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Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
14. Natural bundles and operators 139

˜
de¬ned by F f (p, ) = F fp , p ∈ P , is smooth, i.e. smoothly parameterized systems
of local di¬eomorphisms are transformed into smoothly parameterized systems
of ¬bered local isomorphisms.
In sections 10 and 12 we met several bundle functors on Mfm .
14.2. Now let F be a natural bundle. We shall denote by tx : Rm ’ Rm the
translation y ’ y + x and for any manifold M and point x ∈ M we shall write
Fx M for the pre image p’1 (x). In particular, F0 Rm will be called the standard
M
¬ber of the bundle functor F . Every bundle functor F : Mfm ’ FM determines
an action „ of the abelian group Rm on F Rm via „x = F tx .
Proposition. Let F : Mfm ’ FM be a bundle functor on Mfm and let S :=
F0 Rm be the standard ¬ber of F . Then there is a canonical isomorphism Rm —
S ∼ F Rm , (x, z) ’ F tx (z), and for every m-dimensional manifold M the value
=
F M is a locally trivial ¬ber bundle with standard ¬ber S.
Proof. The map ψ : F Rm ’ Rm — S de¬ned by z ’ (x, F t’x (z)), x = p(z), is
the inverse to the map de¬ned in the proposition and both maps are smooth ac-
cording to the regularity condition 14.1.(iii). The rest of the proposition follows
from the locality condition 14.1.(ii). Indeed, a ¬bered atlas of F M is formed by
the values of F on the charts of any atlas of M .
14.3. De¬nition. A natural bundle F : Mfm ’ FM is said to be of ¬nite
order r, 0 ¤ r < ∞, if for all local di¬eomorphisms f , g : M ’ N and every
r r
point x ∈ M , the equality jx f = jx g implies F f |Fx M = F g|Fx M .
14.4. Associated maps. Let us consider a natural bundle F : Mfm ’ FM
of order r. For all m-dimensional manifolds M , N we de¬ne the mapping
FM,N : invJ r (M, N ) —M F M ’ F N , (jx f, y) ’ F f (y). The mappings FM,N
r

are called the associated maps of the bundle functor F .
Proposition. The associated maps are smooth.
Proof. For m = 0 the assertion is trivial. Let us assume m > 0. Since smooth-
ness is a local property, we may restrict ourselves to M = N = Rm . Indeed,
chosen local charts on M and N we get local trivializations on F M and F N and
the induced local chart on invJ r (M, N ). Hence we have

∼ ∼
FU,V
= =
invJ r (Rm , Rm ) —Rm F Rm ’ invJ r (U, V ) —U F U ’ ’ F V ’ F Rm
’ ’’ ’

and we can apply the locality condition.
Now, let us recall that every jet in J r (Rm , Rn ) has a canonical polynomial
representative and that this space coincides with the cartesian product of Rm and
the Euclidean space of coe¬cients of these polynomials, as a smooth manifold. If
we consider the map ev : invJ r (Rm , Rm ) — Rm ’ Rm , evx (j0 f ) = f (x), then the
r
˜
associated map FRm ,Rm coincides with the map F (ev) appearing in the regularity
condition.


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
140 Chapter IV. Jets and natural bundles


14.5. Induced action. According to proposition 14.4 the restriction =
FRm ,Rm |Gr — S is a smooth left action of the jet group Gr on the standard
m m
¬ber S.
Let us de¬ne qM = FRm ,M |invJ0 (Rm , M ) — S : P r M — S ’ F M . For every
r

u = j0 g ∈ invJ0 (Rm , M ), s ∈ S and j0 f ∈ Gr we have
r r r
m


qM (j0 g —¦ j0 f, (j0 f ’1 , s)) = qM (j0 g, s)
r r r r
(1)

r
and the restriction (qM )u := qM (j0 g, ) is a di¬eomorphism. Hence q determines
the structure of the associated ¬ber bundle P r M [S; ] on F M , cf. 10.7.
Proposition. For every bundle functor F : Mfm ’ FM of order r and every
m-dimensional manifold M there is a canonical structure of an associated bundle
P r M [S; ] on F M given by the map qM and the values of the functor F lie in
the category of bundles with structure group Gr and standard ¬ber S.
m

Proof. The ¬rst part was already proved. Consider a local di¬eomorphism
f : M ’ N . For every j0 g ∈ P r M , s ∈ S we have
r


r r
F f —¦ qM (j0 g, s) = F f —¦ F g(s) = qN (j0 (f —¦ g), s).

