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CHAPTER V. FINITE ORDER THEOREMS
The purpose of this chapter is to develop a general framework for the theory
of geometric objects and operators and to reduce local geometric considerations
to _nite order problems. In general, the latter is a hard analytical problem and
its solution essentially depends on the category in question. Roughly speaking,
our methods are e_cient when we deal with a su_ciently large class of smooth
maps, but they fail e.g. for analytic maps.
We _rst extend the concepts and results from section 14 to a wider class of
categories. Then we present our important analytical tool, a nonlinear generalization
of well known Peetre theorem. In section 20 we prove the regularity
of bundle functors for a class of categories which includes Mf, Mfm, FM,
FMm, FMm;n, and we get near to the _niteness of the order of bundle functors.
It remains to deduce estimates on the possible orders of jet groups acting
on manifolds. We derive such estimates for the actions of jet groups in the category
FMm;n so that we describe all bundle functors on FMm;n. For n = 0
this reproves in a di_erent way the classical results due to [Palais, Terng, 77]
and [Epstein, Thurston, 79] on the regularity and the _niteness of the order of
natural bundles.
The end of the chapter is devoted to a discussion on the order of natural
operators. Also here we essentially pro_t from the nonlinear Peetre theorem.
First of all, its trivial consequence is that every (even not natural) local operator
depends on in_nite jets only. So instead of natural transformations between the
in_nite dimensional spaces of sections of the bundles in question, we have to deal
with natural transformations between the (in_nite) jet prolongations. The full
version of Peetre theorem implies that in fact the order is _nite on large subsets
of the in_nite jet spaces and, by naturality, the order is invariant under the
action of local isomorphisms on the in_nite jets. In many concrete situations the
whole in_nite jet prolongation happens to be the orbit of such a subset. Then all
natural operators from the bundle in question are of _nite order and the problem
of _nding a full list of them can be attacked by the methods developed in the
next chapter.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
18. Bundle functors and natural operators 169
18. Bundle functors and natural operators
Roughly speaking, the objects of a di_erential geometric category should be
manifolds with an additional structure and the morphisms should be smooth
maps. The following approach is somewhat abstract, but this is a direct modi_-
cation of the contemporary point of view to the concept of a concrete category,
which is de_ned as a category over the category of sets.
18.1. De_nition. A category over manifolds is a category C endowed with a
faithful functor m: C !Mf. The manifold mA is called the underlying manifold
of C-object A and A is said to be a C-object over mA.
The assumption that the functor m is faithful means that every induced map
mA;B : C(A;B) ! C1(mA;mB), A, B 2 ObC, is injective. Taking into account
this inclusion C(A;B) _ C1(mA;mB), we shall use the standard abuse of
language identifying every smooth map f : mA ! mB in mA;B(C(A;B)) with a
C-morphism f : A ! B.
The best known examples of categories over manifolds are the categoriesMfm
orMf, the categories FM, FMm, FMm;n of _bered manifolds, oriented manifolds,
symplectic manifolds, manifolds with _xed volume forms, Riemannian
manifolds, etc., with appropriate morphisms.
For a category over manifolds m: C !Mf, we can de_ne a bundle functor on
C as a functor F : C ! FM satisfying B _ F = m where B: FM!Mf is the
base functor. However, we have seen that the localization property of a natural
bundle over m-dimensional manifolds plays an important role. To incorporate it
into our theory, we adapt the general concept of a local category by [Eilenberg,
57] and [Ehresmann, 57] to the case of a category over manifolds.
18.2. De_nition. A category over manifolds m: C !Mf is said to be local , if
every A 2 ObC and every open subset U _ mA determine a C-subobject L(A;U)
of A over U, called the localization of A over U, such that
(a) L(A;mA) = A, L(L(A;U); V ) = L(A; V ) for every A 2 ObC and every
open subsets V _ U _ mA,
(b) (aggregation of morphisms) if (U_), _ 2 I, is an open cover of mA and f 2
C1(mA;mB) has the property that every f_iU_ is a C-morphism L(A;U_) ! B,
then f is a C-morphism A ! B,
(c) (aggregation of objects) if (U_), _ 2 I, is an open cover of a manifold M
and (A_), _ 2 I, is a system of C-objects such that mA_ = U_ and L(A_;U_ \
U_) = L(A_;U_ \ U_) for all _, _ 2 I, then there exists a unique C-object A
over M such that A_ = L(A;U_).
We recall that the requirement L(A;U) is a C-subobject of A means
(i) the inclusion iU : U ! mA is a C-morphism L(A;U) ! A,
(ii) if for a smooth map f : mB ! U the composition iU _ f is a C-morphism
B ! A, then f is a C-morphism B ! L(A;U).
There are categories like the category VB of vector bundles with no localization
of the above type, i.e. we cannot localize to an arbitrary open subset of the
total space. From our point of view it is more appropriate to consider VB (and
other similar categories) as a category over _bered manifolds, see 51.4.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
170 Chapter V. Finite order theorems
18.3. De_nition. Given a local category C over manifolds, a bundle functor on
C is a functor F : C ! FMsatisfying B_F = m and the localization condition:
(i) for every inclusion of an open subset iU : U ,! mA, F(L(A;U)) is the
restriction p1
A (U) of the value pA : FA ! mA over U and FiU is the
inclusion p1
A (U) ,! FA.
In particular, the projections pA, A 2 ObC, form a natural transformation
p: F ! m. We shall see later on that for a large class of categories one can
equivalently de_ne bundle functors as functors F : C !Mf endowed with such
a natural transformation and satisfying the above localization condition.
18.4. De_nition. A locally de_ned C-morphism of A into B is a C-morphism
f : L(A;U) ! L(B; V ) for some open subsets U _ mA, V _ mB. A C-object A
is said to be locally homogeneous, if for every x, y 2 mA there exists a locally
de_ned C-isomorphism f of A into A such that f(x) = y. The category C is called
locally homogeneous, if each C-object is locally homogeneous. A local skeleton of
a locally homogeneous category C is a system (C_), _ 2 I, of C-objects such that
locally every C-object A is isomorphic to a unique C_. In such a case we say
that A is an object of type _. The set I is called the type set of C. A pointed local
skeleton of a locally homogeneous category C is a local skeleton (C_), _ 2 I,
with a distinguished point 0_ 2 mC_ for each _ 2 I.
A C-morphism f : A ! B is said to be a local isomorphism, if for every
x 2 mA there are neighborhoods U of x and V of f(x) such that the restricted
map U ! V is a C-isomorphism L(A;U) ! L(B; V ). We underline that a local
isomorphism is a globally de_ned map, which should be carefully distinguished
from a locally de_ned isomorphism.
18.5. Examples. All the categories Mfm, Mf, FMm;n, FMm, FM are locally
homogeneous. A pointed local skeleton of the categoryMf is the sequence
(Rm; 0), m = 0; 1; 2; : : : , while a pointed local skeleton of the category FM is
the double sequence (Rm+n ! Rm; 0), m, n = 0; 1; 2 : : : .
18.6. De_nition. The space Jr(A;B) of all r-jets of a C-object A into a Cobject
B is the subset of the space Jr(mA;mB) of all r-jets of mA into mB
generated by the locally de_ned C-morphisms of A into B. If it is useful to
underline the category C, we write CJr(A;B) for Jr(A;B).
18.7. De_nition. A locally homogeneous category C is called in_nitesimally
admissible, if we have
(a) Jr(A;B) is a submanifold of Jr(mA;mB),
(b) the jet projections _rk
: Jr(A;B) ! Jk(A;B), 0 _ k < r, are surjective
submersions,
(c) if X 2 Jr(A;B) is an invertible r-jet of mA into mB, then X is generated
by a locally de_ned C-isomorphism.
Taking into account (c), we write
invJr(A;B) = Jr(A;B) \ invJr(mA;mB):
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
18. Bundle functors and natural operators 171
18.8. Assume C is in_nitesimally admissible and _x a pointed local skeleton
(C_; 0_), _ 2 I. Let us write Cr(_; _) = Jr
0_(C_;C_)0_ for the set of all r-jets
of C_ into C_ with source 0_ and target 0_. De_nition 18.7 implies that every
Cr(_; _) is a smooth manifold, so that the restrictions of the jet composition
Cr(_; _) _ Cr(_; ) ! Cr(_; ) are smooth maps. Thus we obtain a category Cr
over I called the r-th order skeleton of C.
By de_nition 18.7, Gr
_ := invJr
0_(C_;C_)0_ is a Lie group with respect to the
jet composition, which is called the r-th jet group (or the r-th di_erential group)
of type _. Moreover, if A is a C-object of type _, then PrA := invJr
0_(C_;A) is
a principal _ber bundle over mA with structure group Gr
_, which is called the
r-th order frame bundle of A. Let us remark that every jet group Gr
_ is a Lie
subgroup in the usual jet group Gr
m, m = dimC_.
For example, all objects of the category FMm;n are of the same type, so that
FMm;n determines a unique r-th jet group Gr
m;n
_ Gr
m+n in every order r. In
other words, Gr
m;n is the group of all r-jets at 0 2 Rm+n of _bered manifold
isomorphisms f : (Rm+n ! Rm) ! (Rm+n ! Rm) satisfying f(0) = 0.
18.9. The following assumption, which deals with the local skeleton of C only,
has purely technical character.
A category C is said to have the smooth splitting property, if for every smooth
curve : R ! Jr(C_;C_), _, _ 2 I, there exists a smooth map : R _ mC_ !
mC_ such that (t) = jr
c(t)(t; ), where c(t) is the source of r-jet (t).
Since (t) is a curve on Jr(C_;C_), we know that (t) is generated by a
system of locally de_ned C-morphisms. So we require that on the local skeleton
this can be done globally and in a smooth way. In all our concrete examples
the underlying manifolds of the objects of the canonical skeleton are numerical
spaces and each polynomial map determined by a jet of Jr(C_;C_) belongs to
C. This implies immediately that C has the smooth splitting property.
De_nition. An in_nitesimally admissible category C with the smooth splitting
property is called admissible.
18.10. Regularity. From now on we assume that C is an admissible category.
A family of C-morphisms f : M ! C(A;B) parameterized by a manifold M is
said to be smoothly parameterized, if the map M_mA ! mB, (u; x) 7! f(u)(x),
is smooth.
De_nition. A bundle functor F : C ! FM is called regular, if F transforms
every smoothly parameterized family of C-morphisms into a smoothly parameterized
family of FM-morphisms.
18.11. De_nition. A bundle functor F : C ! FM is said to be of order r,
r 2 N, if for any two locally de_ned C-morphisms f and g of A into B, the
equality jrx
f = jrx
g implies that the restrictions of Ff and Fg to the _ber FxA
of FA over x 2 mA coincide.
18.12. Associated maps. An r-th order bundle functor F de_nes the so-called
associated maps
FA;B : Jr(A;B) _mA FA ! FB; (jrx
f; y) 7! Ff(y)
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
172 Chapter V. Finite order theorems
where the _bered product is constructed with respect to the source projection
Jr(A;B) ! mA.
