alence F T ’ T F with suitable properties. However, the ¬‚ow-natural transfor-

mation F T ’ T F from 39.2 is a natural equivalence if and only if F preserves

products, i.e. F is a Weil functor. We remark that we do not know any natural

operator transforming general connections on Y ’ M into general connections

on F Y ’ F M for any concrete non-product-preserving functor F on Mf .

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

370 Chapter X. Prolongation of vector ¬elds and connections

46.3. Remark. Slov´k has proved in [Slov´k, 87a] that if “ is a linear con-

a a

nection on a vector bundle p : E ’ M , then TA “ is also a linear connection on

the induced vector bundle TA p : TA E ’ TA M . Furthermore, if p : P ’ M is a

principal bundle with structure group G, then TA p : TA P ’ TA M is a principal

bundle with structure group TA G. Using the ideas from 37.16 one deduces di-

rectly that for every principal connection “ on P ’ M the induced connection

TA “ is also principal on TA P ’ TA M .

46.4. We deduce one geometric property of the connection TA “. If we consider

a general connection “ on Y ’ M in the form “ : Y • T M ’ T Y , the “-lift “ξ

of a vector ¬eld ξ : M ’ T M is given by

(“ξ)(y) = “(y, ξ(p(y))), i.e. “ξ = “ —¦ (idY , ξ —¦ p).

(1)

On one hand, “ξ is a vector ¬eld on Y and we can construct its ¬‚ow prolongation

TA (“ξ) = κY —¦ TA (“ξ). On the other hand, the ¬‚ow prolongation TA ξ = κM —¦

TA ξ of ξ is a vector ¬eld on TA M and we construct its TA “-lift (TA “)(TA ξ).

The following assertion is based on the fact that we have used a ¬‚ow-natural

equivalence in the de¬nition of TA “.

Proposition. For every vector ¬eld ξ on M , we have (TA “)(TA ξ) = TA (“ξ).

Proof. By (1), we have TA “(TA ξ) = TA “ —¦ (idTA Y , TA ξ —¦ TA p) = κY —¦ TA “ —¦

(idTA Y , κ’1 —¦ κM —¦ TA ξ —¦ TA p) = κY —¦ TA (“ —¦ (idY , ξ —¦ p)) = TA (“ξ).

M

We remark that several further geometric properties of TA “ are deduced in

[Slov´k, 87a].

a

¯ ¯

46.5. Let “ be another connection on another ¬bered manifold Y and let

¯ ¯

f : Y ’ Y be a connection morphism of “ into “, i.e. the following diagram

commutes

w TY

u u¯

Tf

TY

¯

(1) “ “

w Y • T BY

f • T Bf ¯ ¯

Y • T BY

¯ ¯

Proposition. If f : Y ’ Y is a connection morphism of “ into “, then TA f :

¯ ¯

TA Y ’ TA Y is a connection morphism of TA “ into TA “.

¯

Proof. Applying TA to (1), we obtain TA T f —¦ TA “ = (TA “) —¦ (TA f • TA T Bf ).

¯

From 46.1.(3) we then deduce directly T TA f —¦ TA “ = TA “ —¦ (TA f • T TA Bf ).

46.6. The problem of ¬nding all natural operators transforming connections on

Y ’ M into connections on TA Y ’ TA M seems to be much more complicated

than e.g. the problem of ¬nding all natural operators T T TA discussed in

section 42. We shall clarify the situation in the case that TA is the classical

tangent functor T and we restrict ourselves to the ¬rst order natural operators.

Let T be the operator from proposition 46.1 in the case TA = T . Hence

T transforms every element of C ∞ (J 1 Y ) into C ∞ (J 1 (T Y ’ T BY )), where

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

46. The cases F Y ’ F M and F Y ’ Y 371

J 1 and J 1 (T ’ T B) are considered as bundle functors on FMm,n . Further we

construct a natural ˜di¬erence tensor ¬eld™ [CY “] for connections on T Y ’ T BY

from the curvature of a connection “ on Y . Write BY = M . In general, the

di¬erence of two connections on Y is a section of V Y — T — M , which can be

interpreted as a map Y • T M ’ V Y . In the case of T Y ’ T M we have T Y •

T T M ’ V (T Y ’ T M ). To de¬ne the operator [C], consider both canonical

projections pT M , T pM : T T M ’ T M . If we compose (pT M , T pM ) : T T M ’

T M • T M with the antisymmetric tensor power and take the ¬bered product

of the result with the bundle projection T Y ’ Y , we obtain a map µY : T Y •

T T M ’ Y • Λ2 T M . Since CY “ : Y • Λ2 T M ’ V Y , the values of CY “ —¦ µY

lie in V Y . Every vector A ∈ V Y is identi¬ed with a vector i(A) ∈ V (V Y ’ Y )

tangent to the curve of the scalar multiples of A. Then we construct [CY “](U, Z),

U ∈ T Y , Z ∈ T T M by translating i(CY “(µY (U, Z))) to the point U in the same

¬ber of V (T Y ’ T M ). This yields a map [CY “] : T Y •T T M ’ V (T Y ’ T M )

of the required type.