So we identify F f with {P r f, idS } : P r M —Gr S ’ P r N —Gr S.
m m


14.6. Description of r-th order natural bundles. Every smooth left action
of Gr on a manifold S determines a covariant functor L : PB(Gr ) ’ FMm ,
m m
LP = P [S; ], Lf = {f, idS }. An r-th order bundle functor F with standard
¬ber S induces an action of Gr on S and we can construct a natural bundle
m
G = L —¦ P r : Mfm ’ FM.
r r
We claim that F is naturally equivalent to G. For every u = j0 g ∈ Px M
there is the di¬eomorphism (qM )u : S ’ Fx M which we shall denote F u. Hence
we can de¬ne maps χM : GM ’ F M by

r
χM ({u, s}) = F u(s) = qM (j0 g, s) = F g(s).

According to 14.5.(1), this is a correct de¬nition, and by the construction, the
maps χM are ¬bered isomorphisms. Since Gf = {P r f, idS } for every local
r
di¬eomorphism f : M ’ N , we have F f —¦ χM ({j0 g, s}) = F (f —¦ g)(s) = χN —¦
r
Gf ({j0 g, s}).
From the geometrical point of view, naturally equivalent functors can be
identi¬ed. Hence we have proved
Theorem. There is a bijective correspondence between the set of all r-th order
natural bundles on m-dimensional manifolds and the set of smooth left actions
of the jet group Gr on smooth manifolds.
m

In the next examples, we demonstrate on well known natural bundles, that
the identi¬cation in the theorem is exactly what the geometers usually do.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
14. Natural bundles and operators 141


14.7. Examples.
1. The reader should reconsider that in the case of frame bundles P r the
identi¬cation used in 14.6, i.e. the relation of the functor P r to the functor
G constructed from the induced action, is exactly the usual identi¬cation of
principal ¬ber bundles (P, p, M, G) with their associated bundles P [G, »], where
» is the left action of G on itself.
1 1
2. For the tangent bundle T , the map (qM )u with u = j0 g ∈ Px M is just the
linear map T0 g : T0 Rm ’ Tx M determined by j0 g, i.e. the linear coordinates
1

on Tx M induced by local chart g. Hence the tangent bundle corresponds to the
canonical action of G1 = GL(m, R) on Rm .
m
r
3. Further well known natural bundles are the functors Tk of r-th order k-
velocities. More precisely, we consider the restrictions of the functors de¬ned in
12.8 to the category Mfm . Let us recall that Tk M = J0 (Rk , M ) and the action
r r

on morphisms is given by the composition of jets. Hence, in this case, for every
r r
u = j0 g ∈ Px M the map (qM )u transforms the classes of r-equivalent maps
(Rk , 0) ’ (M, x) into their induced coordinate expressions in the local chart g,
i.e. (qM )’1 (j0 f ) = j0 (g ’1 —¦ f ).
r r
u

14.8. Vector bundle functors. In accordance with 6.14, a bundle functor
F : Mfm ’ FM is called a vector bundle functor on Mfm , or natural vector
bundle, if there is a canonical vector bundle structure on each value F M and
the values F f on morphisms are morphisms of vector bundles. Let F be an
r-th order natural vector bundle with standard ¬ber V and with induced action
: Gr — V ’ V . Then is a group homomorphism Gr ’ GL(V ) and so V
m m
carries a structure of Gm -module. On the other hand, every Gr -module V gives
r
m
rise to a natural bundle F , see the construction in 14.6, and an application of F
to charts of any atlas on a manifold M yields a vector bundle atlas on the value
F M ’ M . Therefore proposition 14.6 implies
Proposition. There is a bijective correspondence between r-th order vector
bundle functors on Mfm and Gr -modules.
m

