CHAPTER IX. BUNDLE FUNCTORS ON MANIFOLDS

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The description of the product preserving bundle functors on Mf in terms

of Weil algebras reects their general properties in a rather complete way. In

the present chapter we use some other procedures to deduce the basic geometric

properties of arbitrary bundle functors on Mf. Hence the basic subject of this

theory is a bundle functor on Mf that does not preserve products. Sometimes

we also contrast certain properties of the product-preserving and non-productpreserving

bundle functors on Mf. First we study the bundle functors with

the so-called point property, i.e. the image of a one-point set is a one-point

set. In particular, we deduce that their _bers are numerical spaces and that

they preserve products if and only if the dimensions of their values behave well.

Then we show that an arbitrary bundle functor on manifolds is, in a certain

sense, a `bundle' of functors with the point property. For an arbitrary vector

bundle functor F on Mf with the point property we also derive a canonical Lie

group structure on the prolongation FG of a Lie group G.

Next we introduce the concept of a ow-natural transformation of a bundle

functor F on manifolds. This is a natural transformation FT ! TF with the

property that for every vector _eld X: M ! TM its functorial prolongation

FX: FM ! FTM is transformed into the ow prolongation FX: FM !

TFM. We deduce that every bundle functor F on manifolds has a canonical ownatural

transformation, which is a natural equivalence if and only if F preserves

products. Then we point out some special features of natural transformations

from a Weil functor into an arbitrary bundle functor onMf. This gives a rather

e_ective method for their description. We also deduce that the homotheties are

the only natural transformations of the r-th order tangent bundle T(r) into itself.

This demonstrates that some properties of T(r) are quite di_erent from those of

Weil bundles, where such natural transformations are in bijection with a usually

much larger set of all endomorphisms of the corresponding Weil algebras. In the

last section we describe basic properties of the so-called star bundle functors,

which reect some constructions of contravariant character on Mf.

38. The point property

38.1. Examples. First we mention some examples of vector bundle functors

which do not preserve products. In 37.2 we deduced that every product preserving

vector bundle functor on Mf is the _bered product of a _nite number

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

330 Chapter IX. Bundle functors on manifolds

of copies of the tangent bundle T. In particular, every such functor is of order

one. Hence all tensor powers pT, p > 1, their sub bundles like SpT, _pT and

any combinations of them do not preserve products. This is also easily veri_ed

by counting dimensions. An important example of an r-th order vector bundle

functor is the r-th tangent functor T(r) described in 12.14 and 41.8. Let us mention

that another interesting example of an r-th order vector bundle functor, the

bundle of sector r-forms, will be discussed in 48.4.

38.2. Proposition. Every bundle functor F : Mf ! Mf transforms embeddings

into embeddings and immersions into immersions.

Proof. According to 1.14, a smooth mapping f : M ! N is an embedding if

and only if there is an open neighborhood U of f(M) in N and a smooth map

g : U ! M such that g _ f = idM. Hence if f is an embedding, then FU _ FN

is an open neighborhood of Ff(FM) and Fg _ Ff = idFM.

The locality of bundle functors now implies the assertion on immersions.

However this can be also proved easily considering the canonical local form

i : Rm ! Rm+n, x 7! (x; 0), of immersions, cf. 2.6, and applying F to the

composition of i and the projections pr1 : Rm+n ! Rm. _

38.3. The point property. Let us write pt for a one-point manifold. A bundle

functor F on Mf is said to have the point property if F(pt) = pt. Given such

functor F let us consider the maps ix : pt ! M, ix(pt) = x, for all manifolds

M and points x 2 M. The regularity of bundle functors on Mf proved in 20.7

implies that the maps cM : M ! FM, cM(x) = Fix(pt) are smooth sections of

pM : FM ! M. By de_nition, cN _f = Ff _cM for all smooth maps f : M ! N,

so that we have found a natural transformation c : IdMf ! F.

If F = TA for a Weil algebra A, this natural transformation corresponds to

the algebra homomorphism idR

_ 0: R ! R _ N = A. The r-th order tangent

functor has the point property, i.e. we have found a bundle functor which does not

preserve the products in any dimension except dimension zero. The technique

from example 22.2 yields easily bundle functors onMf which preserve products

just in all dimensions less then any _xed n 2 N.

38.4. Lemma. Let S be an m-dimensional manifold and s 2 S be a point.

If there is a smoothly parameterized system ht of maps, t 2 R, such that all

ht are di_eomorphisms except for t = 0, h0(S) = fsg and h1 = idS, then S is

di_eomorphic to RdimS.

Proof. Let us recall that if S = [1

k=0Sk where Sk are open submanifolds diffeomorphic

to Rm and Sk _ Sk+1 for all k, then S is di_eomorphic to Rm, see

[Hirsch, 76, Chapter 1, Section 2]. So let us choose an increasing sequence of

relatively compact open submanifolds Kn _ Kn+1 _ S with S = [1

k=1Kn and a

relatively compact neighborhood U of s di_eomorphic to Rm. Put S0 = U. Since

S0 is relatively compact, there is an integer n1 with Kn1

_ S0 and a t1 > 0 with

ht1 (Kn1 ) _ U. Then we de_ne S1 = (ht1 )􀀀1(U) so that we have S1 _ Kn1

_ S0

and S1 is relatively compact and di_eomorphic to Rm. Iterating this procedure,

we construct sequences Sk and nk satisfying Sk _ Knk

_ Sk􀀀1, nk > nk􀀀1. _

Let us denote by km the dimensions of standard _bers Sm = F0Rm.

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

38. The point property 331

38.5. Proposition. The standard _bers Sm of every bundle functor F onMf

with the point property are di_eomorphic to Rkm.

Proof. Let us write s = cRm(0), 0 2 Rm, and let gt : Rm ! Rm be the homotheties

gt(x) = tx, t 2 R. Since g0(Rm) = f0g, the smoothly parameterized family

ht = FgtjSm: Sm ! Sm satis_es all assumptions of the previous lemma. _

For a product M

p

􀀀 M _ N

q 􀀀!

N

the values FM

Fp

􀀀􀀀 F(M _ N) Fq

􀀀􀀀! FN

determine a canonical map _ : F(M _ N) ! FM _ FN.

38.6. Lemma. For every bundle functor F on Mf with the point property all

the maps _ : F(M _ N) ! FM _ FN are surjective submersions.

Proof. By locality of F it su_ces to discuss the case M = Rm, N = Rn. Write

0k = cRk (0) 2 FRk, k = 0; 1; : : : , and denote i : Rm ! Rm+n, i(x) = (x; 0),

and j : Rn ! Rm+n, j(y) = (0; y). In the tangent space T0m+nFRm+n, there are

subspaces V = TFi(T0mFRm) and W = TFj(T0nFRn). We claim V \ W =

0. Indeed, if A 2 V \ W, i.e. A = TFi(B) = TFj(C) with B 2 T0mFRm

and C 2 T0nFRn, then TFp(A) = TFp(TFi(B)) = B, but at the same time

TFp(A) = TFp_TFj(C) = 0m, for p_ j is the constant map of Rn into 0 2 Rm,

and A = TFi(B) = 0 follows.

Hence T_j(V _W) : V _W ! T0mFRm _T0nFRn is invertible and so _ is a

submersion at 0m+n and consequently on a neighborhood U _ FRm+n of 0m+n.