Proposition. The associated maps of an r-th order bundle functor F on an
admissible category C are smooth if and only if F is regular.
Proof. By locality, it su_ces to discuss
FC_;C_ : Jr(C_;C_) _mC_ FC_ ! FC_:
Consider a smooth curve ((t); _(t)) on Jr(C_;C_)_mC_FC_, so that pC__(t) =
c(t), where c(t) is the source of r-jet (t). Since C has the smooth splitting
property, there exists a smooth map : R _ mC_ ! mC_ such that (t) =
jr
c(t)(t; ). The regularity of F implies "(t) := F((t; ))(_(t)) is a smooth curve
on FC_. By the de_nition of the associated map, it holds FC_;C_ ((t); _(t)) =
"(t). Hence FC_;C_ transforms smooth curves into smooth curves. Now, we can
use the following theorem due to [Boman, 67]
A mapping f : Rm ! Rn is smooth if and only if for every smooth curve
c : R ! Rm the composition f _ c is smooth.
Then we conclude FC_;C_ is a smooth map. The other implication is obvious.
_
18.13. The induced action. Consider an r-th order regular bundle functor
F on an admissible category C. The _bers S_ = F0_C_, _ 2 I, will be called the
standard _bers of F. Write F__ for the restriction of FC_;C_ to Cr(_; _)_S_ !
S_. In the following de_nition we consider an arbitrary system (S_), _ 2 I, of
manifolds with indices from the type set of C.
De_nition. A smooth action of Cr on a system (S_), _ 2 I, of manifolds is a
system '__ : Cr(_; _) _ S_ ! S_ of smooth maps satisfying
'_(b; '__(a; s)) = '_(b _ a; s)
for all _, _, 2 I, a 2 Cr(_; _), b 2 Cr(_; ), s 2 S_.
By proposition 18.12, F__ are smooth maps so that they form a smooth action
of Cr on the system of standard _bers.
18.14. Theorem. There is a canonical bijection between the regular r-th order
bundle functors on C and the smooth actions of the r-th order skeleton of C.
Proof. For every regular r-th order bundle functor F on C, F__ is a smooth
action of Cr on (F0_C_), _ 2 I. Conversely, let ('__) be a smooth action of
Cr on a system of manifolds (S_), _ 2 I. The inclusion Gr
_ ,! Cr(_; _) gives a
smooth left action of Gr
_ on S_. For a C-object A of type _ we de_ne GA to be
the _ber bundle associated to PrA with standard _ber S_. For a C-morphism
f : A ! B we de_ne Gf : GA ! GB by
Gf(fu; sg) = fv; '__(v1 _ jrx
f _ u; s)g
x 2 mA, u 2 Pr
xA, v 2 Pr
f(x)B, s 2 S_. One veri_es easily that G is a wellde
_ned regular r-th order bundle functor on C, cf. 14.22. Clearly, if we apply
the latter construction to the action F__, we get a bundle functor naturally
equivalent to the original functor F. _
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
18. Bundle functors and natural operators 173
18.15. Natural transformations. Given two bundle functors F, G: C !
FM, by a natural transformation T : F ! G we shall mean a system of basepreserving
morphisms TA : FA ! GA, A 2 ObC, satisfying Gf _ TA = TB _ Ff
for every C-morphism f : A ! B. (We remark that for a large class of admissible
categories every natural transformation between any two bundle functors is
formed by base-preserving morphisms, see 14.11.)
Given two smooth actions ('__; S_) and ( __;Z_), a Cr-map
_ : ('__; S_) ! ( __;Z_)
is a system of smooth maps __ : S_ ! Z_, _ 2 I, satisfying
__('__(a; s)) = __(a; __(s))
for all s 2 S_, a 2 Cr(_; _).
Theorem. Natural transformations F ! G between two r-th order regular
bundle functors on C are in a canonical bijection with the Cr-maps between the
corresponding actions of Cr.
Proof. Given T : F ! G, we de_ne __ : F0_C_ ! G0_C_ by __(s) = TC_(s).
One veri_es directly that (__) is a Cr-map (F__; F0_C_) ! (G__;G0_C_). Conversely,
let (__) : ('__; S_) ! ( __;Z_) be a Cr-map between two smooth actions
of Cr. Then the induced bundle functors transform A 2 ObC of type _ into
the _ber bundle associated with PrA with standard _bers S_ and Z_ and we
de_ne TA = (idPrA; __). One veri_es easily that T is a natural transformation
between the induced bundle functors. _
18.16. Morphism operators. We are going to generalize the concept of natural
operator from 14.15 in the following three directions: 1. We replace the
category Mfm by an admissible category C over manifolds. 2. We consider the
operators de_ned on morphisms of _bered manifolds. 3. We study an operator
de_ned on some morphisms only, not on all of them. We start with the general
concept of a morphism operator.
If Y1 ! M and Y2 ! M are two _bered manifolds, we denote by C1
M(Y1; Y2)
the space of all base-preserving morphisms Y1 ! Y2. Given another pair Z1 !
M and Z2 ! M of _bered manifolds, a morphism operator D is a map D: E _
C1
M(Y1; Y2) ! C1
M(Z1;Z2). In the case Z1 is a _bered manifold over Y1, i.e. we
have a surjective submersion q : Z1 ! Y1, we also say that D is a base extending
operator.
In general, if we have four manifolds N1, N2, N3, N4, a map _ : N3 ! N1 and
a subset E _ C1(N1;N2), an operator A: E ! C1(N3;N4) is called _-local,
if the value As(x) depends only on the germ of s at _(x) for all s 2 E, x 2 N3.
Such an operator is said to be of order k, 0 _ k _ 1, if jk
_(x)s1 = jk
_(x)s2 implies
As1(x) = As2(x) for all s1, s2 2 E, x 2 N3. We call A regular if smoothly parameterized
families in E are transformed into smoothly parameterized families
in C1(N3;N4).
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
174 Chapter V. Finite order theorems
Assume we have a surjective submersion q : Z1 ! Y1. Then we have de_ned
both local and k-th order operators C1
M(Y1; Y2) ! C1
M(Z1;Z2) with respect to
q. Such a k-th order operator D determines the associated map
(1) D: JkM
(Y1; Y2) _Y1 Z1 ! Z2; (jk
y s; z) 7! Ds(z); y = q(z);
where JkM
(Y1; Y2) means the space of all k-jets of the maps of C1
M(Y1; Y2). If
D is regular, then D is smooth. Conversely, every smooth map (1) de_nes a
regular operator C1
M(Y1; Y2) ! C1
M(Z1;Z2), s 7! D((jks) _ q; ) : Z1 ! Z2,
s 2 C1
M(Y1; Y2).
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 : C1
mA(F1A; F2A) ! C1
mA(G1A;G2A), A 2 ObC,
such that for all s1 2 C1
mA(F1A; F2A), s2 2 C1
mB(F1B; F2B) and f 2 C(A;B)
the right-hand diagram commutes whenever the left-hand one does.
F2A
u
F2f
u F1A s1
u
F1f
G1A w
DAs1
u
G1f
G2A
u
G2f
F2B u F1B s2 G1B w
DBs2 G2B
This implies the localization property
DL(A;U)(sj(pF1 )1(U)) = (DAs)j(pG1 )1(U)
for every A 2 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 C1
mA(mA; F2A) = C1(F2A), so that DA
transforms sections of F2A into base-preserving morphisms G1A ! G2A; in this
case we write D: F2 (G1;G2). Then D is always pG1 -local by de_nition. If
we have a natural surjective submersion qM : G2M ! G1M and we require the
values of operator D to be sections of q, we write D: (F1; F2) (G2 ! G1)
and D: F2 (G2 ! G1) in the special case F1 = m. In particular, if G2 is
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
D: F2 G2 transforming sections of F2A into sections of G2A for all A 2 ObC.
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 categoryMf. 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 FMX on FM. 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 reects the situation when the operators
are de_ned on some morphisms only.
De_nition. A system of subsets EA _ C1
mA(F1A; F2A), A 2 ObC, is called a
natural domain, if
(i) the restriction of every s 2 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) =
F2f _ s _ (F1f)1, s 2 EA.
If we replace C1
mA(F1A; F2A) 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 TF de_ned on all C-_elds. Indeed,
if we apply F to the ow of a C-_eld X 2 X(mA), we get a ow of a vector
_eld FX on FA. 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 Frolicher-Nijenhuis bracket is a natural operator [ ; ] : T
_kT_ _ T _lT_ ! T _k+lT_ with respect to local di_eomorphisms by the
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
EkM
= k(M; TM) _ C1
M(_kTM; TM). Then we can view the bracket as
an operator [ ; ] : (_kT _ _lT; T _ T) (_k+lT; T) with the natural domain
(EM = EkM
_ElM )M2ObMf 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
m, where m is the dimension of the only object C of a local skeleton of
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
A 7! Jk
mA(F1A; F2A) _F1A G1A =: 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 bijection
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 reected directly in our model. For example, in the case F1 = m assume
we have a natural surjective submersion p: G2 ! G1 and require every DAs
to be a section of pA. Then the k-order operators of this type are in bijection
with the Gs-maps f : (JkF2)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
(i) the space EkA
_ Jk
mA(F1A; F2A) of all k-jets of the maps from EA is a
_bered submanifold of Jk
mA(F1A; F2A) ! F1A,
(ii) for every smooth curve (t) : R ! EkC
there is a smoothly parametrized
family st 2 EC such that (t) = jk
c(t)st, where c(t) is the source of (t).
The second condition has a similar technical character as the smooth splitting
property in 18.9.
Then the rule
A 7! EkA
_F1A G1A =: 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
operators E (G1;G2) are in bijection with the natural transformations H !
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 fx 2 M; s(x) 6= 0g and for every operator
D: C1(L1) ! C1(L2) support non-increasing means suppDs _ supp s
for all sections s 2 C1(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: C1(L1) !
C1(L2). Then for every compact set K _ M there is a natural number r such
that for all sections s1; s2 2 C1(L1) and every point x 2 K the condition
jrs1(x) = jrs2(x) implies Ds1(x) = Ds2(x).
Briey, 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: C1(Rm;Rn) ! C1(Rm;Rp).
19.2. Lemma. Let D: C1(Rm;Rn) ! C1(Rm;Rp) be a support non increasing
linear operator. Then for every point x 2 Rm and every real constant C > 0,
there is a neighborhood V of x and an order r 2 N, such that for all y 2 V n fxg,
s 2 C1(Rm;Rn) the condition jrs(y) = 0 implies jDs(y)j _ C.
Proof. Let us assume the lemma is not true for some x and C. Then we can
construct sequences sk 2 C1(Rm;Rn) and xk ! x, xk 6= x with jksk(xk) = 0
and jDsk(xk)j > C and we can even require jxk xj j _ 4jxk xj for all k > j.