14.9. Examples.
1. In our setting, the p-covariant and q-contravariant tensor ¬elds on a man-
ifold M are just the smooth global sections of F M ’ M , where F is the vector
bundle functor corresponding to the GL(m)-module —p Rm— — —q Rm , cf. 7.2.
2. In 6.7 we discussed constructions with vector bundles corresponding to a
smooth covariant functor F on the category of ¬nite dimensional vector spaces
and these constructions can be applied to the values of any natural vector bundle
to get new natural vector bundles, cf. 6.14. There we applied F to the cocycle of
transition functions. Let us look what happens on the level of the corresponding
Gr -modules. If we apply F to a Gr -module V with action : Gr ’ GL(V ),
m m m
˜: Gr ’ GL(FV ), ˜(g) = F( (g)), i.e.
we get a vector space FV with action m
a new Gr -module FV . Let us assume that G and FG are the natural vector
m
bundles corresponding to V and FV . The canonical vector bundle structure on
(FG)M = P r M —Gr FV coincides with that on F(GM ) by 10.7.(4). Similarly,
m
we can handle contravariant functors and bifunctors on the category of vector

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
142 Chapter IV. Jets and natural bundles


spaces, cf. 6.7. In particular, the values of natural vector bundles corresponding
to direct sums of the modules are just ¬bered products over the base manifolds
of the individual bundles. Let us also note that C ∞ •i Fi M = •i (C ∞ (Fi M )).
3. There are also well known examples of higher order natural vector bun-
dles. First of all, we recall the functor of r-th order k-dimensional covelocities
J r ( , Rk )0 = Tk introduced in 12.8. If r, k = 1, we get the dual bundles to
r—
1
the tangent bundles J0 (R, M ) = T M . So the vector bundle structure on the
cotangent bundle is natural and the tangent spaces are the duals, from our point
of view. But we can apply the construction of a dual module to any Gr -module
m
and this leads to dual natural vector bundles according to 14.6. In this way we
get the r-th order tangent bundles T (r) := (T r— )— or, more general the bundle
functors Tk = (Tk )— , see 12.14.
r r—

14.10. A¬ne bundle functors. A bundle functor F : Mfm ’ FM is called
an a¬ne bundle functor on Mfm , or natural a¬ne bundle, if each value F M ’
M is an a¬ne bundle and the values on morphisms are a¬ne maps. Hence the
standard ¬ber V of an r-th order natural a¬ne bundle is an a¬ne space and the
induced action is a representation of Gr in the group of a¬ne transformations
m
r
of V . So for each g ∈ Gm there is a unique linear map (g) : V ’ V satisfying
(g)(y) = (g)(x) + (g)(y ’ x) for all x, y ∈ V . It follows that is a linear rep-
resentation of Gr on the vector space V and there is the corresponding natural
m
vector bundle F . By the construction, for every m-dimensional manifold M the
value F M is just the modelling vector bundle to F M and for every morphism
f : M ’ N , F f is the modelling linear map to F f . Hence two arbitrary sections
of F M ˜di¬er™ by a section of F M . The best known example of a second order
natural a¬ne bundle is the bundle of elements of linear connections QP 1 which
’’’
we shall study in section 17. The modelling natural vector bundle QP 1 is the
tensor bundle T — T — — T — corresponding to GL(m)-module Rm — Rm— — Rm— .
Next we shall describe all natural transformations between natural bundles
in the terms of Gr -equivariant maps.
m

