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¯ j
Rm corresponding to the canonical basis of the Lie algebra of GL(m) are of the

form δi y j ‚yk . Every local coordinates (xi ) on an open subset U ‚ M de¬ne
k

‚ ‚
a section ρ : U ’ P 1 M formed by the coordinate frames ( ‚x1 , . . . , ‚xm ) and it
holds ρ— θM = dxi . On the other hand, from the explicite equation 25.2.(2) of “
i
i
we deduce easily that the restriction of the connection form ω = (ωj ) of “ to ρ

is (’“i (x)dxk ). Thus, if we consider the coordinate expression ξ i (x) ‚xi of ξ in
jk
our coordinate system and we write j ξ i for the additional coordinates of ξ,
we obtain from 17.14
‚ξ i
ξi = ’ “i ξ k .
j kj
j
‚x
Comparing with the classical formula in [Kobayashi, Nomizu, 63, p. 144], we
conclude that our quantities “i di¬er from the classical Christo¬el symbols by
jk
sign and by the order of subscripts.


Remarks
The development of the theory of natural bundles and operators is described
in the preface and in the introduction to this chapter. But let us come back
to the jet groups. As mentioned in [Reinhart, 83], it is remarkable how very
little of existing Lie group theory applies to them. The results deduced in our
exposition are mainly due to [Terng, 78] where the reader can ¬nd some more
information on the classi¬cation of Gr -modules. For the ¬rst order jet groups,
m
it is very useful to study in detail the properties of irreducible representations,
cf. section 34. But in view of 13.15 it is not interesting to extend this approach
to the higher orders. The bundle functors on the whole category Mf were ¬rst
studied by [Janyˇka, 83]. We shall continue the study of such functors in chapter
s
IX.
The basic ideas from section 15 were introduced in a slightly modi¬ed situation
by [Ehresmann, 55]. Every principal ¬ber bundle p : P ’ M with structure
group G determines the associated groupoid P P ’1 which can be de¬ned as the
factor space P — P/ ∼ with respect to the equivalence relation (u, v) ∼ (ug, vg),
u, v ∈ P , g ∈ G. Writing uv ’1 for such an equivalence class, we have two
projections a, b : P P ’1 ’ M , a(uv ’1 ) = p(v), b(uv ’1 ) = p(u). If E is a ¬ber
bundle associated with P with standard ¬ber S, then every θ = uv ’1 ∈ P P ’1
determines a di¬eomorphism qu —¦ (qv )’1 : Eaθ ’ Ebθ , where qv : S ’ Eaθ and
qu : S ’ Ebθ are the ˜frame maps™ introduced in 10.7. This de¬nes an action of
groupoid P P ’1 on ¬ber bundle E. The space P P ’1 is a prototype of a smooth
groupoid over M . In [Ehresmann, 55] the r-th prolongation ¦r of an arbitrary
smooth groupoid ¦ over M is de¬ned and every action of ¦ on a ¬ber bundle
E ’ M is extended into an action of ¦r on the r-th jet prolongation J r E

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
Remarks 167


of E ’ M . This construction was modi¬ed to the principal ¬ber bundles by
[Libermann, 71], [Virsik, 69] and [Kol´ˇ, 71b].
ar
The canonical R -valued form on the ¬rst order frame bundle P 1 M is one
m

of the basic concepts of modern di¬erential geometry. Its generalization to r-th
order frame bundles was introduced by [Kobayashi, 61]. The canonical form
on W 1 P (as well as on W r P ) was de¬ned in [Kol´ˇ, 71b] in connection with
ar
some local considerations by [Laptev, 69] and [Gheorghiev, 68]. Those canonical
forms play an important role in a generalization of the Cartan method of moving
frames, see [Kol´ˇ, 71c, 73a, 73b, 77].
ar




Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
168


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 gener-
alization 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 func-
tors. 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 cat-
egory 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) ’ C ∞ (mA, mB), A, B ∈ ObC, is injective. Taking into account
this inclusion C(A, B) ‚ C ∞ (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 categories Mfm
or Mf , the categories FM, FMm , FMm,n of ¬bered manifolds, oriented man-
ifolds, 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 ∈ 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 ∈ ObC and every
open subsets V ‚ U ‚ mA,
(b) (aggregation of morphisms) if (U± ), ± ∈ I, is an open cover of mA and f ∈