Further, let us choose maps qk 2 C1(Rm;Rn) in such a way that qk(y) = 0 for
jy xkj > 1
2
jxk xj, germ sk(xk) = germ qk(xk), and maxy2Rm j@_qk(y)j _ 2k,
0 _ j_j _ k. This is possible since jksk(xk) = 0 for all k 2 N and we shall not
verify this in detail. Now one can show that the map
q(y) :=
P1
k=0 q2k(y); y 2 Rm;
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
jDq(x2k+1)j = 0 and jDq(x2k)j = jDs2k(x2k)j > C > 0
which is a contradiction with xk ! x and Dq 2 C1(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 2 K, so we can
choose a _nite cover Vx1 ; : : : ; Vxk . Let r be the maximum of the corresponding
orders. Then the condition jrs(x) = 0 implies jDs(x)j _ 1 for all x 2 K,
s 2 C1(Rm;Rn), with a possible exception of points x1; : : : ; xk 2 K. But if
jDs(x)j = " > 0, then jD( 2
" s)(x)j = 2. Hence for all x 2 K n fx1; : : : ; xkg,
Ds(x) = 0 whenever jrs(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 " > 0, there is a neighborhood V of the point x and
an order r 2 N such that the values Dq(y) and Ds(y) do not di_er more then
by " for all y 2 V n fxg with jrq(y) = jrs(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 C1(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: C1(R;R) ! C0(R;R). For all
f 2 C1(R;R) we put
Df(x) =
1X
k=0
2k
_
arctg _
dkf
dxk (x)
_
; x 2 R:
The value Df(x) depends essentially on j1f(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 _-local operators D: E _ C1(X; Y ) ! C1(Z;W)
with a continuous map _ : 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 _ j_j < 1. There exists a
function f 2 C1(Rm) satisfying @_fjK = f_ for all _ if and only if for every
natural number m
(1) f_(b) =
P
j_j_m
1
_!f_+_(a)(b a)_ + o (jb ajm)
holds uniformly for jb aj ! 0, b, a 2 K.
Let us recall that f(x) = o(jxjm) means limx!0 f(x)xm = 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 _ C1(X; Y ) is said to be Whitney-extendible, or
briey W-extendible, if for every map f 2 C1(X; Y ), every convergent sequence
xk ! x in X and each sequence fk 2 E and f0 2 E, satisfying germ f(xk) =
germ fk(xk), k 2 N, j1f0(x) = j1f(x), there exists a map g 2 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 = C1(X; Y ) is Whitney-extendible.
2. Let E _ C1(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
j1f(x) = j1f0(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 ! C1(FRm; FRm) which is a pRm-local operator
with W-extendible domain.
3. Consider a _bered manifold p: Y ! M. The set of all sections E = C1(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 _-local operators, the
map _ should be locally non-constant, i.e. there are at least two di_erent points
in the image _(U) of any open set U.
Theorem. Let _ : Z ! Rm be a locally non-constant continuous map and
let D: E _ C1(Rm;Rn) ! C1(Z;W) be a _-local operator with a Whitneyextendible
domain. Then for every _xed map f 2 E and for every compact subset
K _ Z there exist a natural number r and a smooth function ": _(K) ! R which
is strictly positive, with a possible exception of a _nite set of points in _(K),
such that the following statement holds.
For every point z 2 K and all maps g1, g2 2 E satisfying j@_(gif)(_(z))j _
"(_(z)), i = 1; 2, 0 _ j_j _ r, the condition
jrg1(_(z)) = jrg2(_(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, _ : Z ! X a locally nonconstant
continuous map and let D: E _ C1(X; Y ) ! C1(Z;W) be a _-local
operator with Whitney-extendible domain. Then for every _xed map f 2 E and
for every compact set K _ Z, there exists r 2 N such that for every x 2 _(K),
g 2 E the condition jrf(x) = jrg(x) implies
Dfj(_1(x) \ K) = Dgj(_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 2 N such that
Ds(x) = 0 whenever jrs(x) = 0, x 2 K, s 2 C1(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 _ : N ! M be continuous and locally nonconstant.
If D: C1(L1)__ _ __C1(Lk) ! C1(L) is a k-linear _-local operator,
then for every compact set K _ N there is a natural number r such that for every
x 2 _(K) and all sections s, q 2 C1(L1__ _ __Lk) the condition jrs(x) = jrq(x)
implies
Dsj(_1(x) \ K) = Dqj(_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 ": _(K) ! R. Consider
arbitrary sections q, s 2 C1(L1 _ _ _ _ _ Lk) and a point x 2 _(K), "(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 "(x). Hence if
jrq(x) = jrs(x), then for a suitable c > 0, c 2 R, it holds
ck _ Ds(z) = D(c _ s)(z) = D(c _ q)(z) = ck _ Dq(z)
for all z 2 K \ _1(x). According to 19.7, the function " can be chosen in such
a way that the set fx 2 _(K); "(x) = 0g 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 operator
constructed in 19.3 and let us restrict its domain to the subset E _ C1(R;R)
of all polynomials. We get an operator D: E ! C1(R;R) essentially depending
on in_nite jets. Also the requirement on _ 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 C1-topology
on C1(Rm;Rn), see e.g. [Hirsch, 76] for de_nition. It would be su_cient if we
might always choose a strictly positive function ": _(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_ak, 88]. Therefore, beside the regularity, we need
that ~E is also W-extendible. This is not obvious in general, but it is evident if
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, _ : Z ! X a locally
non-constant map and let D: E = C1(Y ! X) ! C1(Z;W) be a regular
_-local operator. Then for every _xed map f 2 E and for every compact set
K _ Z, there exist an order r 2 N and a neighborhood V of f in the compact
open C1-topology such that for every x 2 _(K) and all g1, g2 2 V \ E the
condition
jrg1(x) = jrg2(x)
implies
Dg1j(_1(x) \ K) = Dg2j(_1(x) \ K).
Similar, but essentially weaker, results can also be deduced dealing with operators
with continuous values, see [Chrastina, 87], [Slov_ak, 87 b].
Let us pass to the proof of 19.7. In the sequel, we _x manifolds Z, W, a
locally non-constant continuous map _ : Z ! Rm, a Whitney-extendible subset
E _ C1(Rm;Rn) and a _-local operator D: E ! C1(Z;W). The proof is
based on two lemmas.
19.11. Lemma. Let z0 2 Z be a point, x0 := _(z0), f 2 E, and let us de_ne
a function ": Rm ! R by "(x) = exp(jx x0j1) if x 6= x0 and "(x0) = 0.
Then there is a neighborhood V of the point z0 2 Z and a natural number
r such that for every z 2 V _1(x0) and all maps g1, g2 2 E satisfying
j@_(gi f)(_(z))j _ "(_(z)), i = 1,2, 0 _ j_j _ r, the condition jrg1(_(z)) =
jrg2(_(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 := _(zk) ! x0 and maps fk, gk 2 E satisfying for all k 2 N
j@_(fk f)((1) xk)j _ "(xk) for all 0 _ j_j _ k
(2) jkfk(xk) = jkgk(xk)
(3) Dfk(zk) 6= Dgk(zk):
Since all xk are di_erent from x0, by passing to subsequences we can assume
(4) jxk+1 x0j <
1
4
jxk x0j:
Let us _x Riemannian metrics _Z or _W on Z or W, respectively, and choose
further points _zk 2 Z, _zk ! z0, _xk := _(_zk) and neighborhoods Uk or Vk of xk
or _xk, respectively, in such a way that for all k 2 N the following six conditions
hold
(5) jxk x0j _ 2ja bj for all a 2 Uk [ Vk, b 2 Uj [ Vj , k 6= j
(6) j@_(fk f)(a)j _ 2"(xk) for all a 2 Uk [ Vk, 0 _ j_j _ k
j@_(gk f)(a)j _ 2"(xk) for all a 2 Uk (7) [ Vk, 0 _ j_j _ k
(8) _W(Dgk(zk);Dfk(z_k)) _ k_Z(zk; z_k)
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
182 Chapter V. Finite order theorems
and for all m, k 2 N, and multi-indices _ with j_j+2m _ k, a 2 Uk, and b 2 Vk
we require
1
jb ajm
______
X
j_j_m
1
_!@_+_fk(b)(a b)__
@_gk(a)
______
_
1
k
(9)
1
jb ajm
______
X
j_j_m
1
_!@_+_gk(a)(b a)__
@_fk(b)
______
_
1
k
(10) :
All these requirements can be satis_ed. Indeed, the equalities (5), (6), (7) are
valid for all points a, b from some suitable neighborhoods Wk of the points xk.
By the Taylor formula, for any _xed k and j_j + m _ k, (2) implies @_gk(a) =
@_fk(a) + o(ja xkjm). Therefore, if we consider only points a, b 2 Wk such
that
(11) jb xkj _ 2jb aj; ja xkj _ 2jb aj;
then under the condition j_j + 2m _ k we get ( note that o(ja xkjm) or
o(jb xkjm) now implies o(ja bjm))
X
j_j_m
1
_!@_+_fk(b)(a b)_ =
X
j_j_m
1
_!@_+_gk(b)(a b)_ + o(ja bjm)
= @_gk(a) + o(jb ajm)
X
j_j_m
1
_!@_+_gk(a)(b a)_ =
X
j_j_m
1
_!@_+_fk(a)(b a)_ + o(ja bjm)
= @_fk(b) + o(jb ajm):
Hence also conditions (9), (10) are realizable if we take Uk, Vk in su_ciently small
neighborhoods Wk of xk in such a way that (11) holds for all a 2 Uk, b 2 Vk. By
virtue of (3), there are also neighborhoods of the points zk in Z ensuring (8).
Finally, we are able to choose appropriate points _zk and neighborhoods Uk, Vk
using the fact that _ is continuous and locally non-constant.
The aim of conditions (1), (4){(7), (9), (10) is to guarantee the existence of
a map h 2 C1(Rm;Rn) satisfying
(12) germ h(xk) = germ gk(xk) and germ h(_xk) = germ fk(_xk):
Then, by virtue of our requirements on E, we may assume h 2 E, provided we
use (12) for large indices k, only. But applying D to h, the _-locality and (8)
imply
_W(Dh(zk);Dh(_zk)) _ k_Z(zk; _zk)
for large k's, and this is a contradiction with Dh 2 C1(Z;W) and (zk; _zk) !
(z0; z0).
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
19. Peetre-like theorems 183
So it remains to verify condition 19.4.(1) in the Whitney extension theorem
with K =
S
k(_Uk [ _ Vk) [ fx0g and f_(x) = @_gk(x) if x 2 _Uk, f_(x) = @_fk(x)
if x 2 _ Vk and f_(x0) = @_f(x0). This follows by our construction for all couples
(a; b) 2
S
k(_Uk _ _ Vk), see (9), (10). In all other cases and for all m 2 N we have
to use (6) and (7), (5), the Taylor formula, (6) and (7), and (5) to get
P
j_j_m
1
_!f_+_(a)(b a)_ =
P
j_j_m
1
_!
