to the subgroup ai = δj yields

i

j

f i (X j , Xlk , um , unp ) = f i (X j , Xlk + ak X q , um + am urs , unp ).

lq rs

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

43. The case of the second order tangent vectors 359

This gives f i = f i (X j , ukl ). Using the homotheties in i(G1 ) ‚ G3 , we obtain

m m

f i = f i (X j ). Example 24.14 then implies

Y i = kX i .

(7)

Consider further the di¬erence A ’ kT (2) with k taken from (7) and denote

by hi , hij its components. We evaluate easily

ai aj hkl (X m , Xp , Xrs , ut , uuv ) = hij (X m , Xp , Xrs , ut , uuv ).

¯ ¯n ¯q ¯ ¯

n q

(8) kl

Quite similarly as in the ¬rst step we deduce hij = hij (X k , ulm ). By homogene-

ity and the invariant tensor theorem, we then obtain

hij = cuij + aX i X j .

(9)

For hi , we ¬nd

(10) ai hj (X k , Xm , Xpq , ur , ust ) + cai ujk + aai X j X k =

l n

j jk jk

¯ ¯l ¯n ¯ ¯

= hi (X k , Xm , Xpq , ur , ust ).

By (3), hi is independent of Xjk . Then the homogeneity condition implies

i

hi = fj (Xlk )X j + gj (Xlk )uj .

i i

(11)

For X i = 0, the equivariance of (11) with respect to the subgroup ai = δj reads

i

j

gj (Xlk )uj + cai ujk = gj (Xlk )(uj + aj ukl ).

i i

(12) jk kl

Hence gj (Xlk ) = cδj . The remaining equivariance condition is

i i

fj (Xlk )X j + aai X j X k = fj (Xlk + ak X m )X j .

i i

(13) jk lm

This implies that all the ¬rst order partial derivatives of fj (Xlk ) are constant, so

i

that fj are at most linear in Xlk . By the invariant tensor theorem, fj (Xlk )X j =

i i

eX j Xj + bX i . Then (13) yields e = a, i.e.

i

hi = cui + bX i + aX j Xj .

i

(14)

This gives the coordinate expression of (1).

43.3. Remark. For a Weil functor TB , all natural operators T T TB are of

the form H —¦ TB , where H is a natural transformation T TB ’ T TB over the

identity of TB . For T (2) , one evaluates easily that all natural transformations

H : T T (2) ’ T T (2) over the identity of T (2) form the following 3-parameter

family

Y i =k1 Y i ,

U i =k1 U i + k2 Y i + k3 ui ,

U ij =k1 U ij + k3 uij ,

see [Doupovec, 90]. Hence the operators of the form H —¦T (2) form a 3-parameter

family only, in which the operator D is not included.

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360 Chapter X. Prolongation of vector ¬elds and connections

43.4. Remark. In the case of Weil bundles, theorem 42.12 implies that the

di¬erence between a natural operator T T TB and its associated absolute

operator is a linear operator. This is no more true for the non-product-preserving

functors, where the operator D is the simplest counter-example.

43.5. Remark. The operators T (2) , V and L transform every vector ¬eld on a

manifold M into a vector ¬eld on T (2) M tangent to the subbundle T M ‚ T (2) M ,

but D does not. With a little surprise we can express it by saying that the

T T (2) is not compatible with the natural inclusion

natural operator D : T

T M ‚ T (2) M .

43.6. Remark. Recently [Mikulski, to appear b], has solved the general prob-

T T (r) , r ∈ N. All such operators

lem of determining all natural operators T

form an (r + 2)-parameter family linearly generated by the ¬‚ow operator, by the

Liouville vector ¬eld of T (r) and by the analogies of the operator D from 43.1

de¬ned by f ’ X · · · X f , k = 1, . . . , r.

k-times

44. Induced vector ¬elds on jet bundles

44.1. Let F be a bundle functor on FMm,n . The idea of the ¬‚ow prolongation

of vector ¬elds can be applied to the projectable vector ¬elds on every object

p : Y ’ M of FMm,n . The ¬‚ow Fl· of a projectable vector ¬eld · on Y is

t

formed by the local isomorphisms of Y and we de¬ne the ¬‚ow operator F of F

by

FY · = ‚t 0 F (Fl· ).

