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

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

n

Bq (· r , yi , yju , ykl ) = Bq (k· r , kyi , yju , kykl )

p st v p st v

so that Bq depends on yis only. Then the homotheties in i(G1 ) give Bq (yis ) =

p r pr

m

p r p p p

Bq (kyis ), which implies Bq = const. By the invariant tensor theorem, Bq = kδq .

The invariance under the subgroup K reads

kδq + ap · r = kδq .

p p

qr

This cannot be satis¬ed for any k. Thus, there is no ¬rst order operator J 1

J 1 (V ’ Id) natural on the category FMm,n .

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

Remarks 375

46.10. However, the obstruction is ap and the condition ap = 0 characterizes

qr qr

the a¬ne bundles (with vector bundles as a special case). Let us restrict ourselves

to the a¬ne bundles and continue in the previous consideration. By 46.9.(3),

the action of i(G1 — G1 ) on Ap (· q , yi , yjt , ykl ) is tensorial. Using homotheties

rs u

m n i

in i(G1 ), we ¬nd that Ap is linear in yi , yiq , but the coe¬cients are smooth

p p

m i

functions in · p . Using homotheties in i(G1 ), we deduce that the coe¬cients by

n

p p

yi are constant and the coe¬cients by yiq are linear in · p . By the generalized

invariant tensor theorem, we obtain

Ap = ayi + byqi · p + cyiq · q

p q p

a, b, c ∈ R.

(1) i

The equivariance of (1) on the subgroup K implies a = ’k, b = 0, c = 1. Thus

we have proved

Proposition. All ¬rst order operators J 1 J 1 (V ’ Id) which are natural on

the local isomorphisms of a¬ne bundles form the following one-parameter family

p p

d· p = yiq · q dxi + k(dy p ’ yi dxi ), k ∈ R.

Remarks

Section 42 is based on [Kol´ˇ, 88a]. The order estimate in 42.4 follows an idea

ar

by [Zajtz, 88b] and the proof of lemma 42.7 was communicated by the second

author. The results of section 43 were deduced by [Doupovec, 90]. Section 44

is based on [Kol´ˇ, Slov´k, 90]. The construction of the connection F(“, Λ)

ar a

from 45.4 was ¬rst presented in [Kol´ˇ, 82b]. Proposition 46.7 was proved by

ar

[Doupovec, Kol´ˇ, 88]. The relation of proposition 46.10 to Finslerian geometry

ar

was pointed out by B. Kis.

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

376

CHAPTER XI.

GENERAL THEORY

OF LIE DERIVATIVES

It has been clari¬ed recently that one can de¬ne the generalized Lie derivative

˜(ξ,·) f of any smooth map f : M ’ N with respect to a pair of vector ¬elds

L

ξ on M and · on N . Given a section s of a vector bundle E ’ M and a

projectable vector ¬eld · on E over a vector ¬eld ξ on M , the second component

˜

L· s : M ’ E of the generalized Lie derivative L(ξ,·) s is called the Lie derivative

of s with respect to ·. We ¬rst show how this approach generalizes the classical

cases of Lie di¬erentiation. We also present a simple, but useful comparison

of the generalized Lie derivative with the absolute derivative with respect to a

general connection. Then we prove that every linear natural operator commutes

with Lie di¬erentiation. We deduce a similar condition in the non linear case

as well. An operator satisfying the latter condition is said to be in¬nitesimally

natural. We prove that every in¬nitesimally natural operator is natural on the

category of oriented m-dimensional manifolds and orientation preserving local

di¬eomorphisms.

A signi¬cant advantage of our general theory is that it enables us to study

the Lie derivatives of the morphisms of ¬bered manifolds (our feeling is that the

morphisms of ¬bered manifolds are going to play an important role in di¬erential

geometry). To give a deeper example we discuss the Euler operator in the higher

order variational calculus on an arbitrary ¬bered manifold. In the last section

we extend the classical formula for the Lie derivative with respect to the bracket

of two vector ¬elds to the generalized Lie derivatives.