Since the actions of R de_ned by the homotheties gt on Rm, Rn and Rm+n

commute with the product projections p and q, the induced actions on FRm,

FRn, FRm+n commute with _ as well (draw a diagram if necessary). The family

Fgt is smoothly parameterized and Fg0(FRm+n) = f0m+ng, so that every point

of FRm+n is mapped into U by a suitable Fgt, t > 0. Further all Fgt with t > 0

are di_eomorphisms and so _ is a submersion globally. Therefore the image

_(FRm+n) is an open neighborhood of (0m; 0n) 2 FRm _ FRn. But similarly

as above, every point of FRm _ FRn can be mapped into this neighborhood by

a suitable Fgt, t > 0. This implies that _ is surjective. _

It should be an easy exercise for the reader to extend the lemma to arbitrary

_nite products of manifolds.

38.7. Corollary. Every bundle functor F on Mf with the point property

transforms submersions into submersions.

Proof. The local canonical form of any submersion is p: Rn_Rk ! Rn, p(x; y) =

x, cf. 2.2. Then Fp = pr1 _ _ is a composition of two submersions _ : F(Rn _

Rk) ! FRn _FRk and pr1 : FRn _FRk ! FRn. Since every bundle functor is

local, this concludes the proof. _

38.8. Proposition. If a bundle functor F onMf has the point property, then

the dimensions of its standard _bers satisfy km+n _ km+kn for all 0 _ m+n <

1. Equality holds if and only if F preserves products in dimensions m and n.

Proof. By lemma 38.6, we have the submersions _ : F(Rm_Rn) ! FRm_FRn

which implies km+n _ km+kn. If the equality holds, then _ is a local di_eomorphism

at each point. Since _ commutes with the action of the homotheties, it

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

332 Chapter IX. Bundle functors on manifolds

must be bijective on each _ber over Rm+n, and therefore _ must be a global diffeomorphism.

Given arbitrary manifolds M and N of the proper dimensions, the

locality of bundle functors and a standard diagram chasing lead to the conclusion

that

FM

Fp

􀀀􀀀 F(M _ N) Fq

􀀀􀀀! FN

is a product. _

In view of the results of the previous chapter we get

38.9. Corollary. For every bundle functor F on Mf with the point property

the dimensions of its values satisfy dimFRm = mdimFR if and only if there is

a Weil algebra A such that F is naturally equivalent to the Weil bundle TA.

38.10. For every Weil algebra A and every Lie group G there is a canonical Lie

group structure on TAG obtained by the application of the Weil bundle TA to

all operations on G, cf. 37.16. If we replace TA by an arbitrary bundle functor

onMf, we are not able to repeat this construction. However, in the special case

of a vector bundle functor F on Mf with the point property we can perform

another procedure.

For all manifolds M, N the inclusions iy : M ! M _ N, iy(x) = (x; y),

jx : N ! M _ N, jx(y) = (x; y), (x; y) 2 M _ N, form smoothly parameterized

families of morphisms and so we can de_ne a morphism _M;N : FM _ FN !

F(M _N) by _M;N (z;w) = FipN(w)(z)+FjpM(z)(w), where pM : FM ! M are

the canonical projections. One veri_es easily that the diagram

FM _ FN w

_M;N

u

Ff _ Fg

F(M _ N)

u

F(f _ g)

F _M _ F _N w

_ _M; _N F( _M _ _N )

commutes for all maps f : M ! _M , g : N ! _N . So we have constructed a

natural transformation _ : Prod_(F; F) ! F _Prod, where Prod is the bifunctor

corresponding to the products of manifolds and maps. The projections p: M _

N ! M, q : M_N ! N determine the map (Fp; Fq) : F(M_N) ! FM_FN

and by the de_nition of _M;N , we get (Fp; Fq) _ _M;N = idFM_FN. Now, given

a Lie group G with the operations _: G_G ! G, _ : G ! G and e: pt ! G, we

de_ne _FG = F_ _ _G;G, _FG = F_ and eFG = Fe = cG(e) where cG : G ! FG

is the canonical section. By the de_nition of _ , we get for every element (z;w) 2

FG _ FG over (x; y) 2 G _ G

_FG(z;w) = F(_( ; y))(z) + F(_(x; ))(w)

and it is easy to check all axioms of Lie groups for the operations _FG, _FG and

eFG on FG. In particular, we have a canonical Lie group structure on the r-th

order tangent bundles T(r)G over any Lie group G and on all tensor bundles

over G.

Since _ is the identity if F equals to the tangent bundle T, we have generalized

the canonical Lie group structure on tangent bundles over Lie groups to all vector

bundle functors with the point property, cf. 37.2.

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

38. The point property 333

38.11. Remark. Given a bundle functor F onMf and a principal _ber bundle

(P; p;M;G) we might be interested in a natural principal bundle structure on

Fp: FP ! FM with structure group FG. If F is a Weil bundle, this structure

can be de_ned by application of F to all maps in question, cf. 37.16. Though we

have found a natural Lie group structure on FG for vector bundle functors with

the point property which do not preserve products, there is still no structure of

principal _ber bundle (FP; Fp; FM; FG) for dimension reasons, see 38.8.

38.12. Let us now consider a general bundle functor F on Mf and write

Q = F(pt). For every manifold M the unique map qM : M ! pt induces

FqM : FM ! Q and similarly to 38.3, every point a 2 Q determines a canonical

natural section c(a)M(x) = Fix(a). Let G be the bundle functor onMf de_ned

by GM = M _ Q on manifolds and Gf = f _ idQ on maps.

Lemma. The maps _M(x; a) = c(a)M(x), (x; a) 2 M _ Q, and _M(z) =

(pM(z); FqM(z)), z 2 FM, de_ne natural transformations _ : G ! F and

_: F ! G satisfying _ _ _ = id. Moreover the _M are embeddings and the

_M are submersions for all manifolds M. In particular, for every a 2 Q the rule

FaM = (FqM)􀀀1(a), Faf = FfjFaM determines a bundle functor on Mf with

the point property.

Proof. It is easy to verify that _ and _ are natural transformations satisfying

_ _ _ = id. This equality implies that _M is an embedding and also that _M

is a surjective map which has maximal rank on a neighborhood U of the image

_M(M _ Q). It su_ces to prove that every _Rm is a submersion. Consider the

homotheties gt(x) = tx on Rm. Then Fgt is a smoothly parameterized family

with Fg1 = idRm and Fg0(FRm) = Fi0 _ FqRm(FRm) _ _M(Rm _ Q). Hence

every point of FRm is mapped into U by some Fgt with t > 0 and so _Rm has

maximal rank everywhere.

Since FqM is the second component of the surjective submersion _M, all the

subsets FaM _ FM are submanifolds and one easily checks all the axioms of

bundle functors. _

38.13. Proposition. Every bundle functor on Mf transforms submersions

into submersions.

Proof. By the previous lemma, every value Ff : FM ! FN is a _bered morphism

of FqM : FM ! Q into FqN : FN ! Q over the identity on Q. If f

is a submersion, then every Faf : FaM ! FaN is a submersion according to

38.7. _

38.14. Proposition. The dimensions of the standard _bers of every bundle

functor F on Mf satisfy km+n _ km + kn 􀀀 dimF(pt). Equality holds if and

only if all bundle functors Fa preserve products in dimensions m and n. _

38.15. Remarks. If the standard _bers of a bundle functor F on Mf are

compact, then all the functors Fa must coincide with the identity functor on

Mf according to 38.5. But then the natural transformations _ and _ from

38.12 are natural equivalences.