C (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± ), ± ∈ I, is an open cover of a manifold M
and (A± ), ± ∈ I, is a system of C-objects such that mA± = U± and L(A± , U± ©
Uβ ) = L(Aβ , U± © Uβ ) for all ±, β ∈ 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 localiza-
tion 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 ’ FM satisfying 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 p’1 (U ) of the value pA : F A ’ mA over U and F iU is the
A
inclusion p’1 (U ) ’ F A.
A
In particular, the projections pA , A ∈ 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 ∈ 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± ), ± ∈ 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± ), ± ∈ I,
with a distinguished point 0± ∈ mC± for each ± ∈ I.
A C-morphism f : A ’ B is said to be a local isomorphism, if for every
x ∈ 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 lo-
cally homogeneous. A pointed local skeleton of the category Mf 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 J r (A, B) of all r-jets of a C-object A into a C-
object B is the subset of the space J r (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 CJ r (A, B) for J r (A, B).
18.7. De¬nition. A locally homogeneous category C is called in¬nitesimally
admissible, if we have
(a) J r (A, B) is a submanifold of J r (mA, mB),
(b) the jet projections πk : J r (A, B) ’ J k (A, B), 0 ¤ k < r, are surjective
r

submersions,
(c) if X ∈ J r (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

invJ r (A, B) = J r (A, B) © invJ r (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± ), ± ∈ I. Let us write C r (±, β) = J0± (C± , Cβ )0β for the set of all r-jets
r

of C± into Cβ with source 0± and target 0β . De¬nition 18.7 implies that every
C r (±, β) is a smooth manifold, so that the restrictions of the jet composition
C r (±, β) — C r (β, γ) ’ C r (±, γ) are smooth maps. Thus we obtain a category C r
over I called the r-th order skeleton of C.
By de¬nition 18.7, Gr := invJ0± (C± , C± )0± is a Lie group with respect to the
r
±
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 P r A := invJ0± (C± , A) is
r

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±
r
subgroup in the usual jet group Gm , 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 ‚ Gr m+n in every order r. In
m,n
other words, Gm,n is the group of all r-jets at 0 ∈ Rm+n of ¬bered manifold
r

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 γ : R ’ J r (C± , Cβ ), ±, β ∈ I, there exists a smooth map “ : R — mC± ’
r
mCβ such that γ(t) = jc(t) “(t, ), where c(t) is the source of r-jet γ(t).
Since γ(t) is a curve on J r (C± , Cβ ), we know that γ(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 J r (C± , Cβ ) 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) ’ 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 parame-
terized family of FM-morphisms.
18.11. De¬nition. A bundle functor F : C ’ FM is said to be of order r,
r ∈ N, if for any two locally de¬ned C-morphisms f and g of A into B, the
r r
equality jx f = jx g implies that the restrictions of F f and F g to the ¬ber Fx A
of F A over x ∈ mA coincide.
18.12. Associated maps. An r-th order bundle functor F de¬nes the so-called
associated maps
FA,B : J r (A, B) —mA F A ’ F B, r
(jx f, y) ’ F f (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
J r (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± ,Cβ : J r (C± , Cβ ) —mC± F C± ’ F Cβ .
Consider a smooth curve (γ(t), δ(t)) on J r (C± , Cβ )—mC± F C± , so that pC± δ(t) =
c(t), where c(t) is the source of r-jet γ(t). Since C has the smooth splitting
property, there exists a smooth map “ : R — mC± ’ mCβ such that γ(t) =
r
jc(t) “(t, ). The regularity of F implies µ(t) := F (“(t, ))(δ(t)) is a smooth curve
on F Cβ . By the de¬nition of the associated map, it holds FC± ,Cβ (γ(t), δ(t)) =
µ(t). Hence FC± ,Cβ 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± ,Cβ is a smooth map. The other implication is obvi-
ous.
18.13. The induced action. Consider an r-th order regular bundle functor
F on an admissible category C. The ¬bers S± = F0± C± , ± ∈ I, will be called the
standard ¬bers of F . Write F±β for the restriction of FC± ,Cβ to C r (±, β) — S± ’
Sβ . In the following de¬nition we consider an arbitrary system (S± ), ± ∈ I, of
manifolds with indices from the type set of C.
De¬nition. A smooth action of C r on a system (S± ), ± ∈ I, of manifolds is a
system •±β : C r (±, β) — S± ’ Sβ of smooth maps satisfying
•βγ (b, •±β (a, s)) = •±γ (b —¦ a, s)
for all ±, β, γ ∈ I, a ∈ C r (±, β), b ∈ C r (β, γ), s ∈ S± .
By proposition 18.12, F±β are smooth maps so that they form a smooth action
of C r 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±β is a smooth
action of C r on (F0± C± ), ± ∈ I. Conversely, let (•±β ) be a smooth action of
C r on a system of manifolds (S± ), ± ∈ I. The inclusion Gr ’ C r (±, ±) gives a
±
r
smooth left action of G± on S± . For a C-object A of type ± we de¬ne GA to be
the ¬ber bundle associated to P r A with standard ¬ber S± . For a C-morphism
f : A ’ B we de¬ne Gf : GA ’ GB by
Gf ({u, s}) = {v, •±β (v ’1 —¦ jx f —¦ u, s)}
r