46.7. Proposition. All ¬rst order natural operators J 1 J 1 (T ’ T B) form

the following one-parameter family

T + k[C], k ∈ R.

Proof. Let

dy p = Fip (x, y)dxi

(1)

be the equations of “. Evaluating 46.1.(3) in the case TA = T , one ¬nds that

the equations of T “ are (1) and

p p

‚Fi j ‚Fi q

dxi + Fip (x, y)dξ i

p

(2) d· = ‚xj ξ + ‚y q ·

where ξ i = dxi , · p = dy p are the induced coordinates on T Y . The equations of

[CY “]

p p

‚Fi ‚Fi q

dy p = 0, d· p = ξ j § dxi

(3) + q Fj

j

‚x ‚y

follow directly from the de¬nition.

Let S1 = J0 (J 1 (Rm+n ’ Rm ) ’ Rm+n ), Q = T0 (Rm+n ), Z = J0 (T Rm+n

1 1

’ T Rm ) be the standard ¬bers in question and q : Z ’ Q be the canonical pro-

jection. According to 18.19, the ¬rst order natural operators A : J 1 J 1 (T ’

T B) are in bijection with the G2 -maps A : S1 — Q ’ Z satisfying q —¦ A = pr2 .

m,n

p p p

The canonical coordinates yi , yiq , yij on S1 and the action of G2 on S1 are

m,n

p p p

described in 27.3. It will be useful to replace yij by Sij and Rij in the same way

as in 28.2. One sees directly that the action of G2 on Q with coordinates ξ i ,

m,n

p

· is

· p = ap ξ i + ap · q .

¯

ξ i = ai ξ j ,

(4) ¯

j q

i

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

372 Chapter X. Prolongation of vector ¬elds and connections

The coordinates on Z are ξ i , · p and the quantities Ap , Bi , Ci , Di determined

p p p

i

by

dy p = Ap dxi + Bi dξ i ,

p p p

d· p = Ci dxi + Di dξ i .

(5) i

A direct calculation yields that the action of G2 on Z is (4) and

m,n

Ap = ap Aq aj ’ aq + Bj aj ak ξ l

q

¯

j ˜i ˜i ˜ik l

q

i

Bi = ap Bj aj

¯p q

˜i

q

Ci = ap ’˜q ξ j ’ aq ξ j Ar ’ aq · r ’ aq · r As

¯p aij ¯ ˜jr ¯ ¯i ˜ir ¯ ˜rs ¯ ¯i

q

(6)

+ C q aj + Dq aj ξ k

˜¯

˜ ji j ik

¯p ap ’˜jr ak ξ as Bl ai ’ aq ’ aq ar ξ k as Bj aj

q jkr sl u

Di = a ˜ ˜i ˜rs k ˜i

q u

’ aq ar · t as Bj aj + Dj aj .

q

u

˜rs t ˜i ˜i

u

p p p p

Write ξ = (ξ i ), · = (· p ), y = (yi ), y1 = (yiq ), S = (Sij ), R = (Rij ).

p

I. Consider ¬rst the coordinate functions Bi (ξ, ·, y, y1 , S, R) of A. The com-

mon kernel L of π1 : G2

2 1 2 2 2

m,n ’ Gm,n and of the projection Gm,n ’ Gm — Gn

described in 28.2 is characterized by ai = δj , ap = δq , ap = 0, ai = 0, ap = 0.

i p

q qr

j i jk

p p

The equivariance of Bi with respect to L implies that Bi are independent of y1

and S. Then the homotheties in i(G1 ) ‚ G2 yield a homogeneity condition

n m,n

p p

kBi = Bi (ξ, k·, ky, kR).

Therefore we have

p p pj q pjk q

Bi = fiq (ξ)· q + fiq (ξ)yj + fiq (ξ)Rjk

with some smooth functions of ξ. Now the homotheties in i(G1 ) give

m

p p pj q pjk q

k ’1 Bi = fiq (kξ)· q + fiq (kξ)k ’1 yj + fiq (kξ)k ’2 Rjk .

p p pj pj pjk pjk

Hence it holds a) fiq (ξ) = kfiq (kξ), b) fiq (ξ) = fiq (kξ), c) kfiq (ξ) = fiq (kξ).

p pj

If we let k ’ 0 in a) and b), we obtain fiq = 0 and fiq = const. The relation

pjk p

c) yields that fiq is linear in ξ. The equivariance of Bi with respect to the

pj pjk

whole group i(G1 — G1 ) implies that fiq and fiq correspond to the generalized

m n

invariant tensors. By theorem 27.1 we obtain

p p p

Bi = c1 Rij ξ j + c2 yi

p

with real parameters c1 , c2 . Consider further the equivariance of Bi with respect

to the subgroup K ‚ G2 characterized by ai = δj , ap = δq . This yields

i p

m,n q

j

c1 Rij ξ j + c2 yi = c1 Rij ξ j + c2 (yi + ap ).

p p p p

i

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

46. The cases F Y ’ F M and F Y ’ Y 373

This relation implies c2 = 0.