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

334 Chapter IX. Bundle functors on manifolds

38.16. Example. Taking any bundle functor G onMf with the point property

and any manifold Q, we can de_ne FM = GM _ Q and Ff = Gf _ idQ

to get a bundle functor with F(pt) = Q. We present an example showing

that not all bundle functors on Mf are of this type. The basic idea is that

some of the individual `_ber components' Fa of F coincide with the functor T2

1

of 1-dimensional velocities of the second order while some other ones are the

Whitney sums T _ T in dependence on the zero values of a smooth function

on Q. According to the general theory developed in section 14, it su_ces to

construct a functor on the second order skeleton of Mf. So we take the system

of standard _bers Sn = Q _ Rn _ Rn, n 2 N0, and we have to de_ne the action

of all jets from J2

0 (Rm;Rn)0 on Sm. Let us write ap

i , ap

ij for the coe_cients of

canonical polynomial representatives of the jets in question. Given any smooth

function f : Q ! R we de_ne a map J2

0 (Rm;Rn)0 _ Sm ! Sn by

(ap

i ; arj

k)(q; y`; zm) = (q; ap

i yi; f(q)ar

ijyiyj + ari

zi):

One veri_es easily that this is an action of the second order skeleton on the

system Sn. Obviously, the corresponding bundle functor F satis_es F(pt) = Q

and the bundle functors Fq coincide with T _ T for all q 2 Q with f(q) = 0.

If f(q) 6= 0, then Fq is naturally equivalent to the functor T2

1 . Indeed, the

maps R2n ! R2n, yi 7! yi, and zi 7! f(q)zi are invertible and de_ne a natural

equivalence of T2

1 into Fq, see 18.15 for a help in a more detailed veri_cation.

38.17. Consider a submersion f : Y ! M and denote by _: FY ! FM _M Y

the induced pullback map, cf. 2.19.

Proposition. The pullback map _: FY ! FM _M Y of every submersion

f : Y ! M is a submersion as well.

We remark that this property represents a special case of the so-called prolongation

axiom which was introduced in [Pradines, 74b] for a more general

situation.

Proof. In view of 38.12 we may restrict ourselves to bundle functors with point

property (in general FqM : FM ! F(pt) and FqY : FY ! F(pt) are _bered

manifolds and _ is a _bered morphism so that we can verify our assertion

_berwise). Further we may consider the submersion f in its local form, i.e.

f : Rm+n ! Rm, (x; y) 7! x, for then the claim follows from the locality of the

functors. Now we can easily choose a smoothly parametrized family of local

sections s: Y _M ! Y with s(y; f(y)) = y, sy 2 C1(Y ), e.g. s(x;y)(_x) = (_x; y).

Then we de_ne a mapping _: FM_M Y ! FY , _(z; y) := Fsy(z). Since locally

Ff _Fsy = idM and pFY

_Fsy = sy_pF

M, we have constructed a section of _. Since

the canonical sections cM : M ! FM are natural, we get _(cM(x); y) = cY (y).

Hence the section goes through the values of the canonical section cY and _ has

the maximal rank on a neighborhood of this section. Now the action of homotheties

on Y = Rm+n and M = Rm commute with the canonical local form of f

and therefore the rank of _ is maximal globally. _

In particular, given two bundle functors F, G on Mf, the natural transformation

_: FG ! F _ G de_ned as the product of the natural transformations

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

38. The point property 335

F(pG) : FG ! F and pF _ G: FG ! G is formed by surjective submersions

_M : F(GM) ! FM _M GM.

38.18. At the end of this section, we shall indicate how the above results can

be extended to bundle functors on FMm. The point property still plays an

important role. Since any manifold M can be viewed as the _bered manifold

idM : M ! M, we can say that a bundle functor F : FMm ! FMhas the point

property if FM = M for all m-dimensional manifolds. Bundle functors on FMm

with the point property do not admit canonical sections in general, but for every

_bered manifold qY : Y ! M in FMm we have the _bration FqY : FY ! M and

FqY = qY _ pY , where pY : FY ! Y is the bundle projection of FY . Moreover,

the mapping C1(qY : Y ! M) ! C1(FqY : FY ! M), s 7! Fs is natural with

respect to _bered isomorphisms. This enables us to generalize easily the proof

of proposition 38.5 to our more general situation, for we can use the image of

the section i : Rm ! Rm+n, x 7! (x; 0) instead of the canonical sections cM from

38.5. So the standard _bers Sn = FRm+n of a bundle functor with the point

property are di_eomorphic to Rkn.

Proposition. The dimensions kn of standard _bers of every bundle functor

F : FMm ! FM with the point property satisfy kn+p _ kn + kp and for every

FMm-objects qY : Y ! M, q _ Y : _ Y ! M the canonical map _ : F(Y _M _ Y ) !

FY _M F _ Y is a surjective submersion. Equality holds if and only if F preserves

_bered products in dimensions n and p of the _bers. So F preserves _bered

products if and only if k(n) = n:k(1) for all n 2 N0.

Proof. Consider the diagram

F(Y _M _ Y )

NNNNNNNNNNNNNNP

F _p _ _ [[[[[[[]

Fp FY _M F _ Y w pr2

u

pr1

F _ Y

u

Fq _ Y

FY w FqY M

where p and _p are the projections on Y _M _ Y .

By locality of bundle functors it su_ces to restrict ourselves to objects from

a local pointed skeleton. In particular, we shall deal with the values of F on

trivial bundles Y = M _ S. In the special case m = 0, the proposition was

proved above.

For every point x 2 M we write (FY )x := (FqY )􀀀1(x) and we de_ne a functor

G = Gx : Mf ! FM as follows. We set G(Yx) := (FY )x and for every map

f = idM _ f1 : Y ! _ Y , f1 : Yx ! _ Yx we de_ne Gf1 := Ffj(FY )x : GYx ! G_ Yx.

If we restrict all the maps in the diagram to the appropriate preimages, we get

the product (FY )x

pr1 􀀀􀀀 (FY )x _ (F _ Y )x

pr2 􀀀􀀀! (F _ Y )x and _x : G(Yx _ _ Yx) !

GYx _ G_ Yx. Since G has the point property, _x is a surjective submersion.

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

336 Chapter IX. Bundle functors on manifolds

Hence _ is a _bered morphism over the identity on M which is _ber wise

a surjective submersion. Consequently _ is a surjective submersion and the

inequality kn+p _ kn + kp follows.

Now similarly to 38.8, if the equality holds, then _ is a global isomorphism. _

38.19. Vertical Weil bundles. Let A be a Weil algebra. We de_ne a functor

VA : FMm ! FM as follows. For every qY : Y ! M, we put VAY : =

[x2MTAYx and given f 2 FMm(Y; _ Y ) we write fx = fjYx, x 2 M, and we set

VAfj(VAY )x := TAfx. Since VA(Rm+n ! Rm) = Rm _ TARn carries a canonical

smooth structure, every _bered atlas on Y ! M induces a _bered atlas

on VAY ! Y . It is easy to verify that VA is a bundle functor which preserves

_bered products. In the special case of the algebra D of dual numbers we get

the vertical tangent bundle V .

Consider a bundle functor F : FMm ! FM with the point property which

preserves _bered products, and a trivial bundle Y = M _ S. If we repeat the

construction of the product preserving functors G = Gx, x 2 M, from the proof

of proposition 38.18 we have Gx = TAx for certain Weil algebras A = Ax. So

we conclude that F(idM _ f1)j(FY )x = Gx(f1) = VAx(idM _ f1)j(FY )x. At

the same time the general theory of bundle functors implies (we take A = A0)

FRm+n = Rm _ Rn _ Sn = Rm _ An = VARm+n for all n 2 N (including the

actions of jets of maps of the form idRm_f1). So all the algebras Ax coincide and

since the bundles in question are trivial, we can always _nd an atlas (U_; '_)

on Y such that the chart changings are over the identity on M. But a cocycle

de_ning the topological structure of FY is obtained if we apply F to these chart

changings and therefore the resulting cocycle coincides with that obtained from

the functor VA.