47. The general geometric approach

47.1. Let M , N be two manifolds and f : M ’ N be a map. We recall that

a vector ¬eld along f is a map • : M ’ T N satisfying pN —¦ • = f , where

pN : T N ’ N is the bundle projection.

Consider further a vector ¬eld ξ on M and a vector ¬eld · on N .

˜

De¬nition. The generalized Lie derivative L(ξ,·) f of f : M ’ N with respect

to ξ and · is the vector ¬eld along f de¬ned by

˜

L(ξ,·) f : T f —¦ ξ ’ · —¦ f.

(1)

˜

By the very de¬nition, L(ξ,·) is the zero vector ¬eld along f if and only if the

vector ¬elds · and ξ are f -related.

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

47. The general geometric approach 377

47.2. De¬nition 47.1 is closely related with the kinematic approach to Lie dif-

ferentiation. Using the ¬‚ows Flξ and Fl· of vector ¬elds ξ and ·, we construct

t t

a curve

t ’ (Fl· —¦f —¦ Flξ )(x)

(1) ’t t

for every x ∈ M . Di¬erentiating it with respect to t for t = 0 we obtain the

following

˜

Lemma. L(ξ,·) f (x) is the tangent vector to the curve (1) at t = 0, i.e.

(Fl· —¦f —¦ Flξ )(x).

˜ ‚

L(ξ,·) f (x) = ’t t

‚t 0

47.3. In the greater part of di¬erential geometry one meets de¬nition 47.1 in

certain more speci¬c situations. Consider ¬rst an arbitrary ¬bered manifold

p : Y ’ M , a section s : M ’ Y and a projectable vector ¬eld · on Y over a

vector ¬eld ξ on M . Then it holds T p —¦ (T s —¦ ξ ’ · —¦ s) = 0M , where 0M means

˜

the zero vector ¬eld on M . Hence L(ξ,·) s is a section of the vertical tangent

bundle of Y . We shall write

˜ ˜

L(ξ,·) s =: L· s : M ’ V Y

˜

and say that L· is the generalized Lie derivative of s with respect to ·. If we

have a vector bundle E ’ M , then its vertical tangent bundle V E coincides

with the ¬bered product E —M E, see 6.11. Then the generalized Lie derivative

˜

L· s has the form

˜

L· s = (s, L· s)

where L· s is a section of E.

47.4. De¬nition. Given a vector bundle E ’ M and a projectable vector ¬eld

· on E, the second component L· s : M ’ E of the generalized Lie derivative

˜

L· s is called the Lie derivative of s with respect to the ¬eld ·.

If we intend to contrast the Lie derivative L· s with the generalized Lie deriv-

˜

ative L· s, we shall say that L· s is the restricted Lie derivative. Using the fact

that the second component of L· s is the derivative of Fl· —¦s —¦ Flξ for t = 0 in

˜

’t t

the classical sense, we can express the restricted Lie derivative in the form

(L· s)(x) = lim 1 (Fl· —¦s —¦ Flξ ’s)(x).

(1) ’t t

t

t’0

47.5. It is useful to compare the general Lie di¬erentiation with the covariant

di¬erentiation with respect to a general connection “ : Y ’ J 1 Y on an arbitrary

¬bered manifold p : Y ’ M . For every ξ0 ∈ Tx M , let “(y)(ξ0 ) be its lift to the

horizontal subspace of “ at p(y) = x. For a vector ¬eld ξ on M , we obtain in this

way its “-lift “ξ, which is a projectable vector ¬eld on Y over ξ. The connection

map ω“ : T Y ’ V Y means the projection in the direction of the horizontal

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

378 Chapter XI. General theory of Lie derivatives

subspaces of “. The generalized covariant di¬erential ˜ “ s of a section s of Y is

de¬ned as the composition of ω“ with T s. This gives a linear map Tx M ’ Vs(x) Y

for every x ∈ M , so that ˜ “ s can be viewed as a section M ’ V Y —T — M , which

was introduced in another way in 17.8. The generalized covariant derivative ˜ “ s

ξ

of s with respect to a vector ¬eld ξ on M is then de¬ned by the evaluation

˜ “ s := ξ, ˜ “ s : M ’ V Y.

