bijection with the natural transformations H ’

operators E

G2 .

19. Peetre-like theorems

We ¬rst present the well known Peetre theorem on the ¬niteness of the order

of linear support non-increasing operators. After sketching a non-traditional

proof of this theorem, we discuss the way to its generalization and the most of

this section is occupied by the proof and corollaries of a nonlinear version of the

Peetre theorem formulated in 19.7.

19.1. Let us recall that the support supps of a section s : M ’ L of a vector

bundle L over M is the closure of the set {x ∈ M ; s(x) = 0} and for every op-

erator D : C ∞ (L1 ) ’ C ∞ (L2 ) support non-increasing means supp Ds ‚ supp s

for all sections s ∈ C ∞ (L1 ) .

Theorem, [Peetre, 60]. Consider vector bundles L1 ’ M and L2 ’ M over

the same base M and a linear support non-increasing operator D : C ∞ (L1 ) ’

C ∞ (L2 ). Then for every compact set K ‚ M there is a natural number r such

that for all sections s1 , s2 ∈ C ∞ (L1 ) and every point x ∈ K the condition

j r s1 (x) = j r s2 (x) implies Ds1 (x) = Ds2 (x).

Brie¬‚y, for any compact set K ‚ M , D is a di¬erential operator of some ¬nite

order r on K.

We shall see later that the theorem follows easily from more general results.

However the following direct (but rather sketched) proof based on lemma 19.2.

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19. Peetre-like theorems 177

contains the basic ideas of the forthcoming generalization. By the standard

compactness argument, we may restrict ourselves to M = Rm , L1 = Rm — Rn ,

L2 = Rm — Rp and to view D as a linear map D : C ∞ (Rm , Rn ) ’ C ∞ (Rm , Rp ).

19.2. Lemma. Let D : C ∞ (Rm , Rn ) ’ C ∞ (Rm , Rp ) be a support non increas-

ing linear operator. Then for every point x ∈ Rm and every real constant C > 0,

there is a neighborhood V of x and an order r ∈ N, such that for all y ∈ V \ {x},

s ∈ C ∞ (Rm , Rn ) the condition j r s(y) = 0 implies |Ds(y)| ¤ C.

Proof. Let us assume the lemma is not true for some x and C. Then we can

construct sequences sk ∈ C ∞ (Rm , Rn ) and xk ’ x, xk = x with j k sk (xk ) = 0

and |Dsk (xk )| > C and we can even require |xk ’ xj | ≥ 4|xk ’ x| for all k > j.

Further, let us choose maps qk ∈ C ∞ (Rm , Rn ) in such a way that qk (y) = 0 for

|y ’ xk | > 1 |xk ’ x|, germ sk (xk ) = germ qk (xk ), and maxy∈Rm |‚ ± qk (y)| ¤ 2’k ,

2

0 ¤ |±| ¤ k. This is possible since j k sk (xk ) = 0 for all k ∈ N and we shall not

verify this in detail. Now one can show that the map

∞

y ∈ Rm ,

q(y) := k=0 q2k (y),

is well de¬ned and smooth (note that the supports of the maps qk are disjoint). It

holds germ q(x2k ) = germ s2k (x2k ) and germ q(x2k+1 ) = 0. Since the operator D

is support non-increasing and linear, its values depend on germs only. Therefore

|Dq(x2k+1 )| = 0 and |Dq(x2k )| = |Ds2k (x2k )| > C > 0

which is a contradiction with xk ’ x and Dq ∈ C ∞ (Rm , Rp ).

Proof of theorem 19.1. Given a compact subset K we choose C = 1 and apply

lemma 19.2. We get an open cover of K by neighborhoods Vx , x ∈ K, so we can

choose a ¬nite cover Vx1 , . . . , Vxk . Let r be the maximum of the corresponding

orders. Then the condition j r s(x) = 0 implies |Ds(x)| ¤ 1 for all x ∈ K,

s ∈ C ∞ (Rm , Rn ), with a possible exception of points x1 , . . . , xk ∈ K. But if

|Ds(x)| = µ > 0, then |D( 2 s)(x)| = 2. Hence for all x ∈ K \ {x1 , . . . , xk },

µ

Ds(x) = 0 whenever j r s(x) = 0. The linearity expressed in local coordinates

implies, that this is true for the points x1 , . . . , xk as well.