Hence we have deduced the following characterization (which is not a complete

description as in 36.1) of the _bered product preserving bundle functors on

FMm.

Proposition. Let F : FMm ! FM be a bundle functor with the point property.

The following conditions are equivalent.

(i) F preserves _bered products

(ii) For all n 2 N it holds dimSn = n(dimS1)

(iii) There is a Weil algebra A such that FY = VAY for every trivial bundle

Y = M _ S and for every mapping f1 : S ! _ S we have F(idM _ f1) =

VA(idM _ f1) : F(M _ S) ! F(M _ _ S).

39. The ow-natural transformation

39.1. De_nition. Consider a bundle functor F : Mf ! FM and the tangent

functor T : Mf ! FM. A natural transformation _ : FT ! TF is called a ownatural

transformation if the following diagram commutes for all m-dimensional

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

39. The ow-natural transformation 337

manifolds M and all vector _elds X 2 X(M) on M.

(1)

FTM w

F_M 44446 _M

FM

FM

u

FX

w FX TFM

u

_FM

39.2. Given a map f : Q _M ! N, we have denoted by ~ Ff : Q _ FM ! FN

the `collection' of F(f(q; )) for all q 2 Q, see 14.1. Write (x;X) = Y 2 TRm =

Rm _ Rm and de_ne _Rm : R _ TRm ! Rm, _Rm(t; Y ) = (x + tX) for t 2 R,.

Theorem. Every bundle functor F : Mf ! FM admits a canonical ownatural

transformation _ : FT ! TF determined by

_Rm(z) = j1

0

~ F_Rm( ; z):

If F has the point property, then _ is a natural equivalence if and only if F is a

Weil functor TA. In this case _ coincides with the canonical natural equivalence

TAT ! TTA corresponding to the exchange homomorphism A  D ! D  A

between the tensor products of Weil algebras.

39.3. The proof requires several steps. We start with a general lemma.

Lemma. Let M, N, Q be smooth manifolds and let f, g : Q _ M ! N be

smooth maps. If jk

q f( ; y) = jk

q g( ; y) for some q 2 Q and all y 2 M, then

for every bundle functor F on Mf the maps ~ Ff, ~ Fg : Q _ FM ! FN satisfy

jk

q

~ Ff( ; z) = jk

q

~ Fg( ; z) for all z 2 FM.

Proof. It su_ces to restrict ourselves to objects from the local skeleton (Rm),

m = 0; 1; : : : , of Mf. Let r be the order of F valid for maps with source Rm,

cf. 22.3, and write p for the bundle projection pRm. By the general theory of

bundle functors the values of F on morphisms f : Rm ! Rn are determined by

the smooth associated map FRm;Rn : Jr(Rm;Rn) _

Rm FRm ! FRn, see section

14. Hence the map ~ Ff : Q _ FRm ! FRn is de_ned by the composition of

FRm;Rn with the smooth map fr : Q _ FRm ! Jr(Rm;Rn) _

Rm FRm, (q; z) 7!

(jr

p(z)f(q; ); z). Our assumption implies that fr( ; z) and gr( ; z) have the same

k-jet at q, which proves the lemma. _

39.4. Now we deduce that the maps _Rm determine a natural transformation

_ : FT ! TF such that the upper triangle in 39.1.(1) commutes. These maps

de_ne a natural transformation between the bundle functors in question if they

obey the necessary commutativity with respect to the actions of morphisms

between the objects of the local skeleton Rm, m = 0; 1; : : : . Given such a

morphism f : Rm ! Rn we have

_Rn(FTf(z)) = j1

0

~ F_Rn( ; FTf(z)) = j1

0F((_Rn)t _ Tf)(z)

TFf(_Rm(z)) = TFf(j1

0

~ F_Rm( ; z)) = j1

0 (Ff _ F(_Rm)t(z)) =

= j1

0F(f _ (_Rm)t)(z):

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

338 Chapter IX. Bundle functors on manifolds

So in view of lemma 39.3 it is su_cient to prove for all Y 2 TRm, f : Rm ! Rn

j1

0 ((f _ (_Rm)t)(Y )) = j1

0 ((_Rn)t _ Tf)(Y ):

By the de_nition of _, the values of both sides are Tf(Y ).

Since (_Rm)0 = _Rm : TRm ! Rm, we have _FRm _ _Rm = F(_Rm)0 = F_Rm.

39.5. Let us now discuss the bottom triangle in 39.1.(1). Given a bundle functor

F on Mf, both the arrows FX and FX are values of natural operators and _

is a natural transformation. If we _x dimension of the manifold M then these

operators are of _nite order. Therefore it su_ces to restrict ourselves to the

_bers over the distinguished points from the objects of a local pointed skeleton.

Moreover, if we verify _Rm _ FX = FX on the _ber (FT)0Rm for a jet of a

suitable order of a _eld X at 0 2 Rm, then this equality holds on the whole

orbit of this jet under the action of the corresponding jet group. Further, the

operators in question are regular and so the equality follows for the closure of

the orbit.

Lemma. The vector _eld X = @

@x1 on (Rm; 0) has the following two properties.

(1) Its ow satis_es FlX = _Rm _ (idR

_ X) : R _ Rm ! Rm.

(2) The orbit of the jet jr

0X under the action of the jet group Gr+1

m is dense

in the space of r-jets of vector _elds at 0 2 Rm.

Proof. We have FlXt

(x) = x+t(1; 0; : : : ; 0) = _Rm(t;X(x)). The second assertion

is proved in section 42 below. _

By the lemma, the mappings _Rm determine a ow-natural transformation

_ : FT ! TF.

Assume further that F has the point property and write kn for the dimension

of the standard _ber of FRn. If _ is a natural equivalence, then k2n = 2kn for all

n. Hence proposition 38.8 implies that F preserves products and so it must be

naturally equivalent to a Weil bundle. On the other hand, assume F = TA for

some Weil algebra A and denote 1 and e the generators of the algebra D of dual

numbers. For every jAf 2 TATR, with f : Rk ! TR = D, f(x) = g(x) + h(x):e,

take q : R _ Rk ! R, q(t; x) = g(x) + th(x), i.e. f(x) = j1

0q( ; x). Then we get

_R(jAf) = j1

0TA(_R)t(jAf) = j1

0 jA(g( ) + th( )) = j1

0 jAq(t; ):

Hence _R coincides with the canonical exchange homomorphism AD ! DA

and so _ is the canonical natural equivalence TAT ! TTA. _

39.6. Let us now modify the idea from 39.1 to bundle functors on FMm.

De_nition. Consider a bundle functor F : FMm ! FM and the vertical tangent

functor V : FMm ! FM. A natural transformation _ : FV ! V F is called

a ow-natural transformation if the diagram

(1)

FV Y w

F_Y 44446 _Y

FY

FY

u

FX

w FX V FY

u

_FY

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

39. The ow-natural transformation 339

commutes for all _bered manifolds Y with m-dimensional basis and for all vertical

vector _elds X on Y .

For every _bered manifold q : Y ! M in ObFMm, the _bration q__Y : V Y !