14.11. Lemma. For every natural transformation χ : F ’ G between two
natural bundles on Mfm all mappings χM : F M ’ GM cover the identities
idM .
Proof. Let χ : F ’ G be a natural transformation and let us write p : F M ’ M
and q : GM ’ M for the canonical projections onto an m-dimensional manifold
M . If y ∈ F M is a point with z := q(χM (y)) = p(y), then there is a local
di¬eomorphism f : M ’ M such that germp(y) f = germp(y) idM and f (z) = z , ¯
z = z. But now the localization condition implies q—¦χM —¦F f (y) = q—¦Gf —¦χM (y),
¯
for q —¦ Gf = f —¦ q. This is a contradiction.
14.12. Theorem. There is a bijective correspondence between the set of all
natural transformations between two r-th order natural bundles on Mfm and
the set of smooth Gr -equivariant maps between their standard ¬bers.
m

Proof. Let F and G be natural bundles with standard ¬bers S and Q and let
χ : F ’ G be a natural transformation. According to 14.11, we have the restric-

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
14. Natural bundles and operators 143


tion χRm |S : S ’ Q and we claim that this is Gr -equivariant with respect to the
m
induced actions. Indeed, for any j0 f ∈ Gr we get (χRm |S) —¦ F f = Gf —¦ (χRm |S),
r
m
r
but F f : S ’ S and Gf : Q ’ Q are just the induced actions of j0 f on S and
Q. Now we have to show that the whole transformation χ is determined by the
map χRm |S. First, using translations tx : Rm ’ Rm we see this for the map
χRm . Then, if we choose any atlas (U± , u± ) on a manifold M , the maps F u±
form a ¬ber bundle atlas on F M and we know χM —¦ F u± = Gu± —¦ χRm . Hence
the locality of bundle functors implies χM |(pF )’1 (U± ) = Gu± —¦ χRm —¦ (F u± )’1 .
M
On the other hand, let χ0 : S ’ Q be an arbitrary Gr -equivariant smooth
m
map. According to 14.6, the functors F or G are canonically naturally equivalent
to the functors L —¦ P r or K —¦ P r , where L or K are the functors corresponding
to the induced Gr -actions or k on the standard ¬bers S or Q, respectively.
m
So it su¬ces to de¬ne a natural transformation χ : L —¦ P r ’ K —¦ P r . We
set χM = {idP r M , χ0 }. It is an easy exercise to verify that χ is a natural
transformation. Moreover, we have χRm |S = χ0 .
In general, an operator is a rule transforming sections of a ¬bered manifold
¯ ¯
Y ’ M into sections of another ¬bered manifold Y ’ M . We shall deal with
¯ ∞
the case M = M in this section. Let us recall that C Y means the set of all
smooth sections of a ¬bered manifold Y ’ M .
p p
¯
¯’
14.13. De¬nition. Let Y ’ M , Y ’ M be ¬bered manifolds. A local

¯
operator A : C ∞ Y ’ C ∞ Y is a map such that for every section s : M ’ Y
and every point x ∈ M the value As(x) depends on the germ of s at x only.
k k
If, moreover, for certain k ∈ N or k = ∞ the condition jx s = jx q implies
¯
As(x) = Aq(x), then A is said to be of order k. An operator A : C ∞ Y ’ C ∞ Y
is called a regular operator if every smoothly parameterized family of sections of
¯
Y is transformed into a smoothly parameterized family of sections of Y .
14.14. Associated maps to an k-th order operator. Consider an operator
¯ ¯
A : C ∞ Y ’ C ∞ Y of order k. We de¬ne a map A : J k Y ’ Y by A(jx s) = As(x)
k

which is called the associated map to the k-th order operator A.
Proposition. The associated map to any ¬nite order operator A is smooth if
and only if A is regular.
¯
Proof. Let A : C ∞ Y ’ C ∞ Y be an operator of order k. If we choose local ¬bered
coordinates on Y , we also get the induced ¬bered coordinates on J k Y . But
in these local coordinates, the jets of sections are identi¬ed with (polynomial)
sections. Thus, a chart on J k Y can be viewed as a smoothly parameterized
family of sections in C ∞ Y and so the smoothness of A follows from the regularity.
The converse implication is obvious.
14.15. Natural operators. A natural operator A : F G between two
natural bundles F and G is a system of regular operators AM : C ∞ (F M ) ’
C ∞ (GM ), M ∈ ObMfm , satisfying
(i) for every section s ∈ C ∞ (F M ’ M ) and every di¬eomorphism f : M ’ N
it holds
AN (F f —¦ s —¦ f ’1 ) = Gf —¦ AM s —¦ f ’1