If we look carefully at the proof of lemma 19.2, we see that the result does

not essentially depend on the linearity of the operator. Dealing with a nonlinear

operator, the assertion can be formulated as follows. For all sections s, q, each

point x and real constant µ > 0, there is a neighborhood V of the point x and

an order r ∈ N such that the values Dq(y) and Ds(y) do not di¬er more then

by µ for all y ∈ V \ {x} with j r q(y) = j r s(y). At the same time, there are two

essential assumptions in the proof only. First, the operator D depends on germs,

and second, the domain of D is the whole C ∞ (Rm , Rn ). Moreover, let us note

that we have used only the continuity of the values in the proof of 19.2. But the

next example shows, that having no additional assumptions on the values of the

operators, there is no reason for any ¬niteness of the order.

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

178 Chapter V. Finite order theorems

19.3. Example. We de¬ne an operator D : C ∞ (R, R) ’ C 0 (R, R). For all

f ∈ C ∞ (R, R) we put

∞

dk f

’k

arctg —¦ k (x) , x ∈ R.

Df (x) = 2

dx

k=0

The value Df (x) depends essentially on j ∞ f (x).

That is why in the rest of this section we shall deal with operators with smooth

values, only. The technique used in 19.2 can be applied to more general types of

operators. We will study the π-local operators D : E ‚ C ∞ (X, Y ) ’ C ∞ (Z, W )

with a continuous map π : Z ’ X, see 18.19 for the de¬nition.

In the nonlinear case we need a general tool for extending a sequence of germs

of sections to one globally de¬ned section. In our considerations, this role will

be played by the Whitney extension theorem:

19.4. Theorem. Let K ‚ Rm be a compact set and let f± be continuous

functions de¬ned on K for all multi-indices ±, 0 ¤ |±| < ∞. There exists a

function f ∈ C ∞ (Rm ) satisfying ‚ ± f |K = f± for all ± if and only if for every

natural number m

1

’ a)β + o (|b ’ a|m )

(1) f± (b) = |β|¤m β! f±+β (a)(b

holds uniformly for |b ’ a| ’ 0, b, a ∈ K.

Let us recall that f (x) = o(|x|m ) means limx’0 f (x)x’m = 0.

The proof is rather complicated and technical and can be found in [Whitney,

34], [Malgrange, 66] or [Tougeron, 72]. If K is a one-point set, we obtain the

classical Borel theorem. We shall work with a special case of this theorem where

the compact set K consists of a convergent sequence of points in Rm . Therefore

we shall use the following assumptions on the domains of the operators.

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

brie¬‚y W-extendible, if for every map f ∈ C ∞ (X, Y ), every convergent sequence

xk ’ x in X and each sequence fk ∈ E and f0 ∈ E, satisfying germ f (xk ) =

germ fk (xk ), k ∈ N, j ∞ f0 (x) = j ∞ f (x), there exists a map g ∈ E and a natural

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

19.6. Examples.

1. By de¬nition E = C ∞ (X, Y ) is Whitney-extendible.

2. Let E ‚ C ∞ (Rm , Rm ) be the subset of all local di¬eomorphisms. Then E

is W-extendible. Indeed, we need to join given germs on some neighborhood of x

only, but the original map f itself has to be a local di¬eomorphism around x, for

j ∞ f (x) = j ∞ f0 (x) and every germ of a locally de¬ned di¬eomorphism on Rm

is a germ of a globally de¬ned local di¬eomorphism. So every bundle functor F

on Mfm de¬nes a map F : E ’ C ∞ (F Rm , F Rm ) which is a pRm -local operator

with W-extendible domain.