M is an FMm-morphism. Further, consider the local skeleton (Rm+n ! Rm)

of FMm and de_ne

_Rm+n : R _ V Rm+n = R1+m+n+n ! Rm+n; (t; x; y;X) 7! (x; y + tX):

Then every _Rm+n(t; ) is a globally de_ned FMm morphism and we have

j1

0_Rm+n( ; x; y;X) = (x; y;X):

39.7. The proof of 39.3 applies to general categories over manifolds. A bundle

functor on an admissible category C is said to be of a locally _nite order if for

every C-object A there is an order r such that for all C-morphisms f : A ! B

the values Ff(z), z 2 FA, depend on the jets jr

pA(z)f only. Let us recall that all

bundle functors on FMm have locally _nite order, cf. 22.3.

Lemma. Let f, g : Q _ mA ! mB be smoothly parameterized families of Cmorphisms

with jk

q f( ; y) = jk

q g( ; y) for some q 2 Q and all y 2 mA. Then

for every regular bundle functor F on C with locally _nite order, the maps ~ Ff,

~ Fg : Q _ FA ! FB satisfy jk

q

~ Ff( ; z) = jk

q

~ Fg( ; z) for all z 2 FA. _

39.8. Let us de_ne _Rm+n(z) = j1

0

~ F_Rm+n( ; z). If we repeat the considerations

from 39.4 we deduce that our maps _Rm+n determine a natural transformation

_ : FV ! TF. But its values satisfy TpY _ _Y (z) = j1

0pY _ F(_Y )t(z) = pY (z) 2

V Y and so _Y (z) 2 V (FY ! BY ). So _ : FV ! V F and similarly to 39.4 we

show that the upper triangle in 39.6.(1) commutes.

Every non-zero vertical vector _eld on Rm+n ! Rm can be locally transformed

(by means of an FMm-morphism) into a constant one and for all constant

vertical vector _elds X on Rm+n we have FlX = (_Rm+n _ (idRm+n _ X)).

Hence we also have an analogue of lemma 39.5.

Theorem. For every bundle functor F : FMm ! FM there is the canonical

ow-natural transformation _ : FV ! V F. If F has the point property, then _

is a natural equivalence if and only if F preserves _bered products.

We have to point out that we consider the _bered manifold structure FY !

BY for every object Y ! BY 2 ObFMm, i.e. _Y : F(V Y ! BY ) ! V (FY !

BY ).

Proof. We have proved that _ is ow-natural. Assume F has the point property.

If _ is a natural equivalence, then proposition 38.18 implies that F preserves

_bered products. On the other hand, F preserves _bered products if and only if

FRm+n = VARm+n for aWeil algebra A and then also Ff coincides with VAf for

morphisms of the form idRm_g : Rm+n ! Rm+k, see 38.19. But each _Rm+n(t; )

is of this form and any restriction of _Rm+n to a _ber (VAV Rm+n)x

_=

TATRm

coincides with the canonical ow natural equivalence TAT ! TTA, cf. 39.2.

Hence _ is a natural equivalence. _

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

340 Chapter IX. Bundle functors on manifolds

Let us remark that for F = Jr we obtain the well known canonical natural

equivalence JrV ! V Jr, cf. [Goldschmidt, Sternberg, 73], [Mangiarotti, Modugno,

83].

39.9. The action of some bundle functors F : FMm ! FM on morphisms can

be extended in such a way that the proof of theorem 39.8 might go through for

the whole tangent bundle. We shall show that this happens with the functors

Jr : FMm ! FM.

Since Jr(Rm+n ! Rm) is a sub bundle in the bundle Krm

Rm+n of contact

elements of order r formed by the elements transversal to the _bration, the

action of Jrf on a jet jrx

s extends to all local di_eomorphisms transforming jrx

s

into a jet of a section. Of course, we are not able to recover the whole theory

of bundle functors for this extended action of Jr, but one veri_es easily that

lemma 39.3 remains still valid.

So let us de_ne _t : TRm+n ! Rm+n by _t(x; z;X;Z) = (x + tX; z + tZ).

For every section (x; z(x);X(x);Z(x)) of TRm+n ! Rm, its composition with

_t and the _rst projection gives the map x 7! x + tX(x). If we proceed in a

similar way as above, we deduce

Proposition. There is a canonical ow-natural transformation _ : JrT ! TJr

and its restriction JrV ! V Jr is the canonical ow-natural equivalence.

39.10. Remark. Let us notice that _ : TJr ! JrT cannot be an equivalence

for dimension reasons if m > 0. The ow-natural transformations on jet bundles

were presented as a useful tool in [Mangiarotti, Modugno, 83].

It is instructive to derive the coordinate description of _Rm+n at least in the

case r = 1. Let us write a map f : (Rm+n _ 􀀀! Rm) ! (Rm+n _

􀀀! Rm) in the

form zk = fk(xi; yp), wq = fq(xi; yp). In order to get the action of J1f in the

extended sense on j1

0s = (yp; yp

i ) we have to consider the map (_ _ f _ s)􀀀1 = ~ f,

xi = ~ fi(z). So zk = fk( ~ f(z); yp( ~ f(z))) and we evaluate that the matrix @ ~ fi=@zk

is the inverse matrix to @fk=@xi + (@fk=@yp)yp

i (the invertibility of this matrix

is exactly the condition on j1

0s to lie in the domain of J1f). Now the coordinates

wq

k of J1f(j1

0s) are

wq

k = @fq

@xj

@ ~ fj

@zk + @fq

@yp yp

j

@ ~ fj

@zq :

Consider the canonical coordinates xi, yp on Y = Rm+n and the additional

coordinates yp

i or Xi, Y p or yp

i , Xi

j , Y p

i or yp

i , _i, _p, _p

i on J1Y or TY or J1TY

or TJ1Y , respectively. If j1

xs = (xi; yp;Xj ; Y q; yrk

;X`m ; Y s

n ), then

J1(_Y )t(j1

0s) = (xi + tXi; yp + tY p; _yq

j (t))

_yp

i (t)(_ij

+ tXi

j) = (yp

i + tY p

i )_ij

:

Di_erentiating by t at 0 we get

_Rm+n(xi; yp;Xj ; Y q; yrk

;X`m; Y s

n ) = (xi; yp; yrk

;Xj ; Y q; Y s

`

􀀀 ysm

Xm

` ):

This formula corresponds to the de_nition in [Mangiarotti, Modugno, 83].

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

40. Natural transformations 341

40. Natural transformations

40.1. The _rst part of this section is concerned with natural transformations

with a Weil bundle as the source. In this case we get a result similar to the

Yoneda lemma well known from general category theory. Namely, each point in

a Weil bundle TAM is an equivalence class of mappings in C1(Rn;M) where n

is the width of the Weil algebra A, see 35.15, and the canonical projections yield

a natural transformation _: C1(Rn; ) ! TA. Hence given any bundle functor

F on Mf, every natural transformation _: TA ! F gives rise to the natural

transformation _ _ _: C1(Rn; ) ! F and this is determined by the value of

(_ _ _)Rn(idRn). So in order to classify all natural transformations _: TA ! F

we have to distinguish the possible values v := _Rn _ _Rn(idRn) 2 FRn. Let

us recall that for every natural transformation _ between bundle functors on

Mf all maps _M are _bered maps over idM, see 14.11. Hence v 2 F0Rn and

another obvious condition is Ff(v) = Fg(v) for all maps f, g : Rn ! M with

jAf = jAg. On the other hand, having chosen such v 2 F0Rn, we can de_ne

_v

M(jAf) = Ff(v) and if all these maps are smooth, then they form a natural

transformation _v : TA ! F.