‚

t

The general concept of a natural operator A transforming every projectable

vector ¬eld on Y ∈ ObFMm,n into a vector ¬eld on F Y was introduced in

section 18. We shall denote such an operator brie¬‚y by A : Tproj TF.

44.2. Lemma. If F is an r-th order bundle functor on FMm,n , then the order

T F is ¤ r.

of every natural operator Tproj

Proof. This is quite similar to 42.5, see [Kol´ˇ, Slov´k, 90] for the details.

ar a

44.3. We shall discuss the case F is the functor J r of the r-th jet prolongation

of ¬bered manifolds. We remark that a simple evaluation leads to the following

coordinate formula for J 1 ·

‚· p ‚· p q ‚· j p

J 1 · = · i ‚xi + · p ‚yp +

‚ ‚ ‚

’

+ ‚y q yi ‚xi yj p

‚xi ‚yi

‚ ‚

provided · = · i (x) ‚xi + · p (x, y) ‚yp , see [Krupka, 84]. To evaluate J r ·, we

have to iterate this formula and use the canonical inclusion J r (Y ’ M ) ’

J 1 (J r’1 (Y ’ M ) ’ M ).

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

44. Induced vector ¬elds on jet bundles 361

T J r is a constant multiple of

Proposition. Every natural operator A : Tproj

the ¬‚ow operator J r .

Proof. Let V r be the space of all r-jets of the projectable vector ¬elds on

Rn+m ’ Rm with source 0 ∈ Rm+n , let V 0 ‚ V r be the space of all r-jets

of the constant vector ¬elds and V0 ‚ V 0 be the subset of all vector ¬elds with

zero component in Rn . Further, let S r or Z r be the ¬ber of J r (Rm+n ’ Rm )

or T J r (Rm+n ’ Rm ) over 0 ∈ Rm+n , respectively. By lemma 44.2 and by the

general theory, we have to determine all Gr+1 -maps A : V r — S r ’ Z r over the

m,n

r

identity of S . Analogously to section 42, every projectable vector ¬eld on Y

with non-zero projection to the base manifold can locally be transformed into

‚

the vector ¬eld ‚x1 . Hence A is determined by its restriction A0 to V0 — S r .

However, in the ¬rst part of the proof we have to consider the restriction A0 of

A to V 0 — S r for technical reasons.

Having the canonical coordinates xi and y p on Rm+n , let X i , Y p be the

induced coordinates on V 0 , let y± , 1 ¤ |±| ¤ r, be the induced coordinates on

p

S r and Z i = dxi , Z p = dy p , Z± = dy± be the additional coordinates on Z r . The

p p

restriction A0 is given by some functions

Z i = f i (X j , Y q , yβ )

s

Z p = f p (X i , Y q , yβ )

s

Z± = f± (X i , Y q , yβ ).

p p s

Let us denote by g i , g p , g± the restrictions of the corresponding f ™s to V0 — S r .

p

The ¬‚ows of constant vector ¬elds are formed by translations, so that their r-jet

prolongations are the induced translations of J r (Rm+n ’ Rm ) identical on the

‚ ‚

standard ¬ber. Therefore J r ‚x1 = ‚x1 and it su¬ces to prove

g i = kX i , g p = 0, p

g± = 0.