@_+_f(a) + o(jxk(a)
x0jm)
_
(b a)_
=
P
j_j_m
1
_!@_+_f(a)(b a)_ + o(jb ajm)
= @_f(b) + o(jb ajm)
= f_(b) + o(jxk(b)
x0jm) + o(jb ajm)
= f_(b) + o(jb ajm): _
19.12. Lemma. Let z0 2 Z be a point, x = _(z0) and f 2 E. Then there is
a neighborhood V of z0 in _1(x) and a natural number r such that for every
z 2 V and all maps g 2 E the condition jrg(x) = jrf(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 2 E with j1g1(x) = j1g2(x) and
a point y 2 _1(x). Let us choose a sequence yk ! y in Z, _(yk) =: xk 6= x
and neighborhoods Uk of xk satisfying ja xj _ 2ja bj for all a 2 Uk, b 2 Uj ,
k 6= j. Using the Whitney extension theorem 19.4, the Taylor formula, and our
assumptions on E we _nd a map h 2 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, _(zk) = x and maps gk 2 E satisfying for all k 2 N
(1) jkf(x) = jkgk(x)
(2) Dgk(zk) 6= Df(zk):
We choose further points _zk ! z0 in Z, _xk := _(_zk), _xk 6= x, and neighborhoods
Vk of _xk in such a way that
(3) _W(Dgk(z_k);Df(zk)) _ k_Z(z_k; zk) for all k 2 N
(4) ja xj _ 2ja bj for all a 2 Vk, b 2 Vj , k 6= j
j@_(gk f)(a)j
ja xjm
_
1
k
(5) for all a 2 Vk, j_j + 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 2 E satisfying
germ h(_xk) = germ gk(_xk) and j1h(x) = j1f(x)
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
184 Chapter V. Finite order theorems
for large k's. Hence (3) and the _rst part of this proof imply
_W(Dh(_zk);Dh(zk)) = _W(Dgk(_zk);Df(zk)) _ k_Z(_zk; zk)
which is a contradiction with Dh 2 C1(Z;W). _
Proof of theorem 19.7. According to lemmas 19.11 and 19.12, for every point
z 2 K we _nd a neighborhood Vz of z, an order rz and a smooth function
"z : _(Vz) ! R which is strictly positive with a possible exception of the point
_(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 _ C1(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 _ C1(Rm; Y ) is said to be almost Whitney-extendible
if for every map f 2 C1(Rm; Y ), sequence fk 2 E, f0 2 E and every convergent
sequence xk ! x satisfying for all k 2 N, jxk xj _ 2jxk+1 xj, germ f(xk) =
germ fk(xk), j1f(x) = j1f0(x), there is a map g 2 E and a natural number k0
satisfying germ g(xk) = germ fk(xk) for all k _ k0.
19.14. Proposition. Let _ : Z ! Rm be a locally non-constant continuous
map, E _ C1(Rm; Y ) be an almost Whitney-extendible subset and let D: E !
C1(Z;W) be a _-local operator. Then for every _xed map f 2 E, point x 2 Rm,
and for every compact subset K _ _1(x), there exists a natural number r such
that for all maps g 2 E the condition jrg(x) = jrf(x) implies DgjK = DfjK.
Proof. The proposition is implied by lemma 19.12 and by the standard compactness
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: C1(R;R) ! C1(R;R)
such that if we take f = idR, then for any order r and any compact neighborhood
K of 0 2 R, every function ": R ! R from 19.7 satis_es "(0) = 0.
Let g : R2 ! R be a function with the following three properties
(1) g is smooth in all points x 2 R2 n f(0; 1)g
(2) lim supx!1 g(0; x) = 1
(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) 6= 0 if and only if
jxj > t > 0.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
20. The regularity of bundle functors 185
Given f 2 C1(R;R), x 2 R, we de_ne
Df(x) =
1X
k=0
__
(a(k;) _ g _ (f _
df
dx
)(x)
_
dkf
dxk (x)
_
:
The sum is locally _nite if g _ (f _ df
dx ) is locally bounded. Hence Df is well
de_ned and smooth if g _ (f _ df
dx ) is smooth. The only di_culty may happen if
we deal with some f 2 C1(R;R) and x 2 R with f(x) = 0, df
dx (x) = 1. However,
in this case it holds
lim
y!x
df
dx (y) 1
f(y)
= d2f
dx2 (x)
and the property (3) of g implies g _ (f _ df
dx ) = 0 on some neighborhood of x.
On the other hand, for f = idR, arbitrary " > 0 and order r 2 N, there are
functions h1, h2 2 C1(R;R) such that jrh1(0) = jrh2(0), j dk
dxk (h1idR)(0)j < "
for all 0 _ k _ r, and Dh1(0) 6= 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
FC_, _x = F(tx). In section 14 we used this action and the regularity of the
natural bundles to _nd canonical di_eomorphisms FRm _= Rm _ p1
Rm(0). The
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 FC_ 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 functor
endowed with a natural transformation p: F ! m such that the locality
condition 18.3.(i) holds. Then there are canonical di_eomorphisms
(1) mC_ _ p1
C_
(0_) _= FC_; (x; z) 7! Ftx(z)
and for every A 2 ObC of type _ the map pA : FA ! A is a locally trivial _ber
bundle with standard _ber p1
C_
(0_). In particular F is a bundle functor on C.
Proof. Let us _x a type _ and write Rm for mC_. By proposition 20.2, the action
_ : Rm _ FC_ ! FC_ is a continuous action and each map _x : FC_ ! FC_ is
a di_eomorphism. But then a general theorem, see 5.10, implies that this action
is smooth. It follows that for every z 2 p1
C_
(0_) the map s: Rm ! FC_, s(x) =
_x(z) is smooth and pC_
_ s = idRm. Therefore pC_ is a submersion and p1
C_
(0_)
is a manifold. Since both the maps (x; z) 7! _ (x; z) and y 7! _ (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 FA 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 2 A and a compact set K _ p1
A (x) _ FA. We de_ne
QK := [
f2invC(A;A)Ff(K):
Lemma. If C(A;A) _ C1(mA;mA) is almost Whitney-extendible, then for
every compact K as above there is an order r 2 N such that for all invertible
C-morphisms f, g and for every point y 2 A the equality jr
yf = jr
yg implies
Ffj(QK \ p1
A (y)) = Fgj(QK \ p1
A (y)):
Proof. Let us _x the map idA 2 C(A;A) and let us apply proposition 19.14 to
F : C(A;A) ! C1(FA; FA), _ = pA and K. We denote by r the resulting order.
For every z 2 QK there are y 2 K and g 2 invC(A;A) with Fg(y) = z. Consider
f1, f2 2 invC(A;A) such that jrf1(_(z)) = jrf2(_(z)). Then jr(f1 _ g)(_(y)) =
jr(f2 _ g)(_(y)) and therefore jr(g1 _ f1
1
_ f2 _ g)(_(y)) = jridA(_(y)). Hence
Ff1(z) = Ff1 _ Fg(y) = Ff2 _ Fg(y) = Ff2(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 locally
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 2 C(C;C) with parameters in a manifold P. For any z 2 FC, x = p(z),
f 2 C(C;C) we have
(1) Ff(z) = _f(x)
_ F(tf(x)
_ f _ tx) _ _x(z)
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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 Fgs to the standard _ber S = p1(0) are
smoothly parameterized. Since the case m = 0 is trivial, we may assume m > 0.
By lemma 20.4 F is of order 1. We _rst show that the induced action of the
group of in_nite jets G1
_ = invJ1
0 (C;C)0 on S is continuous with respect to the
inverse limit topology.
Consider converging sequences zn ! z in S and j1
0 fn ! j1
0 f0 in G1
_ . We
shall show that any subsequence of Ffn(zn) contains a further subsequence converging
to the point Ff0(z). On replacing fn by fn _ f1
0 , 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 en. Let us
choose positive reals "n < en in such a way that on the open balls B(0; "n)
centered at 0 with diameters "n all the derivatives in question vary at most by
en. Let xn := (2n; 0; : : : ; 0) 2 Rm. By the Whitney extension theorem there
is a local di_eomorphism f : Rm ! Rm such that
fjB(x2n+1; "2n+1) = idC and fjB(x2n; "2n) = tx2n
_ f2n _ tx2n
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
_ Fh _ _xn(zn) = Ffn(zn) if n is even
_xn
_ Fh _ _xn(zn) = zn if n is odd.
Hence, by virtue of proposition 20.2, Ff2n(z2n) converges to z and we have
proved the continuity of the action of G1
_ on S as required.
Now, let us choose a relatively compact open neighborhood V of z and de_ne
QV := ([
f2invC(C;C)Ff(V )) \ S. This is an open submanifold in S and the
functor F de_nes an action of the group G1
_ 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 theorems
20.3 and 20.5.
20.7. Corollary. Every bundle functor on FMm is regular.
Proof. The system (Rm+n ! Rm; 0), n 2 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
(g0 = f0, h1(x; y) = f1(f1
0 (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;nmorphisms.
Hence it remains to discuss the latter type of morphisms. We may restrict
ourselves to families hp : Rm+n ! Rm+k parameterized by p 2 Rq, for some
q 2 N. Let us consider i : Rm+n ! Rm+n _Rq, (x; y) 7! (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 Fhp = Fh _ Ft(0;0;p)
_ Fi.
According to theorem 20.3 Fhp 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 withMfm (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 ofMf. So corollary 20.6 reproves the classical
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 briey write p instead of pRm, _ for the action of Rm on FRm
and we denote by B(x; ") the open ball fy 2 Rm; jy xj < "g _ 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 2 FRm, i = 1, 2,: : : , be a sequence of points converging
to z 2 FRm such that p(zi) 6= p(z). Then there is a sequence of real constants
"i > 0 such that for any point a 2 Rm and any neighborhood W of _a(z) the
inclusion _ (B(a; "i) _ fzig) _ 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 "i of positive real numbers there are a point a 2 Rm , a
neighborhood W of _a(z) and a sequence ai 2 B(a; "i) such that _ (ai; zi) =2 W
for an in_nite set of indices i 2 I0 _ N. Let us denote xi := p(zi), x := p(z).
Passing to a further subset of indices we can arrange that 2jxi xj j > jxi xj
for all i, j 2 I0, i 6= 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 2 I _ I0 and
(2) germ f(xj) = germ ta(xj)
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
20. The regularity of bundle functors 189
for an in_nite subset of indices j 2 J _ I0, then using the almost W-extendibility
of C-morphisms we _nd some g 2 C(Rm;Rm) satisfying Fg(zi) = Ftai (zi) =
_ (ai; zi) for large i 2 I and Fg(zj) = _ (a; zj) for large j 2 J. Hence F(ta) _
Fg(zi) = _ (aia; zi) for large i 2 I while F(ta)_Fg(zj) = zj for large j 2 J and
this implies Fg(z) = _a(z) which is in contradiction with Fg(zi) = _ (ai; zi) =2 W
for large i 2 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 "i 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 2 I0. Then the condition
19.4.(1) reads
lim
j;i!1
j;i2I
jai aj j
jxi xj jk
! 0; lim
i!1
i2I
ja aij
jxi xjk
! 0 lim
j;i!1
i2I;j2J
ja aij
jxi xj jk
! 0
for all k 2 N.