So from the technical point of view, our next considerations consist in a

better description of the points v with the above properties. In particular, we

deduce that it su_ces to verify Ff(v) = Fi(v) for all maps f : Rn ! Rn+1 with

jAf = jAi where i : Rn ! Rn+1, x 7! (x; 0).

40.2. De_nition. For every Weil algebra A of width n and for every bundle

functor F on Mf, an element v 2 F0Rn is called A-admissible if jAf =

jAi implies Ff(v) = Fi(v) for all f 2 C1(Rn;Rn+1). We denote by SA(F) _

S = F0Rn the set of all A-admissible elements.

40.3. Proposition. For every Weil algebra A of width n and every bundle

functor F on Mf, the map

_ 7! _Rn(jAidRn)

is a bijection between the natural transformations _: TA ! F and the subset of

A-admissible elements SA(F) _ F0Rn.

The proof consists in two steps. First we have to prove that each v 2 SA(F)

de_nes the transformation _v : TA ! F at the level of sets, cf. 40.1, and then

we have to verify that all maps _v

M are smooth.

40.4. Lemma. Let F : Mf ! FM be a bundle functor and A be a Weil

algebra of width n. For each point v 2 SA(F) and for all mappings f, g : Rn !

M the equality jAf = jAg implies Ff(v) = Fg(v).

Proof. The proof is a straightforward generalization of the proof of theorem 22.3

with m = 0. Therefore we shall present it in a rather condensed form.

During the whole proof, we may restrict ourselves to mappings f, g : Rn ! Rk

of maximal rank. The reason lies in the regularity of all bundle functors onMf,

cf. 22.3 and 20.7.

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

342 Chapter IX. Bundle functors on manifolds

The canonical local form of a map f : Rn ! Rn+1 of maximal rank is i and

therefore the assertion is trivial for the dimension k = n + 1.

Since the equivalence on the spaces C1(Rn;Rk) determined by A is compatible

with the products of maps, we can complete the proof as in 22.3.(b) and

22.3.(e) with m = 0, jr

0 replaced by jA and Sn replaced by SA(F). _

Let us remark that for m = 0 theorem 22.3 follows easily from this lemma.

Indeed, we can take the Weil algebra A corresponding to the bundle Trn+1

n of

n-dimensional velocities of order rn+1. Then jAf = jAg if and only if jrn+1

0 f =

jrn+1

0 g and according to the assumptions in 22.3, SA(F) = Sn. By the general

theory, the order rn+1 extends from the standard _ber Sn to all objects of

dimension n.

40.5. Lemma. For every Weil algebra A of width n and every smooth curve

c : R ! TARk there is a smoothly parameterized family of maps  : R_Rn ! Rk

such that jAt = c(t).

Proof. There is an ideal A in the algebra of germs En = C1

0 (Rn;R), cf. 35.5,

such that A = En=A. Write Drn

= Mr+1 where M is the maximal ideal in

En, and Dr

n = En=Drn

, i.e. TDr

n = Tr

n. Then A _ Drn

for suitable r and so we

get the linear projection Dr

n

! A, jr

0f 7! jAf. Let us choose a smooth section

s: A ! Dr

n of this projection. Now, given a curve c(t) = jAft in TARk there

are the canonical polynomial representatives gt of the jets s(jAft). If c(t) is

smooth, then gt is a smoothly parameterized family of polynomials and so jAgt

is a smooth curve with jAgt = c(t). _

Proof of proposition 40.3. Given a natural transformation _: TA ! F, the value

_Rn(jAidRn) is an A-admissible element in F0Rn. On the other hand, every

A-admissible element v 2 SA(F) determines the maps _v

Rk : TARk ! FRk,

_v

Rk (jAf) = Ff(v) and all these maps are smooth. By the de_nition, _v

Rn obey

the necessary commutativity relations and so they determine the unique natural

transformation _v : TA ! F with _v

Rn(v) = v. _

40.6. Let us apply proposition 40.3 to the case F = T(r), the r-th order tangent

functor. The elements in the standard _ber of T(r)Rn are the linear forms on

the vector space Jr

0 (Rn;R)0 and for every Weil algebra A of width n one veri_es

easily that such a form ! lies in SA(T(r)) if and only if !(jr

0g) = 0 for all g with

jAg = jA0.

As a simple illustration, we _nd all natural transformations Tq

1

! T(r). Every

element jr

0f 2 Tr_

0 R = Jr

0 (R;R)0 has the canonical representative f(x) = a1x +

a2x2+_ _ _+arxr. Let us de_ne 1-forms vi 2 T(r)

0 R by vi(jr

0f) = ai, i = 1; 2; : : : ; r.

Since jD

q

1f = jq

0f, the forms vi are Dq

1-admissible if and only if i _ q. So

the linear space of all natural transformations Tq

1

! T(r) is generated by the

linearly independent transformations _vi , i = 1; : : : ; minfq; rg. The maps _viM

can be described as follows. Every jq

0g 2 Tq

1M determines a curve g : R ! M

through x = g(0) up to the order q and given any jrx

f 2 Jr

x(M;R)0 the value

_viM (jq

0g)(jrx

f) is obtained by the evaluation of the i-th order term in f _g : R ! R

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

40. Natural transformations 343

at 0 2 R. So _viM (jq

0g) might be viewed as the i-th derivative on Jr

x(M;R)0 in

the direction jq

0g.

In general, given any vector bundle functor F on Mf, the natural transformations

TA ! F carry a vector space structure and the corresponding set SA(F)

is a linear subspace in F0Rn. In particular, the space of all natural transformations

TA ! F is a _nite dimensional vector space with dimension bounded by

the dimension of the standard _ber F0Rn.

As an example let us consider the two natural vector bundle structures given

by _TM : TTM ! TM and T_M : TTM ! TM which form linearly independent

natural transformations TT ! T. For dimension reasons these must form

a basis of the linear space of all natural transformations TT ! T. Analogously

the products T_M ^ _TM : TTM ! _2TM generate the one-dimensional space

of all natural transformations TT ! _2T and there are no non-zero natural

transformations TT ! _pT for p > 2.

40.7. Remark. [Mikulski, to appear a] also determined the natural operators

transforming functions on a manifold M of dimension at least two into functions

on FM for every bundle functor F : Mf ! FM. All of them have the form

f 7! h _ Ff, f 2 C1(M;R), where h is any smooth function h: FR ! R.

40.8. Natural transformations T(r) ! T(r). Now we are going to show that

there are no other natural transformations T(r) ! T(r) beside the real multiples

of the identity. Thus, in this direction the properties of T(r) are quite di_erent

from the higher order product preserving functors where the corresponding Weil

algebras have many endomorphisms as a rule. Let us remark that from the

technical point of view we shall prove the proposition in all dimensions separately

and only then we `join' all these partial results together.

