For example, the order of the vertical functor V is (0, 1), while the functor of

the ¬rst jet prolongation J 1 has order (1, 1).

45.3. Denote by J r,s T Y the space of all (r, s)-jets of the projectable vector

¬elds on Y ’ M . This is a vector bundle over Y . Let F be a bundle functor on

FMm,n and F denote its ¬‚ow operator Tproj TF.

Proposition. If the order of F is (r, s) and · is a projectable vector ¬eld on Y ,

r,s

then the value (F·)(u) at every u ∈ (F Y )y depends only on jy ·. The induced

map

F Y • J r,s T Y ’ T (F Y )

is smooth and linear with respect to J r,s T Y .

Proof. Smoothness can be proved in the same way as in 14.14. Linearity follows

directly from the linearity of the ¬‚ow operator F.

45.4. Let “ be a general connection on p : Y ’ M . Considering the “-lift

r,s r

“ξ of a vector ¬eld ξ on M , one sees directly that jy “ξ depends on jp(y) ξ

only, y ∈ Y . Let F be a bundle functor on FMm,n of order (r, s). If we

combine the map of proposition 45.3 with the lifting map of “, we obtain a

map F “ : F Y • J r T M ’ T F Y linear in J r T M . Let Λ : T M ’ J r T M be

an r-th order linear connection on M , i.e. a linear splitting of the projection

π0 : J r T M ’ T M . By linearity, the composition

r

F “ —¦ (idF Y • Λ) : F Y • T M ’ T F Y

(1)

is a lifting map of a general connection on F Y ’ M .

De¬nition. The general connection F(“, Λ) on F Y ’ M with lifting map (1)

is called the F -prolongation of “ with respect to Λ.

If the order of F is (0, s), we need no connection Λ on M . In particular, every

connection “ on Y ’ M induces in such a way a connection V“ on V Y ’ M ,

which was already mentioned in remark 31.4.

45.5. We show that the construction of F(“, Λ) behaves well with respect to

¯

morphisms of connections. Given an FM-morphism f : Y ’ Y over f0 : M ’

¯ ¯ ¯¯ ¯

M and two general connections “ on p : Y ’ M and “ on p : Y ’ M , one sees

¯

easily that “ and “ are f -related in the sense of 8.15 if and only if the following

diagram commutes

wu

u

Tf ¯

TY TY

¯

“ “

w

f • T f0 ¯ ¯

Y • TM Y • TM

¯

In such a case f is also called a connection morphism of “ into “. Further, two

¯ ¯ ¯

r-th order linear connections Λ : T M ’ J T M and Λ : T M ’ J r T M are called

r

f0 -related, if for every z ∈ Tx M it holds

¯ r r

Λ(T f0 (z)) —¦ (jx f0 ) = (jz T f0 ) —¦ Λ(z).

Let F be as in 45.4.

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

45. Prolongations of connections to F Y ’ M 365

¯ ¯

Proposition. If “ and “ are f -related and Λ and Λ are f0 -related, then F(“, Λ)

¯¯

and F(“, Λ) are F f -related.

Proof. The lifting map of F(“, Λ) can be determined as follows. For every

r

X ∈ Tx M we take a vector ¬eld ξ on M such that jx ξ = Λ(X) and we construct

its “-lift “ξ. Then F(“, Λ)(u) is the value of the ¬‚ow prolongation F(“ξ) at

r¯

¯ ¯

u ∈ Fx Y . Let Λ(T f0 (X)) = jx ξ, x = f0 (x). If Λ and Λ are f0 -related, the

¯

¯

¯ ¯

vector ¬elds ξ and ξ are f0 -related up to order r at x. Since “ and “ are f -

¯

related, the restriction of F(“ξ) over x and the restriction of F(“ξ) over x are

¯

F f -related.

45.6. In many concrete cases, the connection F(“, Λ) is of special kind. We are

going to deduce a general result of this type.

Let C be a category over FM, cf. 51.4. Analogously to example 1 from 18.18,

a projectable vector ¬eld · on Y ∈ ObC is called a C-¬eld, if its ¬‚ow is formed by

local C-morphisms. For example, for the category PB(G) of smooth principal G-

bundles, a projectable vector ¬eld · on a principal ¬ber bundle is a PB(G)-¬eld

if and only if · is right-invariant. For the category VB of smooth vector bundles,

one deduces easily that a projectable vector ¬eld · on a vector bundle E is a

VB-¬eld if and only if · is a linear morphism E ’ T E, see 6.11. A connection “

on (p : Y ’ M ) ∈ ObC is called a C-connection, if “ξ is a C-¬eld for every vector

¬eld ξ on M . Obviously, a PB(G)-connection or a VB-connection is a classical

principal or linear connection, respectively.