We shall proceed by induction on the order r. It is easy to see that the ac-

tion of i(G1 — G1 ) ‚ Gr+1 on all quantities is tensorial. Consider the case

m n m,n

r = 1. Using the equivariance with respect to the homotheties in i(G1 ), we n

q q

obtain f i (X j , Y p , yl ) = f i (X j , kY p , kyl ), so that f i depends on X i only. Then

the equivariance of f i with respect to i(G1 ) yields f i = kX i by 24.7. The equiv-

m

ariance of f with respect to the homotheties in i(G1 ) gives kf p (X i , Y q , yj ) =

p s

n

q

f p (X i , kY q , kyj ). This kind of homogeneity implies f p = hp (X i )Y q + hpj (X i )yj

s

q q

with some smooth functions hp , hpj . Using the homotheties in i(G1 ), we fur-

q q m

ther obtain hp (kX) = hp (X) and hpj (kX) = khpj (X). Hence hp = const

q q q q q

pj i

and hq is linear in X . Then the generalized invariant tensor theorem yields

f p = aY p + byi X i , a, b ∈ R. Applying the same procedure to fip , we ¬nd

p

fip = cyi , c ∈ R.

p

Consider the injection G2 ’ G2 m,n determined by the products with the

n

identities on R . The action of an element (ap , ar ) of the latter subgroup is

m

q st

given by

¯p q

yi = ap yi

(2) q

Zi = ap yi Z t + ap Zi

¯p q q

(3) qt q

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

362 Chapter X. Prolongation of vector ¬elds and connections

and V0 is an invariant subspace. In particular, (3) with ap = δq gives an equiv-

p

q

ariance condition

cyi = bap yi yj X j + cyi .

p qt p

qt

This yields b = 0, so that g p = 0. Further, the subspace V0 is invariant with the

respect to the inclusion of G1 into G2 . The equivariance of fip with respect

m,n m,n

to an element (δj , δq , ap ) ∈ G1 means cyi = c(yi + ap ). Hence c = 0, which

p p

ip

m,n

i i

completes the proof for r = 1.

For r ≥ 2 it su¬ces to discuss the g™s only. Using the homotheties in i(G1 ),

n

p q q

j

we ¬nd that gi1 ···is (X , yβ ), 1 ¤ |β| ¤ r, is linear in yβ . The homotheties in

i(G1 ) and the generalized invariant tensor theorem then yield

m

gi1 ···is = Wip ···is + cs yi1 ···is is+1 ···ir X is+1 . . . X ir

p p

(4) 1

where Wip ···is do not depend on yi1 ···ir , s = 1, . . . , r ’ 1, and

p

1

p p

(5) gi1 ···ir = cr yi1 ···ir

p p

g p = b1 yi X i + · · · + br yi1 ···ir X i1 . . . X ir .

(6)

Similarly to the ¬rst order case, we have an inclusion Gr+1 ’ Gr+1 determined

n m,n

by the products of di¬eomorphisms on Rn with the identity of Rm . One ¬nds

easily the following transformation law

yi1 ···is = ap yi1 ···is + Fip ···is + ap1 ···qs yi1 . . . yis

¯p q q1 qs

(7) q q

1

where Fip ···is is a polynomial expression linear in ap with 2 ¤ |±| ¤ s ’ 1 and

±

1

p

independent of yi1 ···is . This implies

Zi1 ···is = ap Zi1 ···is + Gp1 ···is + ap1 ···qs qs+1 yi1 . . . yis Z qs+1

¯p q q1 qs

(8) q q

i

where Gp1 ···is is a polynomial expression linear in ap with 2 ¤ |±| ¤ s and linear

±

i

p

in Z± , 0 ¤ |±| ¤ s ’ 1.

p p

We deduce that every gi1 ···is , 0 ¤ s ¤ r ’ 1 , is independent of yi1 ···ir . On the

kernel of the jet projection Gr+1 ’ Gr , (8) for r = s gives

n n

q1 qr

0 = ap1 ···qr qr+1 yi1 . . . yir g qr+1 .