Proposition. All natural transformations T(r) ! T(r) form the one-parameter

family

X 7! kX; k 2 R:

Proof. If xi are local coordinates on a manifold M, then the induced _ber coordinates

ui, ui1i2 ; : : : ; ui1:::ir (symmetric in all indices) on Tr_

1 M correspond

to the polynomial representant uixi + ui1i2xi1xi2 + _ _ _ + ui1:::irxi1 : : : xir of a

jet from Tr_

1 M. A linear functional on (Tr_

1 M)x with the _ber coordinates Xi,

Xi1i2 ; : : : ;Xi1:::ir (symmetric in all indices) has the form

(1) Xiui + Xi1i2ui1i2 + _ _ _ + Xi1:::irui1:::ir :

Let yp be some local coordinates on N, let Y p, Y p1p2 ; : : : ; Y p1:::pr be the induced

_ber coordinates on T(r)N and yp = fp(xi) be the coordinate expression of a

map f : M ! N. If we evaluate the jet composition from the de_nition of the

action of the higher order tangent bundles on morphisms, we deduce by (1) the

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

344 Chapter IX. Bundle functors on manifolds

coordinate expression of T(r)f

(2)

Y p = @fp

@xi Xi +

1

2!

@2fp

@xi1@xi2

Xi1i2 + _ _ _ +

1

r!

@rfp

@xi1 : : : @xir

Xi1:::ir

...

Y p1:::ps = @fp1

@xi1

: : :

@fps

@xis

Xi1:::is + : : :

...

Y p1:::pr = @fp1

@xi1

: : :

@fpr

@xir

Xi1:::ir

where the dots in the middle row denote a polynomial expression, each term of

which contains at least one partial derivative of fp of order at least two.

Consider _rst T(r) as a bundle functor on the subcategory Mfm _ Mf.

According to (2), its standard _ber S = T(r)

0 Rm is a Gr

m-space with the following

action

(3)

_X

i = ai

jXj + ai

j1j2Xj1j2 + _ _ _ + ai

j1:::jrXj1:::jr

...

_X

i1:::is = ai1

j1 : : : ais

js

Xj1:::js + : : :

...

_X

i1:::ir = ai1

j1 : : : air

jr

Xj1:::jr

where the dots in the middle row denote a polynomial expression, each term of

which contains at least one of the quantities ai

j1j2 ; : : : ; ai

j1:::jr . Write

(Xi;Xi1i2 ; : : : ;Xi1:::ir ) = (X1;X2; : : : ;Xr):

By the general theory, the natural transformations T(r) ! T(r) correspond

to Gr

m-equivariant maps f = (f1; f2; : : : ; fr) : S ! S. Consider _rst the equivariance

with respect to the homotheties in GL(m) _ Gr

m. Using (3) we obtain

(4)

kf1(X1; : : : ;Xs; : : : ;Xr) = f1(kX1; : : : ; ksXs; : : : ; krXr)

...

ksfs(X1; : : : ;Xs; : : : ;Xr) = fs(kX1; : : : ; ksXs; : : : ; krXr)

...

krfr(X1; : : : ;Xs; : : : ;Xr) = fr(kX1; : : : ; ksXs; : : : ; krXr):

By the homogeneous function theorem (see 24.1), f1 is linear in X1 and independent

of X2; : : : ;Xr, while fs = gs(Xs) + hs(X1; : : : ;Xs􀀀1), where gs is

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

41. Star bundle functors 345

linear in Xs and hs is a polynomial in X1; : : : ;Xs􀀀1, 2 _ s _ r. Further,

the equivariancy of f with respect to the whole subgroup GL(m) implies that

gs is a GL(m)-equivariant map of the s-th symmetric tensor power SsRm into

itself. By the invariant tensor theorem (see 24.4), gs = csXs (or explicitly,

gi1:::is = csXi1:::is ) with cs 2 R.

Now let us use the equivariance with respect to the kernel Br

1 of the jet

projection Gr

m

! GL(m), i.e. ai

j = _ij

. The _rst line of (3) implies

(5) c1Xi + ai

j1j2 (c2Xj1j2 + hj1j2 (X1))+

+ _ _ _ + ai

j1:::jr (crXj1:::jr + hj1:::jr (X1; : : : ;Xr􀀀1)) =

= c1(Xi + ai

j1j2Xj1j2 + _ _ _ + ai

j1:::jrXj1:::jr ):

Setting ai

j1:::js = 0 for all s > 2, we _nd c2 = c1 and hj1j2 (X1) = 0. By a

recurrence procedure of similar type we further deduce

cs = c1; hj1:::js (X1; : : : ;Xs􀀀1) = 0

for all s = 3; : : : ; r.

This implies that the restriction of every natural transformation T(r) ! T(r)

to each subcategory Mfm is a homothety with a coe_cient km. Taking into

account the injection R ! Rm, x 7! (x; 0; : : : ; 0) we _nd km = k1. _

40.9. Remark. We remark that all natural tensors of type

􀀀1

1

_

on both T(r)M

and the so-called extended r-th order tangent bundle (Jr(M;R))_ are determined

in [Gancarzewicz, Kol_a_r, to appear].

41. Star bundle functors

The tangent functor T is a covariant functor on the category Mf, but its

dual T_ can be interpreted as a covariant functor on the subcategory Mfm of

local di_eomorphisms of m-manifolds only. In this section we explain how to

treat functors with a similar kind of contravariant character like T_ on the whole

category Mf.

41.1. The category of star bundles. Consider a _bered manifold Y ! M

and a smooth map f : N ! M. Let us recall that the induced _bered manifold

f_Y ! N is given by the pullback

f_Y w

fY

u

Y

u

N w

f

M

The restrictions of the _bered morphism fY to individual _bers are di_eomorphisms

and we can write

f_Y = f(x; y); x 2 N; y 2 Yf(x)

g; fY (x; y) = y.

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

346 Chapter IX. Bundle functors on manifolds

Clearly (f _ g)_Y _= g_(f_Y ). Let us consider another _bered manifold Y 0 ! M

over the same base, and a base-preserving _bered morphism ': Y ! Y 0. Given

a smooth map f : N ! M, by the pullback property there is a unique _bered

morphism f_': f_Y ! f_Y 0 such that

(1) fY 0 _ f_' = ' _ fY .

The pullbacks appear in many well known constructions in di_erential geometry.

For example, given manifolds M, N and a smooth map f : M ! N, the

cotangent mapping T_f transforms every form ! 2 T_

f(x)N into T_f! 2 T_

xM.

Hence the mapping f_(T_N) ! T_M is a morphism over the identity on M.

We know that the restriction of T_ to manifolds of any _xed dimension and local

di_eomorphisms is a bundle functor on Mfm, see 14.9, and it seems that the

construction could be functorial on the whole category Mf as well. However

the codomain of T_ cannot be the category FM.

De_nition. The category FM_ of star bundles is de_ned as follows. The objects

coincide with those of FM, but morphisms ': (Y ! M) ! (Y 0 ! M0)

are couples ('0; '1) where '0 : M ! M0 is a smooth map and '1 : ('0)_Y 0 ! Y

is a _bered morphism over idM. The composition of morphisms is given by

(2) ( 0; 1) _ ('0; '1) = ( 0 _ '0; '1 _ (('0)_ 1)).

Using the formulas (1) and (2) one veri_es easily that this is a correct de_nition

of a category. The base functor B: FM_ !Mf is de_ned by B(Y ! M) = M,

B('0; '1) = '0.

41.2. Star bundle functors. A star bundle functor on Mf is a covariant

functor F : Mf ! FM_ satisfying

(i) B _ F = IdMf , so that the bundle projections determine a natural transformation

p: F ! IdMf .

(ii) If i : U ! M is an inclusion of an open submanifold, then FU = p􀀀1

M (U)

and Fi = (i; '1) where '1 : i_(FM) ! FU is the canonical identi_cation

i_(FM) _= p􀀀1

M (U) _ FM.

(iii) Every smoothly parameterized family of mappings is transformed into a

smoothly parameterized one.