More generally, a projectable family of tangent vectors along a ¬ber Yx , i.e. a

section σ : Yx ’ T Y such that T p —¦ σ is a constant map, is said to be a C-family,

if there exists a C-¬eld · on Y such that σ is the restriction of · to Yx . We shall

say that the category C is in¬nitesimally regular, if any projectable vector ¬eld

on a C-object the restriction of which to each ¬ber is a C-family is a C-¬eld.

Proposition. If F is a bundle functor of a category C over FM into an in-

¬nitesimally regular category D over FM and “ is a C-connection, then F(“, Λ)

is a D-connection for every Λ.

Proof. By the construction that we used in the proof of proposition 45.5, the

F(“, Λ)-lift of every vector X ∈ T M is a D-family. Since D is in¬nitesimally

regular, the F(“, Λ)-lift of every vector ¬eld on T M is a D-¬eld.

45.7. In the special case F = J 1 we determine all natural operators trans-

forming a general connection on Y ’ M and a ¬rst order linear connection

Λ on M into a general connection on J 1 Y ’ M . Taking into account the

rigidity of the symmetric linear connections on M deduced in 25.3, we ¬rst as-

sume Λ to be without torsion. Thus we are interested in the natural operators

J 1 • Q„ P 1 B J 1 (J 1 ’ B).

On one hand, “ and Λ induce the J 1 -prolongation J 1 (“, Λ) of “ with respect

to Λ. On the other hand, since J 1 Y is an a¬ne bundle with associated vec-

tor bundle V Y — T — M , the section “ : Y ’ J 1 Y determines an identi¬cation

I“ : J 1 Y ∼ V Y — T — M . The vertical prolongation V“ of “ is linear over Y , see

=

31.1.(3), so that we can construct the tensor product V“ — Λ— with the dual

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

366 Chapter X. Prolongation of vector ¬elds and connections

connection Λ— on T — M , see 47.14 and 47.15. The identi¬cation I“ transforms

V“ — Λ— into another connection P (“, Λ) on J 1 Y ’ M .

45.8. Proposition. All natural operators J 1 • Q„ P 1 B J 1 (J 1 ’ B) form

the one-parameter family

tP + (1 ’ t)J 1 , t ∈ R.

(1)

Proof. In usual local coordinates, let

dy p = Fip (x, y)dxi

(2)

be the equations of “ and

dξ i = Λi (x)ξ j dxk

(3) jk

be the equations of Λ. By direct evaluation, one ¬nds the equations of J 1 (“, Λ)

in the form (2) and

p p

‚Fj ‚Fj q

p p p

+ Λk (Fk ’ yk ) dxj

(4) dyi = + ‚y q yi ji

‚xi

while the equations of P (“, Λ) have the form (2) and

p p p

‚Fj ‚Fi ‚Fi

p q

’ Fiq ) + q p p

’ Λk (yk ’ Fk ) dxj .

(5) dyi = q (yi + q Fj ij

j

‚y ‚x ‚y

First we discuss the operators of ¬rst order in “ and of order zero in Λ.

Let S1 = J0 (J 1 (Rn+m ’ Rm ) ’ Rn+m ) be the standard ¬ber from 27.3,

1

S0 = J0 (Rm+n ’ Rm ), Λ = (Q„ P 1 Rm )0 and Z = J0 (J 1 (Rm+n ’ Rm ) ’ Rm ).

1 1

By using the general theory, the operators in question correspond to G2 -maps

m,n

S1 — Λ — S0 ’ Z over the identity of S0 . The canonical coordinates on S1 are

p p p

yi , yiq , yij and the action of G2 is given by 27.3.(1)-(3). On S0 we have the

m,n

well known coordinates Yip and the action

Yip = ap Yjq aj + ap aj .

¯

(6) ˜i j ˜i

q

The standard coordinates on Λ are Λi = Λi and the action is

jk kj

¯

Λi = ai Λl am an + ai al am .