ξ

Proposition. It holds

˜ “ s = L“ξ s.

˜

ξ

Proof. Clearly, the value of ω“ at a vector ·0 ∈ Ty Y can be expressed as

˜

ω“ (·0 ) = ·0 ’ “(y)(T p(·0 )). Hence L“ξ s(x) = T s(ξ(x)) ’ “ξ(s(x)) coincides

with ω“ (T s(ξ(x))).

In the case of a vector bundle E ’ M , we have V E = E • E and ˜ “ s = ξ

“ “

(s, ξ s). The second component ξ : M ’ E is called the covariant derivative

of s with respect to ξ, see 11.12. In such a situation the above proposition implies

“

= L“ξ s.

(1) ξs

47.6. Consider further a natural bundle F : Mfm ’ FM. For every vector

¬eld ξ on M , its ¬‚ow prolongation Fξ is a projectable vector ¬eld on F M over

ξ. If F is a natural vector bundle, we have V F M = F M • F M .

De¬nition. Given a natural bundle F , a vector ¬eld ξ on a manifold M and a

section s of F M , the generalized Lie derivative

˜ ˜

LF ξ s =: Lξ : M ’ V F M

is called the generalized Lie derivative of s with respect to ξ. In the case of a

natural vector bundle F ,

LF ξ s =: Lξ s : M ’ F M

is called the Lie derivative of s with respect to ξ.

47.7. An important feature of our general approach to Lie di¬erentiation is that

it enables us to study the Lie derivatives of the morphisms of ¬bered manifolds.

In general, consider two ¬bered manifolds p : Y ’ M and q : Z ’ M over the

same base, a base preserving morphism f : Y ’ Z and a projectable vector ¬eld

· or ζ on Y or Z over the same vector ¬eld ξ on M . Then T q—¦(T f —¦·’ζ—¦f ) = 0M ,

˜

so that the values of the generalized Lie derivative L(·,ζ) f lie in the vertical

tangent bundle of Z.

De¬nition. If Z is a vector bundle, then the second component

L(·,ζ) f : Y ’ Z

˜

of L(·,ζ) f : Y ’ V Z is called the Lie derivative of f with respect to · and ζ.

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

47. The general geometric approach 379

47.8. Having two natural bundles F M , GM and a base-preserving morphism

f : F M ’ GM , we can de¬ne the Lie derivative of f with respect to a vector

¬eld ξ on M . In the case of an arbitrary G, we write

˜ ˜

L(F ξ,Gξ) f =: Lξ f : F M ’ V GM.

(1)

If G is a natural vector bundle, we set

L(F ξ,Gξ) f =: Lξ f : F M ’ GM.

(2)

47.9. Linear vector ¬elds on vector bundles. Consider a vector bundle

p : E ’ M . By 6.11, T p : T E ’ T M is a vector bundle as well. A projectable

vector ¬eld · on E over ξ on M is called a linear vector ¬eld, if · : E ’ T E is a

linear morphism of E ’ M into T E ’ T M over the base map ξ : M ’ T M .

Proposition. · is a linear vector ¬eld on E if and only if its ¬‚ow is formed by

local linear isomorphisms of E.

Proof. Let xi , y p be some ¬ber coordinates on E such that y p are linear coor-

dinates in each ¬ber. By de¬nition, the coordinate expression of a linear vector

¬eld · is

ξ i (x) ‚xi + ·q (x)y q ‚yp .

p

‚ ‚

(1)

Hence the di¬erential equations of the ¬‚ow of · are

dy p

dxi

= ξ i (x), = ·q (x)y q .

p

dt dt

Their solution represents the linear local isomorphisms of E by virtue of the

linearity in y p . On the other hand, if the ¬‚ow of · is linear and we di¬erentiate

it with respect to t, then · must be of the form (1).

¯

47.10. Let · be another linear vector ¬eld on another vector bundle E ’ M

¯

over the same vector ¬eld ξ on the base manifold M . Using ¬‚ows, we de¬ne a

¯

vector ¬eld · — · on the tensor product E — E by

¯

(Fl· ) — (Fl· ).