Let us choose 0 < "i < e1=(jxixj). Now, if i < j then jai aj j < 2"i and
jxi xj j > 1
2
jxi xj and the _rst estimate follows. Analogously we get the
remaining ones. _
The next lemma is necessary to overcome di_culties with constant sequences
in FRm.
20.10. Lemma. Let zj 2 FRm, j = 1, 2; : : : , be a sequence of points converging
to z 2 FRm. Then there is a sequence of points ai 2 Rm, ai 6= 0, i = 1,
2,: : : , converging to 0 2 Rm and a subsequence zji such that Ftai (zji ) ! z if
i ! 1.
Proof. Let us recall that FRm has a countable basis of open sets and let Uj ,
j 2 N, form a basis of open neighborhoods of the point z satisfying Uj+1 _ Uj .
For each number j 2 N, there is a sequence of points a(j; k) 2 Rm, k 2 N, such
that [
a2Rm
F(ta)(Uj) =
[
k2N
F(ta(j;k))(Uj):
Let bj 2 Rm be such a sequence that for all k 2 N, bj 6= a(j; k). Passing
to subsequences, we may assume zj 2 Uj for all j and consequently we get
F(tbj )(zj) 2
S
k2N F(ta(j;k))(Uj) for all j 2 N. Let us choose a sequence kj 2 N,
such that
F(tbj )(zj) 2 F(ta(j;kj ))(Uj)
for all j 2 N, and denote aj := bj a(j; kj ). Then aj 6= 0 and F(taj )(zj) 2 Uj
for all j 2 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; "i) 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 FRm, p(zi) 6= p(z),
and let W be an open neighborhood of z. Then there exist b 2 Rm and " > 0
such that
_
B(b; ") _ fzg
_
_ W; _
B(b; ") _ fzig
_
_ W
for large i's.
Proof. We _rst deduce that there is some open ball B(y; _) _ Rm satisfying
(1) B(y; _) _ fa 2 Rm; F(ta)(z) 2 Wg:
Let us apply lemma 20.10 to a constant sequence yj := z. So there is a sequence
ai 2 Rm, ai 6= 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) 2 W, there is a sequence
of positive constants _i such that _
B(0; _i) _ fwig
_
_ W for large i's. Let us
choose one of these indices, say i0, and put y := ai0 , _ := _i0 . Now for any
b 2 B(y; _) we have _b(z) = _by _ _y(z) = _by(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 "i such that for any a 2 Rm the condition
_a(z) 2 W implies _
B(a; "i) _ fzig
_
_ W for all large i's. Therefore
B(y; _) _
[
k2N
\
i_k
fa 2 Rm; _
B(a; "i) _ fzig
_
_ Wg:
For any natural number k we de_ne
Bk :=
\
i_k
fa 2 B(y; _); _
B(a; "i) _ fzig
_
_ Wg:
Since [k2NBk = B(y; _), the Baire category theorem implies that there is a
natural number k0 such that int(_Bk0 ) \ B(y; _) 6= ;.
Now, let us choose b 2 Rm and " > 0 such that B(b; ") _ int(_Bk0 ) \ B(y; _).
If x 2 B(b; ") and i _ k0, then there is _x 2 Bk0 with x 2 B(_x; "i) so that we
have _x(zi) 2 W and (1) implies _x(z) 2 W. _
20.12. Proof of proposition 20.2. Let zi ! z be a convergent sequence in
FRm, xi ! x a convergent sequence in Rm. We have to show
(1) _xi (zi) = F(txi )(zi) ! F(tx)(z) = _x(z):
Since we can apply the isomorphism F(tx), we may assume x = 0. Moreover,
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) 6= p(z) or
p(zi) = p(z) for all i 2 N.
Let us _rst deal with the latter case. According to lemma 20.10 there is a
sequence yi 2 Rm and subsequence zij such that _yj (zij ) ! z and yj ! 0,
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20. The regularity of bundle functors 191
yj 6= 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) 6= p(z) for all i 2 N and xi ! 0. Let us moreover assume
that _xi (zi) does not converge to z. Then, for each x 2 Rm, _x+xi (zi) does not
converge to _x(z) as well. Therefore, if we set
A := fx 2 Rm; _x+xi (zi) does not converge to _x(z)g
we _nd A = Rm. Now we use the separability of FRm. Let Vs, s 2 N, be a basis
of open sets in FRm and let
Ls := fx 2 Rm; _x+xi (zi) 2 Vs for large i'sg
Qs := fx 2 Rm; _x(z) 2 Vs and x =2 Lsg:
We know A _ [s2NQs and consequently [s2NQs = Rm. By virtue of the Baire
category theorem there is a natural number k such that int(Qk) 6= ;.
Let us choose a point a 2 Qk \ int(Qk). Then z 2 _a(Vk) and so
W := p1
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; ") _ Rm such that
(2) _
B(b; ") _ fzg
_
_ p1
tp(z)a
int(Qk)
__
(3) _
B(b; ") _ fzig
_
_ _a(Vk)
for all large i's. Inclusion (2) implies p(z) + B(b; ") _ p(z) a + int(Qk) or,
equivalently,
(4) B(b + a; ") _ int
Qk
_
:
Formula (3) is equivalent to
_
B(b + a; ") _ fzig
_
_ Vk
for large i's. Since xi ! 0, we know that for any x 2 B(b + a; ") also (x + xi) 2
B(b + a; ") for large i's and we get the inclusion B(b + a; ") _ Lk. Finally, (4)
implies
B(b + a; ") _ Lk \ int
Qk
_
_ (Rm n 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 Brk
for the kernel of the jet projection _rk
: Gr
_
! Gk
_. For every point
y 2 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 Brk
_ Hy for all points y 2 S. So
if we assume that the order r is essential, i.e. the action does not factorize to
Gr1
_ , then there is a point y 2 S such that Hy does not contain Brk
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
(1) dim S _ dim(Gr
_=Hy)
and we see that dim S is bounded from below by the smallest possible codimension
of Lie subgroups in Gr
_ which do not contain Brk
.
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;n, m _ 1, n _ 0, act continuously on a
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
m;n, k < r. Then
r _ 2s + 1:
Moreover, if m, n > 1, then
r _ maxf
s
m 1;
s
m
+ 1;
s
n 1;
s
n
+ 1g
and if m > 1, n = 0, then
r _ maxf
s
m 1;
s
m
+ 1g.
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 2 S such that its isotropy
group Hy does not contain the normal closed subgroup Br
r1. We shall denote
gr
m;n, brr
1 and h the Lie algebras of Gr
m;n, Br
r1 and Hy, respectively. Since
Br
r1 is a connected and simply connected nilpotent Lie group, its exponential
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
21. Actions of jet groups 193
map is a global di_eomorphism of brr
1 onto Br
r1, cf. 13.16 and 13.4. Therefore
h does not contain brr
1. In this way, our problem reduces to the determination
of a lower bound of the codimensions of subalgebras of gr
m;n that do not contain
the whole brr
1.
Since gr
m;n is a Lie subalgebra in gr
m+n, there is the induced grading
gr
m;n = g0 _ _ _ _ _ gr1
where homogeneous components gp are formed by jets of homogeneous projectable
vector _elds of degrees p + 1, cf. 13.16.
If we consider the intersections of h with the _ltration de_ning the grading
gr
m;n = _pgp, then we get the _ltration
h = h0 _ h1 _ : : : _ hr1 _ 0
and the quotient spaces hp = hp=hp+1 are subalgebras in gp. Therefore we can
construct a new algebra ~h = h0 _ _ _ _ _ hr1 with grading and since
dim h = dim h=h1 + dim h1=h2 + _ _ _ + dim hr1 = dim~h,
both the algebras h and ~h have the same codimension. By the construction,
brr
1
6_ h if and only if hr1 6= gr1, so that~h does not contain brr
1 as well. That
is why in the proof of theorem 21.2 we may restrict ourselves to Lie subalgebras
h _ gr
m;n with grading h = h0__ _ __hr1 satisfying hi _ gi for all 0 _ i _ r1,
and hr1 6= gr1.
Now the proof of the estimate r _ 2s + 1 becomes rather easy. To see this,
let us _x two degrees p 6= q with p + q = r 1 and recall [gp; gq] = gr1,
see 13.16. Hence there is either a 2 gp or a 2 gq with a =2 h, for if not then
[gp; gq] = gr1 _ hr1. It follows
codim h _
1
2
(r 1).
According to 21.1.(1) we get s _ 1
2 (r 1) and consequently r _ 2s + 1.
The remaining case s = 0 follows immediately from the fact that given an
action _: Gr
m;n
! Di_(S) on a zero-dimensional manifold S, then its kernel
ker _ contains the whole connected component of the unit. Since Gr
m;n has two
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 gr1 with kerC _ hr1. Then given p, q, p+q = r1,
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 W0 the annihilators of V
or W related to f, respectively. Let M _ V , N _ W be subspaces satisfying
fj(M _ N) = 0. Then
codimM + codimN _ codim V 0.
Proof. Consider the associated form f_ : V=W0 _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
(1) dim[M] + dim[M]0 = codimW0.
Note that codim V 0 = codimW0. We know dim[M] = dim(M=M \ W0) =
dim(M +W0) dimW0 and similarly for N. Therefore
(2)
dim[M] + dim[N] =
= dim(M +W0) dimW0 + dim(N + V 0) dim V 0
= codimW0 codim(M +W0) + codim V 0 codim(N + V 0)
_ codimW0 + (codim V 0 codimM codimN).
According to our assumptions N _ M0, so that dim[N] _ dim[M]0. But then
(1) implies
dim[M] + dim[N] _ codimW0
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 gr1 and the induced coordinate expressions of linear forms
C on gr1. 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 = (C_
i ), i = 1; : : : ;m+
n, j_j = r. This means C(X) =
P
_;i C_
i ai
_, if X =
P
_;i ai
_x_ @
@xi
2 gr1,
where we sum also over repeated indices. For technical reasons we set C_
i = 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 = j_j, 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 6= j,
and its length will be clear from the context. As before, the symbol 1j denotes
a multiindex _ with _i = _ij
.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
21. Actions of jet groups 195
21.5. Lemma. Let C be a non-zero form on gr1, 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:
(i) C(1)
m 6= 0 and m > 1.
(ii) C(1)
1
6= 0; C_
j = 0 whenever _j = 0 and 1 _ j _ m; C_+11
1 = C_+1j
j (no
summation) for all j_j = r 1, 1 _ j _ m.
(iii) C(m+1)
m+n
6= 0, n > 1, and C_
j = 0 whenever j _ m.
(iv) C(m+1)
m+1
6= 0; C_
j = 0 if j _ m or _j = 0; and C_+1m+1
m+1 = C_+1j
j (no
summation) for all j_j = r 1, j _ m + 1.
