˜ k˜ k˜

F g : Q — F A ’ F B satisfy jq F f ( , z) = jq F g( , z) for all z ∈ F A.

1˜

39.8. Let us de¬ne ιRm+n (z) = j0 F µRm+n ( , z). If we repeat the considerations

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

1

ι : F V ’ T F . But its values satisfy T pY —¦ ιY (z) = j0 pY —¦ F (µY )t (z) = pY (z) ∈

V Y and so ιY (z) ∈ V (F Y ’ BY ). So ι : F V ’ 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 trans-

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

stant vertical vector ¬elds X on Rm+n we have FlX = (µRm+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 ι : F V ’ V F . If F has the point property, then ι

is a natural equivalence if and only if F preserves ¬bered products.

We have to point out that we consider the ¬bered manifold structure F Y ’

BY for every object Y ’ BY ∈ ObFMm , i.e. ιY : F (V Y ’ BY ) ’ V (F Y ’

BY ).

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

If ι 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

F Rm+n = VA Rm+n for a Weil algebra A and then also F f coincides with VA f for

morphisms of the form idRm —g : Rm+n ’ Rm+k , see 38.19. But each µRm+n (t, )

is of this form and any restriction of ιRm+n to a ¬ber (VA V Rm+n )x ∼ TA T Rm

=

coincides with the canonical ¬‚ow natural equivalence TA T ’ T TA , cf. 39.2.

Hence ι 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 = J r we obtain the well known canonical natural

equivalence J r V ’ V J r , cf. [Goldschmidt, Sternberg, 73], [Mangiarotti, Mod-

ugno, 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

J r : FMm ’ FM.

Since J r (Rm+n ’ Rm ) is a sub bundle in the bundle Km Rm+n of contact

r

elements of order r formed by the elements transversal to the ¬bration, the

action of J r f on a jet jx s extends to all local di¬eomorphisms transforming jx s

r r

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 J r , but one veri¬es easily that

lemma 39.3 remains still valid.

So let us de¬ne µt : T Rm+n ’ Rm+n by µt (x, z, X, Z) = (x + tX, z + tZ).

For every section (x, z(x), X(x), Z(x)) of T Rm+n ’ Rm , its composition with

µt and the ¬rst projection gives the map x ’ x + tX(x). If we proceed in a

similar way as above, we deduce

Proposition. There is a canonical ¬‚ow-natural transformation ι : J r T ’ T J r

and its restriction J r V ’ V J r is the canonical ¬‚ow-natural equivalence.

39.10. Remark. Let us notice that ι : T J r ’ J r T 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 ιRm+n at least in the

β

±

case r = 1. Let us write a map f : (Rm+n ’ Rm ) ’ (Rm+n ’ Rm ) in the

’ ’

form z k = f k (xi , y p ), wq = f q (xi , y p ). In order to get the action of J 1 f in the

p ˜

extended sense on j0 s = (y p , yi ) we have to consider the map (β —¦ f —¦ s)’1 = f ,

1

˜ ˜ ˜ ˜

xi = f i (z). So z k = f k (f (z), y p (f (z))) and we evaluate that the matrix ‚ f i /‚z k

p

is the inverse matrix to ‚f k /‚xi + (‚f k /‚y p )yi (the invertibility of this matrix

is exactly the condition on j0 s to lie in the domain of J 1 f ). Now the coordinates

1

q

wk of J 1 f (j0 s) are

1

˜ ˜

‚f q ‚ f j ‚f q p ‚ f j

q

wk = + p yj q .

‚xj ‚z k ‚y ‚z

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

coordinates yi or X i , Y p or yi , Xj , Yip or yi , ξ i , · p , ·i on J 1 Y or T Y or J 1 T Y

p p p p

i

or T J 1 Y , respectively. If jx s = (xi , y p , X j , Y q , yk , Xm , Yn ), then

1 r s

¯q

J 1 (µY )t (j0 s) = (xi + tX i , y p + tY p , yj (t))

1

yi (t)(δj + tXj ) = (yi + tYip )δj .