(7) l mn ˜j ˜k lm ˜j ˜k

jk

p p p

The induced coordinates on Z are zi , Zi , Zij and one evaluates easily that the

p p

action on both zi and Zi has form (6), while

Zij = ap Zkl ak al + ap zk Zlr ak al + ap Zlq ak al

¯p q q

˜i ˜j ˜i ˜j ˜i ˜j

q qr qk

(8)

+ ap zk ak al + ap zk ak + ap ak + ap ak al .

q q

ql ˜i ˜j ˜ij k ˜ij kl ˜i ˜j

q

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

45. Prolongations of connections to F Y ’ M 367

Write Y = (Yip ), y = (yi ), y1 = (yiq ), y2 = (yij ), Λ = (Λi ). Then the

p p p

jk

coordinate form of a map f : S1 — Λ — S0 ’ Z over the identity of S0 is zi = Yip

p

and

Zi = fip (Y, y, y1 , y2 , Λ)

p

(9) p p

Zij = fij (Y, y, y1 , y2 , Λ).

The equivariance of fip with respect to the homotheties in i(G1 ) yields

m

kfip = fip (kY, ky, ky1 , k 2 y2 , kΛ)

so that fip is linear in Y , y, y1 , Λ and independent of y2 . The homotheties in

i(G1 ) give that fip is independent of y1 and Λ. By the generalized invariant

n

tensor theorem 27.1, the equivariance with respect to i(G1 — G1 ) implies

m n

fip = aYip + byi .

p

Then the equivariance with respect to the subgroup K characterized by ai = δj ,

i

j

ap = δq yields

p

q

b = 1 ’ a.

p

For fij the homotheties in i(G1 ) and i(G1 ) give

m n

p p

k 2 fij = fij (kY, ky, ky1 , k 2 y2 , kΛ)

p p

kfij = fij (kY, ky, y1 , ky2 , Λ)

p

so that fij is linear in y2 and bilinear in the pairs (Y, y1 ),(y, y1 ), (Y, Λ), (y, Λ).

p

Considering equivariance with respect to i(G1 — G1 ), we obtain fij in the form

m n

of a 16-parameter family

fij = k1 yij + k2 yji + k3 Yip yqj + k4 Yjp yqi + k5 Yiq yqj + k6 Yjq yqi

p p p q q p p

pq pq qp qp p

+ k7 yi yqj + k8 yj yqi + k9 yi yqj + k10 yj yqi + k11 Yk Λk

ij

+ k12 Yip Λk + k13 Yjp Λk + k14 yk Λk + k15 yi Λk + k16 yj Λk .

p p p

kj ki ij kj ki

Evaluating the equivariance with respect to K, we ¬nd a = 0 and such relations

among k1 , . . . , k16 , which correspond to (1).

Furthermore, 23.7 implies that every natural operator of our type has ¬nite

order. Having a natural operator of order r in “ and of order s in Λ, we shall

deduce r = 1 and s = 0, which corresponds to the above case. Let ± and γ be

multi indices in xi and β be a multi index in y p . The associated map of our

operator has the form zi = Yip and

p

Zi = fip (Y, y±β , Λγ ),

p p p

Zij = fij (Y, y±β , Λγ )

where |±| + |β| ¤ r, |γ| ¤ s. Using the homotheties in i(G1 ), we obtain

m

kfip = fip (kY, k 1+|±| y±β , k 1+|γ| Λγ ).

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

368 Chapter X. Prolongation of vector ¬elds and connections

Hence fip is linear in Y , yβ and Λ, and is independent of the variables with

|±| > 0 or |γ| > 0. The homotheties in i(G1 ) then imply that fip is independent

n

p

of yβ with |β| > 1. For fij , the homotheties in i(G1 ) yield

m

p p

k 2 fij = fij (kY, k 1+|±| y±β , k 1+|γ| Λγ )

(10)

p

so that fij is a polynomial independent of the variables with |±| > 1 or |γ| > 1.

The homotheties in i(G1 ) imply

n

p p

kfij = fij (kY, k 1’|β| y±β , Λγ )

(11)

p

for |±| ¤ 1, |γ| ¤ 1. Combining (10) with (11) we deduce that fij is independent

of y±β for |±| + |β| > 1 and Λγ for |γ| > 0.