Proof. Let C be a non-zero form on gr1 with coordinates C_
j in the canonical
basis of Rm+n ! Rm. Let us consider a matrix A 2 GL(m+ n) whose _rst row
consists of arbitrary real parameters a11
= t1 6= 0, a12
= t2; : : : ; a1
m = tm, a1j
= 0
for j > m, and let all the other elements be like in the unit matrix. Let ~ai
j be the
elements of the inverse matrix A1. If we perform this linear transformation,
we get a new coordinate expression of C, in particular
(1) _ C(1)
j = a1i
1
_ _ _ a1i
rCi1:::ir
s ~asj
.
Hence we get
_ C(1)
1 = ti1
_ _ _ tirCi1:::ir
1
1
t1
(2)
_ C(1)
j = ti1
_ _ _ tir
_
Ci1:::ir
j
tj
t1
Ci1:::ir
1
_
(3) for 1 < j _ m.
Formula (2) implies that either we can obtain _ C(1)
1
6= 0 or C_
1 = 0 for all multi
indices _, j_j = 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 C_
j = 0 whenever _j = 0, j_j = r,
and C_+11
1 = C_+1j
j for all j_j = r 1, 1 _ j _ m. Hence we can summarize:
either (i) can be obtained, or (ii) holds, or _ C_
j = 0 for all 1 _ j _ m, j_j = r, in
suitable a_ne coordinates.
Analogously, let us take a matrix A 2 GL(m + n) whose (m + 1)-st row
consists of real parameters t1; : : : ; tm+n, tm+1 6= 0 and let the other elements be
like in the unit matrix. The new coordinates of C are obtained as above
_ C(m+1)
m+1 =ti1
_ _ _ tirCi1:::ir
m+1
1
tm+1
(4)
_ C(m+1)
j =ti1
_ _ _ tir
_
Ci1:::ir
j
tj
tm+1
Ci1:::ir
m+1
_
(5) .
Now we may assume C_
j = 0 whenever 1 _ j _ m, for if not then (i) or (ii) could
be obtained. As before, either there is a basis relative to which C(m+1)
m+1
6= 0 or
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
196 Chapter V. Finite order theorems
C_
m+1 = 0 for all j_j = r. Further, according to (5) either we can get (iii) or
C_
j = 0 whenever _j = 0, and C_+1m+1
m+1 = C_+1j
j , for all j_j = r 1, j _ m+ 1.
Therefore if both (iii) and (iv) do not hold after arbitrary transformations, then
all C_
j 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 gr1 and suppose [hp; hq] _ kerC. If C_
i , 1 _ i _ m+n,
j_j = r, is a coordinate expression of C satisfying one of the conditions 21.5.(i)-
(iv), then
codim hp + codim hq _
8>>>>>>>><
>>>>>>>>:
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
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) = f0g. 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 2 h0, b 2 gq. We get
C([a; b]) =
X
1_i_m+n
j_j=r
C_
i ([a; b])i
_ = 0:
Using formula for the bracket 13.2.(5) we obtain
0 =
X
1_i;j_m+n
j_j=q+1
j_j=p+1
C_+_1j
i
_
_jbj
_ai
_
_jaj
_bi
_
_
=
X
1_i;j_m+n
j_j=q+1
j_j=p+1
_
C_+_1j
i _j C_+_1i
j _i
_
ai
_bj
_ .
Since b 2 gq is arbitrary, we have got a system of linear equations for the
annihilator h0 containing one equation for each couple (j; _), where 1 _ j _
m + n, j_j = q + 1 and _i = 0 whenever i > m and j _ m. The (j; _)-equation
reads
(1)
X
1_i_m+n
j_j=p+1
_
C_+_1j
i _j C_+_1i
j _i
_
ai
_ = 0.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
21. Actions of jet groups 197
A lower bound of the codimension of h0 is given by any number of linearly
independent (j; _)-equations and we have to discuss this separately for the cases
21.5.(i){(iv).
Let us _rst assume that 21.5.(i) holds, i.e. C(1)
m 6= 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
mX1
s=1
asEs +
mX1
k=1
bkFk = 0 as; bk 2 R.
From (1) we get
(2)
X
1_i_m+n
j_j=p+1
_mX1
s=1
C(1)+_1s
i _s C(1)+_1i
s _1
i (q + 1)
_
as+
+
mX1
k=1
C(1)+1k+_1m
i _m C(1)+1k+_1i
m (_1
i q + _k
i )
_
bk
_
ai
_ = 0.
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.
First, let us substitute _ = (1) and i = m. Then (2) implies C(1)
m (p+1)a1 = 0
and consequently a1 = 0. Now we choose _ = (1) + 1v, i = m, with 1 < v < m,
and we get C(1)
m av = 0 so that as = 0 for 1 _ s _ m 1. Further, take _ = (1)
and 1 < i < m to obtain C(1)
m bi = 0. Finally, the choice i = 1 and _ = (1)
leads to C(1)
m (q+1)b1 = 0. In this way, we have proved that the chosen 2m2
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,
1 _ k _ m. As before, we assume
Pm
s=1 asEs +
Pm
k=1 bkFk = 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; _)-equations with 1 _ j _ m
using the relations from 21.5.(ii). Indeed, (1) reduces to
X
1_i_m
j_j=p+1
C_+_1i
j (_j _i)ai
_ + R = 0.
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
X
1_i_m
j_j=p+1
C(1)+_1i
s (_s _1
i (q + 1))ai
_ + R = 0
X
1_i_m
j_j=p+1
C(1)+1k+1m+_1i
m (_m _1
i (q 1) _k
i
_m
i )ai
_ + R = 0.
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
(see 21.5.(ii)) 0 = 2C(1)+1m
m bm = 2C(1)
1 bm; and 1 < i < m, _ = (1) now
yields C(1)+1m
m bi = 0. Hence bi = 0 for all 1 < i _ m. Further, we take i = m,
_ = (1) + 1v + 1m, v 6= m (note p _ q > 0), so that all the coe_cients in F1 are
zero. In particular, v = 1 implies C(1)
1 a1 = 0 so that a1 = 0. Now, if 1 < v < m,
then C(1)+1v
v av = C(1)
1 av = 0 and what remains are am and b1, only. Taken
_ = (1), i = 1, we see 0 = (q + 1)C(1)
m am qC(1)+1m
m b1 = C(1)
1 b1 and, _nally,
the choice i = m and _ = (1) + 1m + 1m gives C(1)+1m
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 j_j = p + 1 = r _ 2.
If n > 1, then the remaining three parts of the proof are complete recapitulations
of the above ones. This becomes clear if we notice, that we have used
neither any information on C_
j , j > m, nor the fact that C_
j = 0 if j _ m and
_i 6= 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;n, m _ 1, n _ 0, r _ 2, which
does not contain brr
1. Then
(1) codim h _
1
2
(r 1).
Moreover, if m > 1, n > 1, then
(2) codim h _ minfr(m 1); (r 1)m; r(n 1); (r 1)ng
and if m > 1, n = 0, then
(3) codim h _ minfr(m 1); (r 1)mg.
Proof. In 21.3 we deduced that we may suppose h is a subalgebra with grading
h = h0 _ _ _ _ _ hr1, hi _ gi, hr1 6= gr1, and we proved the lower bound
(1). Let us assume m > 1, n = 0 and choose a non-zero form C on gr1 with
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
21. Actions of jet groups 199
kerC _ hr1. Then we know [hj ; hrj1] _ hr1 _ kerC 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 C_
i satis_es 21.5.(i). Then for all j
codim hj + codim hrj1 _ 2m 2
and consequently
codim h =
Xr1
j=0
codim hj _ r(m 1).
If 21.5.(ii) holds, then
codim h0 + codim hr1 _ m
codim hj + codim hrj1 _ 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, m > 1, be de_ned by
h1 = fai
_x_ @
@xi
; aj
(1) = 0 for j = 2; : : : ;m, 1 _ j(1)j _ rg.
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
m of codimension r(m 1). Consider now the
subalgebra h _ gr
m;n consisting of projectable polynomial vector _elds of degree
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
m+n, n > 1, de_ned analogously to the subalgebra
h1
h2 = fai
_x_ @
@xi
; aj
(m+1) = 0 for 1 _ j _ m + n, j 6= m + 1, 1 _ j(m + 1)j _ rg
and de_ne h = h2 \ gr
m;n. Since every polynomial vector _eld in gr
m;n is tangent
to the _ber over zero, this clearly is a Lie subalgebra with coordinate description
h = fai
_x_ @
@xi
; aj
(m+1) = 0 for m + 1 < j _ m + n, 1 _ j(m + 1)j _ rg
and codimension r(n 1).
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
200 Chapter V. Finite order theorems
3. Let us recall that the divergence divX of a polynomial vector _eld X 2 gr
m
can be viewed as the jet jr1
0 (divX), see 13.6. So for an element a = ai
_x_ @
@xi we have
div a =
X
1_i_m
1_j_j_r
_iai
_x_1i .
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 _ _ _ _ _ gr1 (note g0 is missing!)
h3 = fai
_x_ @
@xi
2 g1 _ _ _ _ _ gr1;
div(ai
_x_ @
@xi
)
___
M = 0g:
Of course, h3 is not a Lie subalgebra in gr
m. Let us further consider the Lie
subalgebra h4 _ gr1
m consisting of all polynomial vector _elds without absolute
terms and tangent to M, cf. example 1. Let h5 =
_r
r1
_1
h4 _ gr
m and let us
de_ne a linear subspace
h6 = (h5 \ g0) _ (h3 \ h5).
First we claim that h3 \ h5 is a subalgebra. Indeed, if X, Y 2 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 ])jM = 0.
Now, consider a polynomial vector _eld X from the subalgebra h5 \ g0 and a
_eld Y 2 h3 \ h5. Since every _eld from g0 has constant divergence everywhere
and X is tangent to M, 13.6.(1) implies div([X; Y ])jM = 0. So we have proved
that h6 is a subalgebra. In coordinates, we have
h6 = fai
_x_ @
@xi
2 gr
m; aj
(1) = 0;
Xm
i=1
ai
(1)+1i(1 + _1
i
j(1)j) = 0
for j = 2; : : : ;m, 1 _ j(1)j _ r 1g:
Now, we take the subalgebra h in gr
m;n consisting of polynomial vector _elds
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,
h7 = fai
_x_ @
@xi
2 gr
m+n; aj
(m+1) = 0;
mX+n
i=1
ai
(m+1)+1i(1 + _m+1
i
j(m + 1)j) = 0
for j = 1; : : : ;m + n, j 6= m + 1, 1 _ j(m + 1)j _ r 1g
and let us de_ne h = h7 \ gr
m;n. Then
h = fai
_x_ @
@xi
2 gr
m;n; aj
(m+1) = 0;
mX+n
i=m+1
ai
(m+1)+1i(1 + _m+1
i
j(m + 1)j) = 0
for j = m + 2; : : : ;m + n, 1 _ j(m + 1)j _ r 1g
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
m;n of codimension (r 1)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 isomorphisms
keeping invariant the given lines. These are closed subgroups. In
the remaining two examples, we have to consider analogous subgroups in Gr1
m;n,
then to take their preimages in the group homomorphism _r
r1. Further we consider
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
m;n, r _ 2, which does not factorize to an action of Gr1
m;n, is bounded from
below by the smallest possible codimension of Lie subalgebras h = h0__ _ __hr1,
hi _ gi, hr1 6= gr1, with grading. We also got the lower bound 1
2 (r 1) for
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 _ minfr(m 1); (r 1)m; r(n 1); (r 1)ng
and consequently
r _ maxf
s
m 1;
s
m
+ 1;
s
n 1;
s
n
+ 1g.