Given a smooth map f : M ! N we shall often use the same notation Ff for

the second component ' in Ff = (f; '). We can also view the star bundle functors

as rules transforming any manifoldM into a _ber bundle pM : FM ! M and

any smooth map f : M ! N into a base-preserving morphism Ff : f_(FN) !

FM with F(idM) = idFM and F(g _ f) = Ff _ f_(Fg).

41.3. The associated maps. A star bundle functor F is said to be of order r

if for every maps f, g : M ! N and every point x 2 M, the equality jrx

f = jrx

g

implies Ffj(f_(FN))x = Fgj(g_(FN))x, where we identify the _bers (f_(FN))x

and (g_(FN))x.

Let us consider an r-th order star bundle functor F : Mf ! FM_. For every

r-jet A = jrx

f 2 Jr

x(M;N)y we de_ne a map FA: FyN ! FxM by

(1) FA = Ff _ (fFNj(f_(FN))x)􀀀1,

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

41. Star bundle functors 347

where fFN : f_(FN) ! FN is the canonical map. Given another r-jet B =

jr

yg 2 Jr

y (N; P)z, we have

F(B _ A) = Ff _ (f_(Fg)) _ (fg_(FP)

j(f_g_(FP))x)􀀀1 _ (gFP j(g_(FP))y)􀀀1:

Applying 41.1.(1) to individual _bers, we get

(fFNj(f_(FN))x)􀀀1 _ Fg = f_(Fg) _ (fg_(FP)

j(f_g_(FP))x)􀀀1

and that is why

F(B _ A) = Ff _ (fFNj(f_FN)x)􀀀(2) 1 _ Fg _ (gFP j(g_FP)y)􀀀1

= FA _ FB:

For any two manifolds M, N we de_ne

(3) FM;N : FN _N Jr(M;N) ! FM; (q;A) 7! FA(q):

These maps are called the associated maps to F.

Proposition. The associated maps to any _nite order star bundle functor are

smooth.

Proof. This follows from the regularity and locality conditions in the way shown

in the proof of 14.4. _

41.4. Description of _nite order star bundle functors. Let us consider

an r-th order star bundle functor F. We denote (Lr)op the dual category to

Lr, Sm = F0Rm, m 2 N0, and we call the system S = fS0; S1; : : : g the system

of standard _bers of F, cf. 14.21. The restrictions `m;n : Sn _ Lr

m;n

! Sm,

`m;n(s;A) = FA(s), of the associated maps 41.3.(3) form the induced action of

(Lr)op on S. Indeed, given another jet B 2 Lr(n; p) equality 41.3.(2) implies

`m;p(s;B _ A) = `m;n(`n;p(s;B);A):

On the other hand, let ` be an action of (Lr)op on a system S = fS0; S1; : : : g

of smooth manifolds and denote `m the left actions of Gr

m on Sm given by

`m(A; s) = `m;m(s;A􀀀1). We shall construct a star bundle functor L from these

data. We put LM := PrM[Sm; `m] for all manifolds M and similarly to 14.22 we

also get the action on morphisms. Given a map f : M ! N, x 2 M, f(x) = y,

we de_ne a map FA: FyN ! FxM,

FA(fv; sg) = fu; `m;n(s; v􀀀1 _ A _ u)g;

where m = dimM, n = dimN, A = jrx

f, v 2 Pr

yN, s 2 Sn, and u 2 Pr

xM is

chosen arbitrarily. The veri_cation that this is a correct de_nition of smooth

maps satisfying F(B _A) = FA_FB is quite analogous to the considerations in

14.22 and is left to the reader. Now, we de_ne Lfj(f_(FN))x = FA _ fFN and

it follows directly from 41.1.(1) that L(g _ f) = Lf _ f_(Lg).

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

348 Chapter IX. Bundle functors on manifolds

Theorem. There is a bijective correspondence between the set of all r-th order

star bundle functors on Mf and the set of all smooth actions of the category

(Lr)op on systems S of smooth manifolds.

Proof. In the formulation of the theorem we identify naturally equivalent functors.

Given an r-th order star bundle functor F, we have the induced action `

of (Lr)op on the system of standard _bers. So we can construct the functor L.

Analogously to 14.22, the associated maps de_ne a natural equivalence between

F and L. _

41.5 Remark. We clari_ed in 14.24 that the actions of the category Lr on

systems of manifolds are in fact covariant functors Lr !Mf. In the same way,

actions of (Lr)op correspond to covariant functors (Lr)op !Mf or, equivalently,

to contravariant functors Lr !Mf, which will also be denoted by Finf. Hence

we can summarize: r-th order bundle functors correspond to covariant smooth

functors Lr ! Mf while r-th order star bundle functors to the contravariant

ones.

41.6. Example. Consider a manifold Q and a point q 2 Q. To any manifold M

we associate the _bered manifold FM = Jr(M;Q)q

_ ��! M and a map f : N ! M

is transformed into a map Ff : f_(FM) ! FN de_ned as follows. Given a point

b 2 f_(Jr(M;Q)q), b = (x; jr

f(x)g), we set Ff(b) = jrx

(g _ f) 2 Jr(N;Q)q. One

veri_es easily that F is a star bundle functor of order r. Let us mention the

corresponding contravariant functor Lr !Mf. We have Finf (m) = Jr

0 (Rm;Q)q

and for arbitrary jets jr

0f 2 Lr

m;n, jr

0g 2 Finf (m) it holds Finf (jr

0f)(jr

0g) =

jr

0 (g _ f).

41.7. Vector bundle functors and vector star bundle functors. Let F

be a bundle functor or a star bundle functor on Mf. By the de_nition of the

induced action and by the construction of the (covariant or contravariant) functor

Finf : Lr ! Mf, the values of the functor F belong to the subcategory of

vector bundles if and only if the functor Finf takes values in the category Vect of

_nite dimensional vector spaces and linear mappings. But using the construction

of dual objects and morphisms in the category Vect, we get a duality between

covariant and contravariant functors Finf : Lr ! Vect. The corresponding duality

between vector bundle functors and vector star bundle functors is a source

of interesting geometric objects like r-th order tangent vectors, see 12.14 and

below.

41.8. Examples. Let us continue in example 41.6. If the manifold Q happens

to be a vector space and the point q its origin, we clearly get a vector star bundle

functor. Taking Q = R we get the r-th order cotangent functor Tr_. If we set

Q = Rk, then the corresponding star bundle functor is the functor Tr_

k of the

(k; r)-covelocities, cf. 12.14.

The dual vector bundle functor to Tr_ is the r-th order tangent functor. The

dual functor to the (k; r)-covelocities is the functor Tr_

k , see 12.14.

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

Remarks 349

Remarks

Most of the exposition concerning the bundle functors on Mf is based on

[Kol_a_r, Slov_ak, 89], but the prolongation of Lie groups was described in [Kol_a_r,

83]. The generalization to bundle functors on FMm follows [Slov_ak, 91].

The existence of the canonical ow-natural transformation FT ! TF was

_rst deduced by A. Kock in the framework of the so called synthetic di_erential

geometry, see e.g. [Kock, 81]. His unpublished note originated in a discussion

with the _rst author. Then the latter developed, with consent of the former, the

proof of that result dealing with classical manifolds only.

The description of all natural transformations with the source in aWeil bundle

by means of some special elements in the standard _ber is a generalization of

an idea from [Kol_a_r, 86] due to [Mikulski, 89 b]. The natural transformations

T(r) ! T(r) were _rst classi_ed in [Kol_a_r, Vosmansk_a, 89].

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

350

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