¯

‚

·—· =

¯ t t

‚t 0

¯

Proposition. · — · is the unique linear vector ¬eld on E — E over ξ satisfying

¯

L·—¯(s — s) = (L· s) — s + s — (L· s)

(1) ¯ ¯ ¯¯

·

¯

for all sections s of E and s of E.

¯

Proof. If 47.9.(1) is the coordinate expression of · and y p = sp (x) is the coordi-

nate expression of s, then the coordinate expression of L· s is

‚sp (x) i

’ ·q (x)sq (x).

p

(2) ‚xi ξ (x)

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

380 Chapter XI. General theory of Lie derivatives

Further, let

ξ i (x) ‚xi + ·b (x)z b ‚za

¯a

‚ ‚

be the coordinate expression of · in some linear ¬ber coordinates xi , z a on

¯

¯ If wpa are the induced coordinates on the ¬bers of E — E and xi = •i (x, t),

¯

E. ¯

y = •q (x, t)y or z = •b (x, t)z is the ¬‚ow of · or · , respectively, then Fl· —Fl·

¯

p p q a a b

¯ ¯ ¯ ¯ t t

is

xi = •i (x, t), wpa = •p (x, t)•a (x, t)wqb .

¯ ¯ ¯b

q

By di¬erentiating at t = 0, we obtain

· — · = ξ i (x) ‚xi + (·q (x)δb + δq ·b (x))wqb ‚wpa .

p a pa

‚ ‚

¯ ¯

Thus, if z a = sa (x) is the coordinate expression of s, we have

¯ ¯

‚sp a a

+ sp ‚ s i ξ i ’ ·q sq sa ’ ·b sp sb .

p

¯a ¯

¯

L·—¯(s — s) =

¯ ‚xi s

¯ ¯

· ‚x

This corresponds to the right hand side of (1).

47.11. On the dual vector bundle E — ’ M of E, we de¬ne the vector ¬eld · —

dual to a linear vector ¬eld · on E by

(Fl· )— .

·— = ‚

’t

‚t 0

Having a vector ¬eld ζ on M and a function f : M ’ R, we can take the zero

vector ¬eld 0R on R and construct the generalized Lie derivative

˜

L(ζ,0R ) f = T f —¦ ζ : M ’ T R = R — R.

Its second component is the usual Lie derivative Lζ f = ζf , i.e. the derivative of

f in the direction ζ.

Proposition. · — is the unique linear vector ¬eld on E — over ξ satisfying

Lξ s, σ = L· s, σ + s, L·— σ

for all sections s of E and σ of E — .

Proof. Let vp be the coordinates on E — dual to y p . By de¬nition, the coordinate

expression of · — is

ξ i (x) ‚xi ’ ·p (x)vq ‚vp .

q

‚ ‚

Then we prove the above proposition by a direct evaluation quite similar to the

proof of proposition 47.10.

47.12. A vector ¬eld · on a manifold M is a section of the tangent bundle T M ,

so that we have de¬ned its Lie derivative Lξ · with respect to another vector

¬eld ξ on M as the second component of T · —¦ ξ ’ T ξ —¦ ·. In 3.13 it is deduced

that Lξ · = [ξ, ·]. Then 47.10 and 47.11 imply, that for the classical tensor ¬elds

the geometrical approach to the Lie di¬erentiation coincides with the algebraic

one.

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

48. Commuting with natural operators 381

47.13. In the end of this section we remark that the operations with linear vec-

tor ¬elds discussed here can be used to de¬ne, in a short way, the corresponding

operation with linear connections on vector bundles. We recall that a linear

connection “ on a vector bundle E ’ M is a section “ : E ’ J 1 E which is

a linear morphism from vector bundle E ’ M into vector bundle J 1 E ’ M .

Using local trivializations of E we ¬nd easily that this condition is equivalent to

the fact that the “-lift “ξ of every vector ¬eld ξ on M is a linear vector ¬eld on

E. By 47.9, the coordinate expression of a linear connection “ on E is

dy p = “p (x)y q dxi .

qi

¯ ¯

Let “ be another linear connection on a vector bundle E ’ M over the same

base with the equations

¯ bi

dz a = “a (x)z b dxi .

