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an arbitrary bundle functor F on Mf , we realize that we need a natural equiv-
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

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