3. Consider a ¬bered manifold p : Y ’ M . The set of all sections E = C ∞ (Y )

is W-extendible. Indeed, since we require the extension of given germs on an

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19. Peetre-like theorems 179

arbitrary neighborhood of the limit point x only, we may restrict ourselves to a

local chart Rm — Rn ’ Y . Now, we can work with the coordinate expressions

of the given germs of sections, i.e. with germs of functions. The existence of the

˜extension™ f of given germs implies that the germs of coordinate functions satisfy

condition 19.4.(1), and so there are functions joining these germs. But these

functions represent a coordinate expression of the required section. Therefore

the operators dealt with in 19.1 are idM -local linear operators with W-extendible

domains.

19.7. Nonlinear Peetre theorem. Now we can formulate the main result of

this section. The last technical assumption is that for our π-local operators, the

map π should be locally non-constant, i.e. there are at least two di¬erent points

in the image π(U ) of any open set U .

Theorem. Let π : Z ’ Rm be a locally non-constant continuous map and

let D : E ‚ C ∞ (Rm , Rn ) ’ C ∞ (Z, W ) be a π-local operator with a Whitney-

extendible domain. Then for every ¬xed map f ∈ E and for every compact subset

K ‚ Z there exist a natural number r and a smooth function µ : π(K) ’ R which

is strictly positive, with a possible exception of a ¬nite set of points in π(K),

such that the following statement holds.

For every point z ∈ K and all maps g1 , g2 ∈ E satisfying |‚ ± (gi ’ f )(π(z))| ¤

µ(π(z)), i = 1, 2, 0 ¤ |±| ¤ r, the condition

j r g1 (π(z)) = j r g2 (π(z))

implies

Dg1 (z) = Dg2 (z).

Before going into details of the proof, we present some remarks and corollaries.

19.8. Corollary. Let X, Y , Z, W be manifolds, π : Z ’ X a locally non-

constant continuous map and let D : E ‚ C ∞ (X, Y ) ’ C ∞ (Z, W ) be a π-local

operator with Whitney-extendible domain. Then for every ¬xed map f ∈ E and

for every compact set K ‚ Z, there exists r ∈ N such that for every x ∈ π(K),

g ∈ E the condition j r f (x) = j r g(x) implies

Df |(π ’1 (x) © K) = Dg|(π ’1 (x) © K).

19.9. Multilinear version of Peetre theorem. Let us note that the classical

Peetre theorem 19.1 follows easily from 19.8. Indeed, idM -locality is equivalent to

the condition on supports in 19.1, the sections of a ¬bration form a W-extendible

domain (see 19.6), so we can apply 19.8 to the zero section of the vector bundle

L1 ’ M . Hence for every compact set K ‚ M there is an order r ∈ N such that

Ds(x) = 0 whenever j r s(x) = 0, x ∈ K, s ∈ C ∞ (L), and the classical Peetre

theorem follows.

But applying the full formulation of theorem 19.7, we can prove in a similar

way a ˜multilinear base-extending™ Peetre theorem.

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

180 Chapter V. Finite order theorems

Theorem. Let L1 , . . . , Lk be vector bundles over the same base M , L ’ N

be another vector bundle and let π : N ’ M be continuous and locally non-

constant. If D : C ∞ (L1 ) — · · · — C ∞ (Lk ) ’ C ∞ (L) is a k-linear π-local operator,

then for every compact set K ‚ N there is a natural number r such that for every

x ∈ π(K) and all sections s, q ∈ C ∞ (L1 •· · ·•Lk ) the condition j r s(x) = j r q(x)

implies

Ds|(π ’1 (x) © K) = Dq|(π ’1 (x) © K).

Proof. We may assume Li = Rm — Rni , i = 1, . . . , k. Then all assumptions

of 19.7 are satis¬ed and so, chosen a compact set K ‚ N and the zero section

of L1 • · · · • Lk , we get some order r and a function µ : π(K) ’ R. Consider

arbitrary sections q, s ∈ C ∞ (L1 • · · · • Lk ) and a point x ∈ π(K), µ(x) > 0.