r r
x ∈ mA, u ∈ Px A, v ∈ Pf (x) B, s ∈ S± . One veri¬es easily that G is a well-
de¬ned regular r-th order bundle functor on C, cf. 14.22. Clearly, if we apply
the latter construction to the action F±β , 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 base-
preserving morphisms TA : F A ’ GA, A ∈ ObC, satisfying Gf —¦ TA = TB —¦ F f
for every C-morphism f : A ’ B. (We remark that for a large class of admissi-
ble categories every natural transformation between any two bundle functors is
formed by base-preserving morphisms, see 14.11.)
Given two smooth actions (•±β , S± ) and (ψ±β , Z± ), a C r -map

„ : (•±β , S± ) ’ (ψ±β , Z± )

is a system of smooth maps „± : S± ’ Z± , ± ∈ I, satisfying

„β (•±β (a, s)) = ψ±β (a, „± (s))

for all s ∈ S± , a ∈ C r (±, β).
Theorem. Natural transformations F ’ G between two r-th order regular
bundle functors on C are in a canonical bijection with the C r -maps between the
corresponding actions of C r .
Proof. Given T : F ’ G, we de¬ne „± : F0± C± ’ G0± C± by „± (s) = TC± (s).
One veri¬es directly that („± ) is a C r -map (F±β , F0± C± ) ’ (G±β , G0± C± ). Con-
versely, let („± ) : (•±β , S± ) ’ (ψ±β , Z± ) be a C r -map between two smooth ac-
tions of C r . Then the induced bundle functors transform A ∈ ObC of type ± into
the ¬ber bundle associated with P r A with standard ¬bers S± and Z± and we
de¬ne TA = (idP r A , „± ). 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 nat-
ural 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 CM (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 ‚
∞ ∞
CM (Y1 , Y2 ) ’ CM (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 π : N3 ’ N1 and
a subset E ‚ C ∞ (N1 , N2 ), an operator A : E ’ C ∞ (N3 , N4 ) is called π-local,
if the value As(x) depends only on the germ of s at π(x) for all s ∈ E, x ∈ N3 .
k k
Such an operator is said to be of order k, 0 ¤ k ¤ ∞, if jπ(x) s1 = jπ(x) s2 implies
As1 (x) = As2 (x) for all s1 , s2 ∈ E, x ∈ N3 . We call A regular if smoothly pa-
rameterized families in E are transformed into smoothly parameterized families
in C ∞ (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 CM (Y1 , Y2 ) ’ CM (Z1 , Z2 ) with respect to
q. Such a k-th order operator D determines the associated map
k k
D : JM (Y1 , Y2 ) —Y1 Z1 ’ Z2 , (jy s, z) ’ Ds(z),
(1) y = q(z),

k
where JM (Y1 , Y2 ) means the space of all k-jets of the maps of CM (Y1 , Y2 ). If
D is regular, then D is smooth. Conversely, every smooth map (1) de¬nes a
∞ ∞
regular operator CM (Y1 , Y2 ) ’ CM (Z1 , Z2 ), s ’ D((j k s) —¦ q, ) : Z1 ’ Z2 ,

s ∈ CM (Y1 , Y2 ).

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