¯p p

i i i

Di¬erentiating by t at 0 we get

ιRm+n (xi , y p , X j , Y q , yk , Xm , Yn ) = (xi , y p , yk , X j , Y q , Y s ’ ym X m ).

r s r s

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 TA M is an equivalence class of mappings in C ∞ (Rn , M ) where n

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

a natural transformation ± : C ∞ (Rn , ) ’ TA . Hence given any bundle functor

F on Mf , every natural transformation χ : TA ’ F gives rise to the natural

transformation χ —¦ ± : C ∞ (Rn , ) ’ F and this is determined by the value of

(χ —¦ ±)Rn (idRn ). So in order to classify all natural transformations χ : TA ’ F

we have to distinguish the possible values v := χRn —¦ ±Rn (idRn ) ∈ F Rn . Let

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

Mf all maps χM are ¬bered maps over idM , see 14.11. Hence v ∈ F0 Rn and

another obvious condition is F f (v) = F g(v) for all maps f , g : Rn ’ M with

jA f = jA g. On the other hand, having chosen such v ∈ F0 Rn , we can de¬ne

χv (jA f ) = F f (v) and if all these maps are smooth, then they form a natural

M

transformation χv : 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 F f (v) = F i(v) for all maps f : Rn ’ Rn+1 with

jA f = jA i where i : Rn ’ Rn+1 , x ’ (x, 0).

40.2. De¬nition. For every Weil algebra A of width n and for every bun-

dle functor F on Mf , an element v ∈ F0 Rn is called A-admissible if jA f =

jA i implies F f (v) = F i(v) for all f ∈ C ∞ (Rn , Rn+1 ). We denote by SA (F ) ‚

S = F0 Rn 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

χ ’ χRn (jA idRn )

is a bijection between the natural transformations χ : TA ’ F and the subset of

A-admissible elements SA (F ) ‚ F0 Rn .

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

de¬nes the transformation χv : TA ’ F at the level of sets, cf. 40.1, and then

we have to verify that all maps χv are smooth.

M

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

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

M the equality jA f = jA g implies F f (v) = F g(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 on Mf ,

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 C ∞ (Rn , Rk ) determined by A is compat-

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

r

22.3.(e) with m = 0, j0 replaced by jA and Sn replaced by SA (F ).

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

r

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

r

n-dimensional velocities of order rn+1 . Then jA f = jA g if and only if j0 n+1 f =

r

j0 n+1 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 ’ TA Rk there is a smoothly parameterized family of maps γ : R—Rn ’ Rk

such that jA γt = c(t).

∞

Proof. There is an ideal A in the algebra of germs En = C0 (Rn , R), cf. 35.5,

such that A = En /A. Write Dn = Mr+1 where M is the maximal ideal in

r

En , and Dr = En /Dn , i.e. TDr = Tn . Then A ⊃ Dn for suitable r and so we

r r r

n n

get the linear projection Dr ’ A, j0 f ’ jA f . Let us choose a smooth section

r

n

s : A ’ Dr of this projection. Now, given a curve c(t) = jA ft in TA Rk there

n

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

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

is a smooth curve with jA gt = c(t).

Proof of proposition 40.3. Given a natural transformation χ : TA ’ F , the value

χRn (jA idRn ) is an A-admissible element in F0 Rn . On the other hand, every

A-admissible element v ∈ SA (F ) determines the maps χv k : TA Rk ’ F Rk ,

R

χv k (jA f ) = F f (v) and all these maps are smooth. By the de¬nition, χv n obey

R

R

the necessary commutativity relations and so they determine the unique natural

transformation χv : TA ’ F with χv n (v) = v.

R

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 J0 (Rn , R)0 and for every Weil algebra A of width n one veri¬es

r

easily that such a form ω lies in SA (T (r) ) if and only if ω(j0 g) = 0 for all g with

r

jA g = jA 0.

q

As a simple illustration, we ¬nd all natural transformations T1 ’ T (r) . Every

r r— r

element j0 f ∈ T0 R = J0 (R, R)0 has the canonical representative f (x) = a1 x +

(r)

a2 x2 +· · ·+ar xr . Let us de¬ne 1-forms vi ∈ T0 R by vi (j0 f ) = ai , i = 1, 2, . . . , r.

r

q q

Since jDq f = j0 f , the forms vi are D1 -admissible if and only if i ¤ q. So

1

q

the linear space of all natural transformations T1 ’ T (r) is generated by the

linearly independent transformations χvi , i = 1, . . . , min{q, r}. The maps χvi M

q q

can be described as follows. Every j0 g ∈ T1 M determines a curve g : R ’ M

r r

through x = g(0) up to the order q and given any jx f ∈ Jx (M, R)0 the value

q

χvi (j0 g)(jx f ) is obtained by the evaluation of the i-th order term in f —¦g : R ’ R

r

M

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

40. Natural transformations 343

q

at 0 ∈ R. So χvi (j0 g) might be viewed as the i-th derivative on Jx (M, R)0 in

r

M

q

the direction j0 g.