If n = 0 we get
s _ minfr(m 1); (r 1)mg,
so that
r _ maxf
s
m 1;
s
m
+ 1g:
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
m;n with the extreme values of r.
If m = 1, let us consider h = g0 _ gs _ gs+1 _ _ _ _ _ g2s1 _ g2s+1
1 . Since
[gs; gs] = 0 in dimension one, this is a Lie subalgebra and one can see that the
corresponding subgroup H in G2s+1
1 is closed (in general, every connected Lie
subgroup in a simply connected Lie group is closed, see e.g. [Hochschild, 68, p.
137]). The homogeneous space G2s+1
1 =H has dimension s and G2s+1
1 acts non
trivially. Since there are group homomorphisms Gr
m;n
! Gr
m and Gr
m;n
! Gr
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 := p1
Rm+n(0) is a manifold of dimension s _ 0 and for every
(Y ! M) in ObFMm;n the mapping pY : FY ! 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
r _ maxf
s
m 1;
s
m
+ 1g,
and if m > 1, n > 1, then
r _ maxf
s
m 1;
s
m
+ 1;
s
n 1;
s
n
+ 1g.
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
m;n with the above bounds of r depending on s, 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
m;n for some k 2 N. But then theorem 21.2 yields the necessary
estimates. Moreover, if we consider the Gr
m;n-spaces with the extreme orders
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 2 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 _k operating on category VB of vector bundles. For
every manifold M the value FM is de_ned as the Whitney sum over M
FM =
M
1_k_1
_dkT(k)M
and for every smooth map f : M ! N we set
Ff =
M
1_k_1
_dkT(k)f : FM ! FN.
Since _dkT(k)M = M _ f0g whenever k > dimM, the value FM is a well
de_ned _nite dimensional smooth manifold and Ff is a smooth map. The _ber
projections on T(k)M yield a _bration of FM and all the axioms of bundle
functors are easily veri_ed. Since the order of _dkT(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 ,
the condition jrn+1
x f = jrn+1
x g implies FfjFxY = FgjFxY . If dim _ Y _ 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 2 Rm+k.
(a) First we discuss the case n = k. Let us assume jr
0f = jr
0g, r = rn and
consider families ft = f+tidRm+n, gt = g+tidRm+n, t 2 R. The Jacobians at zero
are certain polynomials in t, so that the maps ft and gt are local di_eomorphisms
at zero except a _nite number of values of t. Since jr
0ft = jr
0gt for all t, we have
FftjSn = FgtjSn except a _nite number of values of t. Hence the regularity of
F implies FfjSn = FgjSn.
Every _bered map f 2 FMm(Rm+n;Rm+k) over f0 : Rm ! Rm locally decomposes
as f = h _ g where g = f0 _ idRn : Rm+n ! Rm+n and h = f _ g1 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
jr
0f = jr
0g with r = rn. Consider _ f = (f; pr2), _g = (g; pr2) : Rm+n ! Rm+n,
where pr2 : Rm+k+q ! Rq is the projection onto the last factor. Since jr
0
_ f = jr
0 _g,
f = pr1
_ _ f and g = pr1
__g, the functoriality and (a) imply FfjSn = FgjSn.
(c) If k = n + 1 and if jr
0f = jr
0g 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 7! (x; 0). For every y 2 Rm+n+1 with
pr1(y) = 0 we have jr
y
_ f = jr
y _g and since f = _ f_i, g = _g_i, we get FfjSn = FgjSn.
(d) Let k = n + q, q > 0, and i : Rm+n ! Rm+n+q, x 7! (x; 0). Analogously
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; f1; : : : ; fk) : Rm+n ! Rm+k, k > n, and assume
jr
0f = jr
0 i with r = rn+1. We de_ne h: Rm+n+1 ! Rm+k
h(x; y) = (idRm; f1(x); : : : ; fn(x); y; fn+2(x); : : : ; fk(x)).
Then we have
h _ (idRm; idRn; fn+1) = f
h _ i = (idRm; f1; : : : ; fn; 0; fn+2; : : : ; fk).
Since jr
0(idRm; idRn; fn+1) = jr
0 i, part (c) of this proof implies
F(idRm+n; fn+1)jSn = FijSn
and we get for every z 2 Sn
Ff(z) = Fh _ Fi(z) = F(idRm; f1; : : : ; fn; 0; fn+2; : : : ; fk)(z).
Now, we shall proceed by induction. Let us assume
Ff(z) = F(idRm; f1; : : : ; fn; 0; : : : ; 0; fn+s; : : : ; fk)(z); s > 1;
for every z 2 Sn and jrn+1
0 f = jrn+1
0 i. Let _ : Rm+n+k ! Rm+n+k be the map
which exchanges the coordinates xn+1 and xn+s, i.e.
_(x; x1; : : : ; xn; xn+1; : : : ; xn+s; : : : ; xk) =
= (x; x1; : : : ; xn; xn+s; : : : ; xn+1; : : : ; xk):
We get
F(idRm;f 1; : : : ; fn; 0; : : : ; 0; fn+s; : : : ; fk)(z) =
= F
_ _ (idRm; f1; : : : ; fn; fn+s; 0; : : : ; 0; fn+s+1; : : : ; fk)
_
(z)
= F_ _ F(idRm; f1; : : : ; fn; 0; : : : ; 0; fn+s+1; : : : ; fk)(z)
= F(idRm; f1; : : : ; fn; 0; : : : ; 0; fn+s+1; : : : ; fk)(z).
So the induction yields Ff(z) = F(idRm; f1; : : : ; fn; 0; : : : ; 0). Since we always
have rn+1 _ rn, (a) implies
F(idRm; f1; : : : ; fn)jSn = F idRm+n jSn.
Finally, we get
FfjSn = F(idRm; f1; : : : ; fn; 0; : : : ; 0)jSn
= F(i _ (idRm; f1; : : : ; fn))jSn = FijSn: _
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_), _ 2 I, and consider bundle functors on C.
23.1. The local order. We call a natural domain E of a natural operator
D: E (G1;G2) W-extendible (or Whitney-extendible) if all the domains EA _
C1
mA(F1A; F2A), A 2 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 _
C1(Y1; Y2) ! C1(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 _ 1. Let us consider a mapping s 2 E, z 2 Z1
and the compact set K = fzg _ Z1. According to 19.8 applied to K and s,
there is the smallest possible order r =: _(j1s(q(z)); z) 2 N such that for all
_s 2 E the condition jrs(q(z)) = jr_s(q(z)) implies Ds(z) = D_s(z). Let us write
Ek _ Jk(Y1; Y2) for the set of all k-jets of mappings from the domain E. The
just de_ned mapping _: E1 _Y1 Z1 ! N is called the local order of D.
For every _-local natural operator D: E (G1;G2) with a natural Wextendible
domain E, the operators
DA : EA _ C1
mA(F1A; F2A) ! C1
mA(G1A;G2A)
are _A-local. The system of local orders (_A)A2ObC is called the local order of
the natural _-local operator D.
Every locally invertible C-morphism f : A ! B acts on E1
A
_F1A G1A by
f_(j1
x s; z) =
j1(F2f _ s _ F1f1)(F1f(x));G1f(z)
_
.
Lemma. Let D: E (G1;G2) be a natural operator with a natural Whitneyextendible
domain E. For every locally invertible C-morphism f : A ! B and
every (j1
x s; z) 2 E1
A
_F1A G1A we have
_B
f_(j1
x s; z)
_
= _A(j1
x s; z).
Proof. Since C is admissible and the domain E is natural, we may restrict
ourselves to A = B = C_, for some _ 2 I. Assume _A(j1
x s; z) = r and
jrq(F1f(x)) = jr(F2f _ s _ F1f1)(F1f(x)) for some x 2 F1C_ and s; q 2 EC_.
Then jr((F2f)1_q_F1f)(x) = jrs(x) and therefore DC_((F2f)1_q_F1f)(z) =
DC_s(z). We have locally for each s 2 EC_
s _ F1f1 = F2f1 _ (F2f _ s _ F1f1)
DC_s = G2f1 _ DC_(F2f _ s _ F1f1) _ G1f.
Hence DC_q(G1f(z)) = G2f _ DC_s(z) and we have proved _B _ f_ _ _A.
Applying the action of the inverse f1 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_ : E1
C_
_F1C_ G1C_ ! G2C_
determined by a natural _-local operator D with a natural W-extendible domain
E. We shall write briey Er_
_ ErC
_
for the subset of jets with sources in the
_ber S_ over 0_ in F1C_, and Z_ := _1
C_
(S_) _ G1C_. By naturality, the whole
operator D is determined by the restrictions
D_ : E1
_
_S_ Z_ ! G2C_
of the maps DC_. Let us write __ : E1
_
_S_ Z_ ! N for the restrictions of _C_.
Lemma. The maps __ are G1
_ -invariant and if __ _ 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' __ is bounded and so it
must be bounded on each orbit under the action of G1
_ . 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 _ C1(FA)). Then E1
_
_ T1
n Q_, where Q_ = F0C_
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 2 EC_
_ C1(FC_) invariant with respect
to all translations, i.e. F(tx) _ s _ t__________x(y) = s(y) for all x 2 mC_ = Rn, y 2 C_
and denote Z the standard _ber (G1)0C_.
Lemma. For every compact set K _ Z there is an order r 2 N and a neighborhood
V _ E1
_
_ T1
n Q_ of j1
0_s in the Cr-topology such that __ _ r on
V _ K.
Proof. Let us apply theorem 19.7 to the translation invariant section s and a
compact set K0 _ K _ C_ _ Z = G1C_, where K0 is a compact neighborhood
of 0_ 2 Rn. We get an order r and a smooth function " > 0 except for _nitely
many points y 2 K0 where "(y) = 0. Let us _x x in the interior of K0 with
"(x) > 0. Hence there is a neighborhood V of s in the Cr-topology on EC_
and a neighborhood U _ C_ of x in K0 such that _U(j1
y q; (y; z)) _ r whenever
(y; z) 2 U _K and q 2 V . Now, let W be a neighborhood of j1
0 s in Cr-topology
on E1
_ 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 G1C_ = Rn _ Z by G1tx = 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 2 EC_ be a translation invariant section and K _ Z a
compact set. Assume that for every order r 2 N, every neighborhood V of jr
0_s
in Er_
_ Tr
nQ_, every relatively compact neighborhood K0 of K and every couple
(jr
0_q; z) 2 Er_
_Z_ there is an element g 2 G1
_ such that g_(jr
0_q; z) 2 V _K0.
Then every natural operator in question has _nite order on all objects of type
_.