Using multiplication of sections by positive real constants, we can arrange that

all their derivatives up to order r at the point x are less then µ(x). Hence if

j r q(x) = j r s(x), then for a suitable c > 0, c ∈ R, it holds

ck · Ds(z) = D(c · s)(z) = D(c · q)(z) = ck · Dq(z)

for all z ∈ K © π ’1 (x). According to 19.7, the function µ can be chosen in such

a way that the set {x ∈ π(K); µ(x) = 0} is discrete. So the theorem follows from

the multilinearity of the operator and the continuity of its values, what is easily

checked looking at the coordinate description of the multilinear operators.

19.10. One could certainly replace the Whitney extendibility by some other

property, but this cannot be completely omitted. To see this, consider the opera-

tor constructed in 19.3 and let us restrict its domain to the subset E ‚ C ∞ (R, R)

of all polynomials. We get an operator D : E ’ C ∞ (R, R) essentially depending

on in¬nite jets. Also the requirement on π is essential because dropping it, any

action of the group of germs of maps f : (Rm , 0) ’ (Rm , 0) on a manifold should

factorize to an action of some jet group Gr . m

Let us notice that the assertion of our theorem is near to local ¬niteness of the

order with respect to the topology on Z and to the compact open C ∞ -topology

on C ∞ (Rm , Rn ), see e.g. [Hirsch, 76] for de¬nition. It would be su¬cient if we

might always choose a strictly positive function µ : π(K) ’ R in the conclusion

of the theorem. However, example 19.15 shows that this need not be possible in

general. On the other hand, if we add a suitable regularity condition, then the

mentioned local ¬niteness can be proved. Regularity will mean that smoothly

parameterized families of maps in the domain are transformed into smoothly

˜

parameterized families. The idea of the proof is to de¬ne a new operator D

˜

with domain E formed by all one-parameter families of maps, then to perform

˜

a similar construction as in the proof of 19.7 and to apply theorem 19.7 to D to

get a contradiction, see [Slov´k, 88]. Therefore, beside the regularity, we need

a

˜ is also W-extendible. This is not obvious in general, but it is evident if

that E

E consists of all sections of a ¬bration. Since we shall mostly deal with regular

operators de¬ned on all sections of a ¬bration, we present the full formulation.

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

19. Peetre-like theorems 181

Theorem. Let Z, W be manifolds, Y ’ X a ¬bration, π : Z ’ X a locally

non-constant map and let D : E = C ∞ (Y ’ X) ’ C ∞ (Z, W ) be a regular

π-local operator. Then for every ¬xed map f ∈ E and for every compact set

K ‚ Z, there exist an order r ∈ N and a neighborhood V of f in the compact

open C ∞ -topology such that for every x ∈ π(K) and all g1 , g2 ∈ V © E the

condition

j r g1 (x) = j r g2 (x)

implies

Dg1 |(π ’1 (x) © K) = Dg2 |(π ’1 (x) © K).

Similar, but essentially weaker, results can also be deduced dealing with op-

erators with continuous values, see [Chrastina, 87], [Slov´k, 87 b].

a

Let us pass to the proof of 19.7. In the sequel, we ¬x manifolds Z, W , a

locally non-constant continuous map π : Z ’ Rm , a Whitney-extendible subset

E ‚ C ∞ (Rm , Rn ) and a π-local operator D : E ’ C ∞ (Z, W ). The proof is

based on two lemmas.

19.11. Lemma. Let z0 ∈ Z be a point, x0 := π(z0 ), f ∈ E, and let us de¬ne

a function µ : Rm ’ R by µ(x) = exp(’|x ’ x0 |’1 ) if x = x0 and µ(x0 ) = 0.

Then there is a neighborhood V of the point z0 ∈ Z and a natural number

r such that for every z ∈ V ’ π ’1 (x0 ) and all maps g1 , g2 ∈ E satisfying

|‚ ± (gi ’ f )(π(z))| ¤ µ(π(z)), i = 1,2, 0 ¤ |±| ¤ r, the condition j r g1 (π(z)) =

j r g2 (π(z)) implies Dg1 (z) = Dg2 (z).

Proof. We assume the lemma does not hold and we shall ¬nd a contradiction.

If the assertion is not true, then we can construct sequences zk ’ z0 in Z,

xk := π(zk ) ’ x0 and maps fk , gk ∈ E satisfying for all k ∈ N