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

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

is a linear subspace in F0 Rn . In particular, the space of all natural transforma-

tions TA ’ F is a ¬nite dimensional vector space with dimension bounded by

the dimension of the standard ¬ber F0 Rn .

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

by πT M : T T M ’ T M and T πM : T T M ’ T M which form linearly indepen-

dent natural transformations T T ’ T . For dimension reasons these must form

a basis of the linear space of all natural transformations T T ’ T . Analogously

the products T πM § πT M : T T M ’ Λ2 T M generate the one-dimensional space

of all natural transformations T T ’ Λ2 T and there are no non-zero natural

transformations T T ’ Λp T 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 F M for every bundle functor F : Mf ’ FM. All of them have the form

f ’ h —¦ F f , f ∈ C ∞ (M, R), where h is any smooth function h : F R ’ 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 ’ kX, k ∈ R.

Proof. If xi are local coordinates on a manifold M , then the induced ¬ber co-

r—

ordinates ui , ui1 i2 , . . . , ui1 ...ir (symmetric in all indices) on T1 M correspond

to the polynomial representant ui xi + ui1 i2 xi1 xi2 + · · · + ui1 ...ir xi1 . . . xir of a

jet from T1 M . A linear functional on (T1 M )x with the ¬ber coordinates X i ,

r— r—

X i1 i2 , . . . , X i1 ...ir (symmetric in all indices) has the form

X i ui + X i1 i2 ui1 i2 + · · · + X i1 ...ir ui1 ...ir .

(1)

Let y p be some local coordinates on N , let Y p , Y p1 p2 , . . . , Y p1 ...pr be the induced

¬ber coordinates on T (r) N and y p = f p (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

‚f p i 1 ‚2f p ‚rf p

1

p i1 i2

X i1 ...ir

+ ··· +

Y= X+ X

i i1 ‚xi2 i1 . . . ‚xir

‚x 2! ‚x r! ‚x

.

.

.

‚f p1 ‚f ps i1 ...is

Y p1 ...ps =

(2) ... X + ...

‚xi1 ‚xis

.

.

.

‚f p1 ‚f pr i1 ...ir

p1 ...pr

Y = ... X

‚xi1 ‚xir

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

which contains at least one partial derivative of f p of order at least two.

Consider ¬rst T (r) as a bundle functor on the subcategory Mfm ‚ Mf .

(r)

According to (2), its standard ¬ber S = T0 Rm is a Gr -space with the following

m

action

¯

X i = ai X j + ai 1 j2 X j1 j2 + · · · + ai 1 ...jr X j1 ...jr

j j j

.

.

.

¯

X i1 ...is = ai1 . . . ais X j1 ...js + . . .

(3) j1 js

.

.

.

¯

X i1 ...ir = ai1 . . . air X j1 ...jr

j1 jr

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

which contains at least one of the quantities ai 1 j2 , . . . , ai 1 ...jr . Write

j j

(X i , X i1 i2 , . . . , X i1 ...ir ) = (X1 , X2 , . . . , Xr ).

By the general theory, the natural transformations T (r) ’ T (r) correspond

to Gr -equivariant maps f = (f1 , f2 , . . . , fr ) : S ’ S. Consider ¬rst the equiv-

m

ariance with respect to the homotheties in GL(m) ‚ Gr . Using (3) we obtain

m

kf1 (X1 , . . . , Xs , . . . , Xr ) = f1 (kX1 , . . . , k s Xs , . . . , k r Xr )

.

.

.

k s fs (X1 , . . . , Xs , . . . , Xr ) = fs (kX1 , . . . , k s Xs , . . . , k r Xr )

(4)

.

.

.

k r fr (X1 , . . . , Xs , . . . , Xr ) = fr (kX1 , . . . , k s Xs , . . . , k r Xr ).

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

dependent 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 S s Rm into

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

g i1 ...is = cs X i1 ...is ) with cs ∈ R.

r

Now let us use the equivariance with respect to the kernel B1 of the jet

projection Gr ’ GL(m), i.e. ai = δj . The ¬rst line of (3) implies

i

m j

(5) c1 X i + ai 1 j2 (c2 X j1 j2 + hj1 j2 (X1 ))+

j

+ · · · + ai 1 ...jr (cr X j1 ...jr + hj1 ...jr (X1 , . . . , Xr’1 )) =

j

= c1 (X i + ai 1 j2 X j1 j2 + · · · + ai 1 ...jr X j1 ...jr ).

j j

Setting ai 1 ...js = 0 for all s > 2, we ¬nd c2 = c1 and hj1 j2 (X1 ) = 0. By a

j