II. For Ap we obtain in the same way as in I

i

Ap = aRij ξ j + c3 yi .

p p

i

The equivariance with respect to subgroup K gives c3 = 1 and c1 = 0.

III. Analogously to I and II we deduce

p p p

Di = bRij ξ j + c4 yi .

p

Taking into account the equivariance of Di with respect to K, we ¬nd c4 = 1.

IV. Here it is useful to summarize. Up to now, we have deduced

Ap = aRij ξ j + yi ,

p p p p p p

Di = bRij ξ j + yi .

(7) Bi = 0,

i

Consider the di¬erence A ’ T , where T is the operator (1) and (2). Write

p p p p

Ei = Ci ’ yij ξ j ’ yiq · q .

(8)

Using ap , we ¬nd easily that Ei does not depend on Sij . By (6) and (8), the

p p

ij

p

action of K on Ei is

’a˜p ξ j Rik ξ k + aap aq ξ j Rik ξ k + aap · q Rij ξ j + Ei ’ bRjk ξ k aj ξ l

q p p

r r

ajq qr j qr il

(9)

= Ei ξ, · q + aq ξ j , yj + ar , ykt + as + as yk , R .

p r s u

j kt tu

j

If we set Ei = ayjq ξ j Rik ξ k + Fip , then (9) implies that Fip is independent of y1 .

p p q

The action of i(G1 — G1 ) on Fip (ξ, ·, y, R) is tensorial. Hence we have the same

m n

situation as for Bi in I. This implies Fip = kRij ξ j + eyi . Using once again (9)

p p p

p p p p p p

we obtain a = b = e = 0. Hence Ei = kRij ξ j and Ci = yij ξ j + yiq · q + kRij ξ j .

Thus we have deduced the coordinate form of our statement.

46.8. Prolongation of connections to F Y ’ Y . Given a bundle functor

F on Mf and a ¬bered manifold Y ’ M , there are three canonical structures

of a ¬bered manifold on F Y , namely F Y ’ M , F Y ’ F M and F Y ’ Y .

Unlike the ¬rst two cases, it seems that there should be only poor results on the

prolongation of connections to F Y ’ Y . We ¬rst present a negative result for

the case of the tangent functor T .

Proposition. There is no ¬rst order natural operator transforming connections

on Y ’ M into connections on T Y ’ Y .

Proof. We shall use the notation from the proof of proposition 46.7. The equa-

tions of a connection on T Y ’ Y are

d· p = Pip dxi + Qp dy q .

dξ i = Mj dxj + Np dy p ,

i i

q

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

374 Chapter X. Prolongation of vector ¬elds and connections

¯

One evaluates easily the action formulae ξ i = ai ξ j and

j

Mj = ai Mlk al + ai Np ap ’ ai al ak ξ m

¯i k

˜j ˜j l ˜jk m

k k

¯i

Np = ai Nq aq .

j

˜p

j

The homotheties in i(G1 ) give

n

Np = kNp (ξ j , k· q , kyk , ytl , kymn ).

i i r s u

Hence Np = 0. For Mj , the homotheties in i(G1 ) imply the independence of Mj

i i i

n

p p

of · p , yi , yij . The equivariance of Mj with respect to the subgroup K means

i

Mj (ξ j , ykq ) + ai ξ k = Mj (ξ j , ykq + ap ).

p p

i i

jk kq

Since the expressions Mj on both sides are independent of ai , the di¬erentiation

i

jk

i i

with respect to ajk yields some relations among ξ only.

46.9. Prolongation of connections to V Y ’ Y . We pay special attention

to this problem because of its relation to Finslerian geometry. We are going to

study the ¬rst order natural operators transforming connections on Y ’ M into

connections on V Y ’ Y , i.e. the natural operators J 1 J 1 (V ’ Id) where Id

means the identity functor. In this case it will be instructive to start from the

computational aspect of the problem.

Using the notation from 46.7, the equations of a connection on V Y ’ Y are

d· p = Ap (xj , y q , · r )dxi + Bq (xj , y r , · s )dy q .

p

(1) i

The induced coordinates on the standard ¬ber Z = J0 (V (Rm+n ’ Rm ) ’

1

Rm+n ) are · p , Ap , Bi and the action of G2 on Z has the form

p

m,n

i

· p = ap · q

(2) ¯ q

Ap = ap aj · q + ap Aq aj ’ ap Br ar as aj ’ ap as · r aq aj

¯ q

(3) qj ˜i q j ˜i ˜s j ˜i rs ˜q j ˜i

q

i

¯p

Bq = ap Bs as + ap as · r .

r

(4) ˜q rs ˜q

r

Our problem is to ¬nd all G2 -maps S1 — Rn ’ Z over the identity on Rn .

m,n

Consider ¬rst the component Bq (· r , yi , yju , ykl ) of such a map. The homotheties

p st v

in i(G1 ) yield