Using 47.10 and 47.11, we obtain immediately the following two assertions.

¯ ¯

47.14. Proposition. There is a unique linear connection “ — “ on E — E

satisfying

¯ ¯

(“ — “)(ξ) = (“ξ) — (“ξ)

for every vector ¬eld ξ on M .

47.15. Proposition. There is a unique linear connection “— on E — satisfying

“— (ξ) = (“ξ)— for every vector ¬eld ξ on M .

¯

Obviously, the equations of “ — “ are

dwpa = (“p (x)δb + δq “a (x))wqb dxi

p¯

a

bi

qi

and the coordinate expression of “— is

dvp = ’“q (x)vq dxi .

pi

48. Commuting with natural operators

48.1. The Lie derivative commutes with the exterior di¬erential, i.e. d(LX ω) =

LX (dω) for every exterior form ω and every vector ¬eld X, see 7.9.(5). Our

geometrical analysis of the concept of the Lie derivative leads to a general result,

which clari¬es that the speci¬c property of the exterior di¬erential used in the

above formula is its linearity.

Proposition. Let F and G be two natural vector bundles and A : F G be a

natural linear operator. Then

AM (LX s) = LX (AM s)

(1)

for every section s of F M and every vector ¬eld X on M .

In the special case of a linear natural transformation this is lemma 6.17.

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

382 Chapter XI. General theory of Lie derivatives

Proof. The explicit meaning of (1) is AM (LF X s) = LGX (AM s). By the Peetre

theorem, AM is locally a di¬erential operator, so that AM commutes with limits.

Hence

1

AM F (FlX ) —¦ s —¦ FlX ’ AM s

AM (LF X s) = lim ’t t

t’0 t

1

= lim G(FlX —¦AM s —¦ FlX ’AM s = LGX (AM s)

’t t

t’0 t

by linearity and naturality.

48.2. A reasonable result of this type can be deduced even in the non linear case.

Let F and G be arbitrary natural bundles on Mfm , D : C ∞ (F M ) ’ C ∞ (GM )

be a local regular operator and s : M ’ F M be a section. The generalized

˜ ˜

Lie derivative LX s is a section of V F M , so that we cannot apply D to LX s.

However, we can consider the so called vertical prolongation V D : C ∞ (V F M ) ’

C ∞ (V GM ) of the operator D. This can be de¬ned as follows.

In general, let N ’ M and N ’ M be arbitrary ¬bered manifolds over the

same base and let D : C ∞ (N ) ’ C ∞ (N ) be a local regular operator. Every

local section q of V N ’ M is of the form ‚t 0 st , st ∈ C ∞ (N ) and we set

‚

(Dst ) ∈ C ∞ (V N ).

‚ ‚

(1) V Dq = V D( ‚t s) =

0t ‚t 0

We have to verify that this is a correct de¬nition. By the nonlinear Peetre

theorem the operator D is induced by a map D : J ∞ N ’ N . Moreover each

in¬nite jet has a neighborhood in the inverse limit topology on J ∞ N on which D

depends only on r-jets for some ¬nite r. Thus, there is neighborhood U of x in M

and a locally de¬ned smooth map Dr : J r N ’ N such that Dst (y) = Dr (jy st )

r

for y ∈ U and for t su¬ciently small. So we get

(Dr (jx st )) = T Dr ( ‚t

r

jr s ) = (T Dr —¦ κ)(jx q)

r

‚ ‚

(V D)q(x) = 0xt

‚t 0

where κ is the canonical exchange map, and thus the de¬nition does not depend

on the choice of the family st .

48.3. A local regular operator D : C ∞ (F M ) ’ C ∞ (GM ) is called in¬nitesi-

mally natural if it holds

˜ ˜

LX (Ds) = V D(LX s)

for all X ∈ X(M ), s ∈ C ∞ (F M ).

Proposition. If A : F G is a natural operator, then all operators AM are

in¬nitesimally natural.