q

Hence g p = 0. On the kernel of the jet projection Gr ’ Gr’1 , (8) with s =

n n

1, . . . , r ’ 1, implies

q1 qr

0 = cs ap1 ...qr yi1 . . . yir X is+1 . . . X ir ,

q

i.e. cs = 0. By projectability, g i and g± , 0 ¤ |±| ¤ r ’ 1, correspond to a

p

Gr -equivariant map V0 — S r’1 ’ Z r’1 . By the induction hypothesis, g± = 0

p

m,n

for all 0 ¤ |±| ¤ r ’ 1. Then on the kernel of the jet projection Gr+1 ’ Gr’1

n n

q1 qr p

(8) gives 0 = cr ap1 ...qr yi1 . . . yir , i.e. gi1 ···ir = 0.

q

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

45. Prolongations of connections to F Y ’ M 363

r

44.4. Bundles of contact elements. Consider the bundle functor Kn on

Mfm of the n-dimensional contact elements of order r de¬ned in 12.15.

r

Proposition. Every natural operator A : T T Kn is a constant multiple of

r

the ¬‚ow operator Kn .

Proof. It su¬ces to discuss the case M = Rm . Consider the canonical ¬bration

Rm = Rn — Rm’n ’ Rn . As remarked at the end of 12.16, there is an identi¬-

cation of an open dense subset in Kn Rm with J r (Rm ’ Rn ). By de¬nition, on

r

this subset it holds J r ξ = Kn ξ for every projectable vector ¬eld ξ on Rm ’ Rn .

r

Since the operator A commutes with the action of all di¬eomorphisms preserving

‚ ‚

¬bration Rm ’ Rn , the restriction of A to ‚x1 is a constant multiple of Kn ( ‚x1 )

r

by proposition 44.3. But every vector ¬eld on Rm can be locally transformed

‚

into ‚x1 in a neighborhood of any point where it does not vanish.

We ¬nd it interesting that we have ¬nished our investigation of the basic

properties of the natural operators T T F for di¬erent bundle functors on

Mfm by an example in which the constant multiples of the ¬‚ow operator are

the only natural operators T TF.

T T — and

44.5. Remark. [Kobak, 91] determined all natural operators T

T (T T — ) for manifolds of dimension at least two. Let T — be the ¬‚ow operator

T

of the cotangent bundle, LM : T — M ’ T T — M be the vector ¬eld generated by

the homotheties of the vector bundle T — M and ωM : T M —M T — M ’ R be the

T T — are of the form f (ω)T — +

evaluation map. Then all natural operators T

g(ω)L, where f , g ∈ C ∞ (R, R) are any smooth functions of one variable. In

the case F = T T — the result is of similar character, but the complete list is

somewhat longer, so that we refer the reader to the above mentioned paper.

45. Prolongations of connections to F Y ’ M

45.1. In 31.1 we deduced that there is exactly one natural operator transforming

every general connection on Y ’ M into a general connection on V Y ’ M .

However, one meets a quite di¬erent situation when replacing ¬bered manifold

V Y ’ M e.g. by the ¬rst jet prolongation J 1 Y ’ M of Y . Pohl has observed in

the vector bundle case, [Pohl, 66], that one needs an auxiliary linear connection

on the base manifold M to construct an induced connection on J 1 Y ’ M . Our

¬rst goal is to clarify this di¬erence from the conceptual point of view.

45.2. Bundle functors of order (r, s). We recall that two maps f , g of a

¬bered manifold p : Y ’ M into another manifold determine the same (r, s)-jet

r,s r,s r r

jy f = jy g at y ∈ Y , s ≥ r, if jy f = jy g and the restrictions of f and g to the

s s

¬ber Yp(y) satisfy jy (f |Yp(y) ) = jy (g|Yp(y) ), see 12.19.

De¬nition. A bundle functor on a category C over FM is said to be of order

¯

(r, s), if for any two C-morphisms f , g of Y into Y

r,s r,s

jy f = jy g implies (F f )|(F Y )y = (F g)|(F Y )y .

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

364 Chapter X. Prolongation of vector ¬elds and connections