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
144 Chapter IV. Jets and natural bundles


(ii) AU (s|U ) = (AM s)|U for every s ∈ C ∞ (F M ) and every open submanifold
U ‚ M.
In particular, condition (ii) implies that natural operators are formed by local
operators.
G is said to be of order k, 0 ¤ k ¤ ∞, if all
A natural operator A : F
operators AM are of order k. The system of associated maps AM : J k F M ’ GM
to the k-th order operators AM is called the system of associated maps to the
natural operator A. The associated maps to ¬nite order natural operators are
smooth.
We can look at condition (i) even from the viewpoint of the local coordinates
on a manifold M . Given a local chart u : U ‚ M ’ V ‚ Rm , the di¬eo-
morphisms f : V ’ W ‚ Rm correspond to the changes of coordinates on U .
Combining this observation with localization property (ii), we conclude that the
natural operators coincide, in fact, with those operators, the local descriptions
of which do not depend on the changes of coordinates.
14.16. Proposition. For every r-th order bundle functor F on Mfm its
composition with the functor of k-th jet prolongations of ¬bered manifolds
J k : FM ’ FM is a natural bundle of order r + k.
Proof. Let f : M ’ N be a local di¬eomorphism. Then, by de¬nition of the
associated maps FM,N , we have

F f = FM,N —¦ (j r f —¦ pM ) — idF M : F M ’ F N .

Hence J k (F f ) depends on (k + r)-jets of f in the underlying points in M only.
It is an easy exercise to verify the axioms of natural bundles.
14.17. Proposition. There is a bijective correspondence between the set of
G between two natural bundles on Mfm
k-th order natural operators A : F
and the set of all natural transformations ± : J k —¦ F ’ G.
Proof. Let AM be the associated maps of an k-th order natural operator A : F
G. We claim that these maps form a natural transformation ± : J k F ’ G. They
are smooth by virtue of 14.14 and we have to verify Gf —¦ AM = AN —¦ J k F f for
an arbitrary local di¬eomorphism f : M ’ N . We have

AN ((J k F f )(jx s)) = AN (j k (F f —¦ s —¦ f ’1 )(f (x)))
k

= AN (F f —¦ s —¦ f ’1 )(f (x)) = Gf —¦ AM s(x)
k
= Gf —¦ AM (jx s).

On the other hand, consider a natural transformation ± : J k F ’ G. We
k
de¬ne operators AM : F M GM by AM s(x) = ±M (jx s) for all sections
s ∈ C ∞ (F M ). Since the maps ±M are smooth ¬bered morphisms and according
to lemma 14.11 they all cover the identities idM , the maps AM s are smooth sec-
tions of GM . The straightforward veri¬cation of the axioms of natural operators
is left to the reader.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
14. Natural bundles and operators 145


14.18. Let F : Mfm ’ FM be an r-th order natural bundle with standard
¬ber S and let : Gr —S ’ S be the induced action. The identi¬cation Rm —S ∼=
m
∼ C ∞ (F Rm ),

m m
F R , (x, s) ’ F (tx )(s), induces the identi¬cation C (R , S) =
(˜ : Rm ’ S) ’ (s(x) = F tx (˜(x))) ∈ C ∞ (F Rm ). Hence the standard ¬ber of
s s
k k
the natural bundle J F equals Tm S. Under these identi¬cations, the action of
F on an arbitrary local di¬eomorphism is of the form

F g(x, s) = (g(x), F (t’g(x) —¦ g —¦ tx )(s))

and the induced action k : Gr+k — Tm S ’ Tm S determined by the functor J k F
k k
m
is expressed by the following formula
r+k r+k
k k k k
(j0 g, j0 (F tx —¦ s(x)))
(1) (j0 g, j0 s) =

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