Proof. For every relatively compact neighborhood K0 of K, there is an r 2 N
and a neighborhood of j1
0_s in the Cr-topology such that __ _ r on V _ K0.
But the assumptions of the corollary ensure that the orbit of V _ K0 coincides
with the whole space E1
_
_ Z_. _
Next we deduce several simple applications of this procedure.
23.5. Proposition. Let F : Mfm ! FM be a bundle functor of order r such
that its standard _ber Q together with the induced action of G1
m
_ Gr
m can be
identi_ed with a linear subspace in a _nite direct sum
Mi Oai
Rm
Obi
Rm__
and bi > ai for all i. Let G1 : Mfm ! FMbe a bundle functor such that either
its standard _ber Z together with the induced G1
m-action can be identi_ed with
a linear subspace in a _nite direct sum
Mj a0
Oj
Rm
b0
Oj
Rm__
and b0
j > a0
j for all j, or Z is compact.
Then every natural operator D: F (G1;G2) de_ned on all sections of the
bundles FM has _nite order.
Proof. Write 't : Rm ! Rm, t 2 R, for the homotheties x 7! tx. Let us consider
the canonical identi_cation FRm = Rm _ Q and the zero section s = (idRm; 0)
in C1(FRm). Further, consider an arbitrary section q : Rm ! FRm and let us
denote qt = F't _ q _ '1
t and qt(x) = (x; qi
t(x)). Under our identi_cation, s is
translation invariant and we can use formula 14.18.(2) to study the derivatives
of the maps qi
t at the origin. For all partial derivatives @_qi
t we get
(1) @_qi
t(0) = taibij_j@_qi(0):
If the standard _ber Z is compact, then we can use lemma 23.4 with K = Z
and the zero section s. Indeed, if we choose an order r and a neighborhood V of
jr
0s in Trm
Q, then taking t large enough we obtain jr
0qt 2 V , so that the bound
r is valid everywhere. But if Z is not compact, then an analogous equality to
(1) holds for the sections of G1Rm with ai bi replaced by a0
j
b0
j and these are
also negative. Hence we can apply the same procedure taking K = f0g, where
0 is the zero element in the tensor space Z. _
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
208 Chapter V. Finite order theorems
23.6. Examples. The assumptions of the proposition are satis_ed by all tensor
bundles with more covariant then contravariant components. But clearly, these
are also satis_ed for all a_ne natural bundles with associated natural vector
bundles formed by the above tensor bundles. So in particular, F can equal to
QP1 : Mfm ! FM, the bundle functor of elements of linear connections, cf.
17.7, or to the bundle functors of elements of exterior forms. If G1 = IdC then Z
is a one-point-manifold, i.e. a compact. Hence we have proved that all natural
operators on connections or on exterior forms that do not extend the bases have
_nite order.
23.7. Let us apply corollary 23.4 to the natural operators on the bundle functor
J1 : FMm;n ! FM, i.e. we want to derive the _niteness of the order for
geometric operations with general connections. For this purpose, consider the
maps 'a;b : Rm+n ! Rm+n, '(x; y) = (ax; by). In words, we will use the inclusion
Gr
m
_ Gr
n ,! Gr
m;n and the jets of homotheties in the jet groups Gr
m
and Gr
n. In canonical coordinates (xi; yp; yp
i ) on J1(Rm+n ! Rm), i = 1; : : : ;m,
j = 1; : : : ; n, we get for every section s = yp
i (xi; yp) and every local _bered
isomorphism ' = ('i; 'p)
J1' _ s _ '1 =
_
@'p
@xj
_ '1
_
@'1
0
@xi +
_
@'p
@yq
_ '1
_
(yq
j
_ '1)@'1
0
@xi :
In particular, for ' = 'a;b we obtain
'a;b
_s(xi; yp) = ba1yp
i
_ '1
a;b:
Hence for every multi index _ = _1 + _2, where _1 includes all the derivatives
with respect to the indices i while _2 those with respect to p's, it holds
(1) @_1+_2 ('a;b
_s)(0) = a1j_1jb1j_2j@_1+_2s(0):
Proposition. Let H: FMm;n ! FM be an arbitrary bundle functor while
G: FMm;n ! FMis either the identity functor or the functor J1 or the vertical
tangent bundle V . Then every natural operator D: J1 (G;H) de_ned on all
sections of the _rst jet prolongations has _nite order.
Proof. If G = IdFMm;n, then we can take b = 1, a > 0 and corollary 23.4
together with (1) imply the assertion. The same choice of a and b leads also to
the case G = J1, for J1'a;b(yp
i ) = a1yp
i on the standard _ber over 0 2 Rm+n.
In the third case we have to be more careful. On the standard _ber Rn of
V Rm+n we have V 'a;b(_p) = (b_p). Let us _x some r 2 N and choose a = br,
0 < b < 1 arbitrary. Then
j@_1+_2 ('a;b
_s)(0)j = br(1+j_1j)+1j_2jj@_1+_2s(0)j
and so for all j_j _ r we get
j@_('a;b
_s)(0)j _ bj@_s(0)j:
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
23. The order of natural operators 209
Hence also in this case corollary 23.4 implies our assertion. _
At the end of this section, we illustrate on two examples how bad things
may be. First we construct a natural operator which essentially depends on
in_nite jets and the next example presents a non-regular natural operator. This
contrasts the results on bundle functors where the regularity follows from the
other axioms.
23.8. Example. Consider the bundle functor F = T _ T_ : Mfm !Mf and
let G be the bundle functor de_ned by GM = M _ R, Gf = f _ idR, for all
m-dimensional manifolds M and local di_eomorphisms f, i.e. `the bundle of real
functions'. The contraction de_nes a natural function, i.e. a natural operator
F G, of order zero. The composition with any _xed real function R ! R
is a natural transformation G ! G and also the addition G _ G + ! G and
multiplication G _ G : ! G are natural transformations. Moreover, there is the
exterior di_erential d: G T_, a natural operator of order 1.
By induction, let us de_ne operators Dk : T _ T_ ! G. We set
(D0)M(X; !) = iX! and (Dk+1)M(X; !) = iX
d
(Dk)M(X; !)
__
for k = 0; 1; : : : . Further, consider a smooth function a: R2 ! R satisfying
a(t; x) 6= 0 if and only if jxj > t > 0. We de_ne
DM(X; !) =
1X
k=0
a(k;) _ (iX!)
_
:
(Dk)M(X; !)
_
.
Since the sum is locally _nite for every (X; !) 2 C1(FM), this is a natural
operator of in_nite order.
23.9. Example. Consider once more the bundle functors F, G and operators
Dk from example 23.8. Let a and g : R2 ! R be the functions used in 19.15.
We shall modify operator D from example 19.15 to get a non-regular natural
operator. Let us de_ne operators DM : C1(FM) C1(GM) by
DM(X; !) =
1X
k=0
a(k;) _ g _ ((iX!) _ (iXd(iX!)))
_
:
(Dk)M(X; !)
_
for all (X; !) 2 C1(T _ T_M). We have used only natural operators in our
construction, but, unfortunately, the values DM(X; !) need not be smooth (or
even de_ned) if dimension m is greater then one. This is caused by the in_nite
value of lim supx!(0;1) g(x). But if m = 1, then all values are smooth and the
system DM satis_es all axioms of natural operators except the regularity. Indeed,
it su_ces to verify the smoothness of the values of DR. But if (iX!)(t0) = 0
and (iXd(iX!))(t0) = 1, i.e. X(t0) d
dx (X!)(t0) = 1, then d
dx (X!)(t0) 6= 0 and
therefore the curve
t 7!
(X!)(t);X
d
dx
(X!)(t)
_
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
210 Chapter V. Finite order theorems
lies on a neighborhood of t0 inside the unit circles centered in (1; 1) and (1; 1).
Hence DR(X; !) = 0 on some neighborhood of t0.
Let us note that our operator D is not only non-regular, but also of in_nite
order and it shows that the assertion of lemma 23.3 does not hold for all maps
s 2 DC_, in general. A non-regular natural operator of order 4 on Riemannian
metrics for dimension m = 2 can be found in [Epstein, 75].
23.10. If we consider natural operators D: F (G1;G2) with domains formed
by all sections of the bundles FM ! M, then we can use the regularity of
D and apply the stronger version of nonlinear Peetre theorem 19.10 instead of
19.7 in the proof of 23.3. Hence we do not need the invariance of the section s.
Consequently, the assertion of lemma 23.3 holds for all sections s 2 EC_. That
is why, under the assumptions of corollary 23.4 we can strengthen its assertion.
Corollary. Let s 2 C1(FC_) be a section and K _ Z be a compact set.
Assume that for every order r 2 N, every neighborhood V of jr
0_s in Er_
_ Tr
nQ_,
every relatively compact neighborhood K0 of K and every couple (jr
0_q; z) 2
Er_
_Z_ there is an element g 2 G1
_ such that g_(jr
0_q; z) 2 V _K0. Then every
natural operator D: F (G1;G2) has a _nite order on all objects of type _.
Remarks
The general setting for bundle functors and natural operators extends the
original categorical approach to geometric objects and operators due to [Nijenhuis,
72] and we follow mainly [Kol_a_r, 90] and partially [Slov_ak, 91].
The multilinear version of Peetre theorem, proved in [Cahen, De Wilde, Gutt,
80], seems to be the _rst non-linear generalization of the famous Peetre theorem,
[Peetre, 60]. The study of general nonlinear operators started in [Chrastina, 87]
and [Slov_ak, 87b]. The original aim of the nonlinear version 19.7, _rst proved in
[Slov_ak, 87b], was the reduction of the problem of _nding natural operators to a
_nite order. The pure analytical results were further generalized and completed
in a setting of Holder-continuous maps and metric spaces in [Slov_ak, 88] and
it became clear that they should help to unify the approach to the _niteness
of the orders of both natural operators and bundle functors and to avoid the
original manipulation with in_nite dimensional Lie algebras, see [Palais, Terng,
77]. Let us remark that nearly all categories over manifolds used in di_erential
geometry are admissible and locally at, however the veri_cation of the Whitney
extendibility might present a serious analytical problem in concrete examples.
In the most technical part of the description of bundle functors, i.e. in the proof
of the regularity, we mainly follow [Mikulski, 85] which generalizes the original
proof due to [Epstein, Thurston, 79] to natural bundles with in_nite dimensional
values. Let us point out that our proof also applies to continuous regularity of
bundle functors on the categories in question with values in in_nite dimensional
manifolds.
Our sharp estimate on the orders of jet groups acting on manifolds is a generalization
of [Zajtz, 87], where similar results are obtained for the full group Gr
m.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
Remarks 211
The results on the order of bundle functors on FMm follow some ideas from
[Kol_a_r, Slov_ak, 89] and [Mikulski, 89 a, b]. The methods used in our discussion
on the order of natural operators never exploit the regularity of the natural operators
which we have incorporated into our de_nition. So the results of section
23 can be applied to non-regular natural operators which can also be classi_ed
in some concrete situations.
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
212
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