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18.17. Natural morphism operators. Let F1 , F2 , G1 , G2 be bundle functors
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.

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

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

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

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

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