45.9. Using a similar procedure as in 45.8 one can prove that the use of a

linear connection on the base manifold for a natural construction of an induced

connection on J 1 Y ’ M is unavoidable. In other words, the following assertion

holds, a complete proof of which can be found in [Kol´ˇ, 87a].

ar

Proposition. There is no natural operator J 1 J 1 (J 1 ’ B).

45.10. If we admit an arbitrary linear connection Λ on the base manifold in

the above problem, the natural operators QP 1 QP 1 from proposition 25.2

must appear in the result. By proposition 25.2, all natural operators QP 1

T — T — — T — form a 3-parameter family

ˆ ˆ

N (Λ) = k1 S + k2 I — S + k3 S — I.

By 12.16, J 1 (J 1 Y ’ M ) is an a¬ne bundle with associated vector bundle

V J 1 Y — T — M . We construct some natural ˜di¬erence tensors™ for this case.

Consider the exact sequence of vector bundles over J 1 Y established in 12.16

Vβ

0 ’ V Y —J 1 Y T — M ’ V J 1 Y ’ ’ V Y ’ 0

’ ’ ’ ’

where —J 1 Y denotes the tensor product of the pullbacks over J 1 Y . The con-

nection “ determines a map δ(“) : J 1 Y ’ V Y — T — M transforming every

u ∈ J 1 Y into the di¬erence u ’ “(βu) ∈ V Y — T — M . Hence for every k1 ,

k2 , k3 we can extend the evaluation map T M • T — M ’ R into a contraction

δ(“), N (Λ) : J 1 Y ’ V Y —J 1 Y T — M —T — M ‚ V J 1 Y —T — M . By the procedure

used in 45.8 one can prove the following assertion, see [Kol´ˇ, 87a].

ar

Proposition. All natural operators transforming a connection “ on Y into a

connection on J 1 Y ’ M by means of a linear connection Λ on the base manifold

form the 4-parameter family

˜

tP (“, Λ) + (1 ’ t)J 1 (“, Λ) + δ(“), N (Λ)

˜

t, k1 , k2 , k3 ∈ R, where Λ means the conjugate connection of Λ.

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

46. The cases F Y ’ F M and F Y ’ Y 369

46. The cases F Y ’ F M and F Y ’ Y

46.1. We ¬rst describe a geometrical construction transforming every connec-

tion “ on a ¬bered manifold p : Y ’ M into a connection TA “ on TA p : TA Y ’

TA M for every Weil functor TA . Consider “ in the form of the lifting map

“ : Y • T M ’ T Y.

(1)

Such a lifting map is characterized by the condition

(πY , T p) —¦ “ = idY •T M

(2)

where π : T ’ Id is the bundle projection of the tangent functor, and by the

fact that, if we interpret (1) as the pullback map

p— T M ’ T Y,

this is a vector bundle morphism over Y . Let κ : TA T ’ T TA be the ¬‚ow-natural

equivalence corresponding to the exchange homomorphism A — D ’ D — A, see

35.17 and 39.2.

Proposition. For every general connection “ : Y • T M ’ T Y , the map

TA “ := κY —¦ (TA “) —¦ (idTA Y • κ’1 ) : TA Y • T TA M ’ T TA Y

(3) M

is a general connection on TA p : TA Y ’ TA M .

Proof. Applying TA to (2), we obtain

(TA πY , TA T p) —¦ TA “ = idTA Y •TA T M .

Since κ is the ¬‚ow-natural equivalence, it holds κM —¦ TA T p —¦ κ’1 = T TA p and

Y

TA πY —¦ κ’1 = πTA Y . This yields

Y

(πTA Y , T TA p) —¦ TA “ = idTA •T TA M

so that TA “ satis¬es the analog of (2). Further, one deduces easily that κY :

TA T Y ’ T TA Y is a vector bundle morphism over TA Y . Even κ’1 : T TA M ’

M

TA T M is a linear morphism over TA M , so that the pullback map (TA p)— κ’1 :

M

(TA p)— T TA M ’ (TA p)— TA T M is also linear. But we have a canonical iden-

ti¬cation (TA p)— TA T M ∼ TA (p— T M ). Hence the pullback form of TA “ on

=

—

(TA p) T TA M ’ T TA Y is a composition of three vector bundle morphisms over

TA Y , so that it is linear as well.

46.2. Remark. If we look for a possible generalization of this construction to