Proof. By lemma 47.2, 48.2.(1) and naturality we have

˜ (F (FlX ) —¦ s —¦ FlX

‚

V AM (LF X s) = V AM ’t t

‚t 0

AM F (FlX ) —¦ s —¦ FlX = G(FlX ) —¦ AM s —¦ FlX

‚ ‚

= ’t ’t

t t

‚t 0 ‚t 0

˜

= LGX AM s.

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

48. Commuting with natural operators 383

+

48.4. Let Mfm be the category of oriented m-dimensional manifolds and ori-

entation preserving local di¬eomorphisms.

+

Theorem. Let F and G be two bundle functors on Mfm , M be an oriented m-

dimensional manifold and let AM : C ∞ (F M ) ’ C ∞ (GM ) be an in¬nitesimally

natural operator. Then AM is the value of a unique natural operator A : F G

on M.

We shall prove this theorem in several steps.

48.5. Let us ¬x an in¬nitesimally natural operator D : C ∞ (F Rm ) ’ C ∞ (GRm )

and let us write S and Q for the standard ¬bers F0 Rm and G0 Rm . Since each

local operator is locally of ¬nite order by the nonlinear Peetre theorem, there is

∞ ∞ ∞

the induced map D : Tm S ’ Q. Moreover, at each j0 s ∈ Tm S the application

∞

of the Peetre theorem (with K = {0}) yields a smallest possible order r = χ(j0 s)

r r

such that for every section q with j0 q = j0 s we have Ds(0) = Dq(0), see 23.1.

˜ ∞ ∞

Let us de¬ne Vr ‚ Tm S as the subset of all jets with χ(j0 s) ¤ r. Let Vr be the

˜ ∞ r

interior of Vr in the inverse limit topology and put Ur := πr (Vr ) ‚ Tm S.

∞

The Peetre theorem implies Tm S = ∪r Vr and so the sets Vr form an open

∞

¬ltration of Tm S. On each Vr , the map D factors to a map Dr : Ur ’ Q.

w Qu ’

’’

”

’

’ ’’ ’’

’’

D3

D2

D1

Uu Uu Uu

1 2 3

∞ ∞ ∞

π1 π2 π3

D

yy xw V 99999w V y

y w ···

I

V1

xx 9 9

2 3

u x B9

x9

∞

Tm S

Since there are the induced actions of the jet groups Gr+k on Tm S (here k is

r

m

the order of F ), we have the fundamental ¬eld mapping ζ (r) : gr+k ’ X(Tm S)

r

m

and we write ζ Q for the fundamental ¬eld mapping on Q. There is an analogy

to 34.3.

Lemma. For all X ∈ gr+k , j0 s ∈ Tm S it holds

r r

m

(r) r r˜

ζX (j0 s) = κ(j0 (L’X s)).

r

Proof. Write » for the action of the jet group on Tm S. We have

(r) r r r X X

‚ ‚

‚t 0 »(exptX)(j0 s) = ‚t 0 j0 (F (Flt ) —¦ s —¦ Fl’t )

ζX (j0 s) =

r˜

κ(j0 ( ‚t 0 (F (FlX ) —¦ s —¦ FlX ))) = κ(j0 (L’X s)).

r‚

= ’t

t

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

384 Chapter XI. General theory of Lie derivatives

(r)

Q

48.6. Lemma. For all r ∈ N and X ∈ gr+k we have T Dr —¦ ζX = ζX —¦ Dr on

m

Ur .

Proof. Recall that (V D)q(0) = (T Dr —¦ κ)(j0 q) for all j0 q ∈ κ’1 (T Ur ). Using

r r

the above lemma and the in¬nitesimal naturality of D we compute

(r) r˜ ˜

r

(T Dr —¦ ζX )(j0 s) = T Dr (κ(j0 (L’X s))) = V D(L’X s)(0) =

Q Q

˜ r

= L’X (Ds)(0) = ζX (Ds(0)) = ζX (Dr (j0 s)).

48.7. Lemma. The map D : Tm S ’ Q is G∞ + -equivariant.

∞

m

Proof. Given a = j0 f ∈ G∞ + and y = j0 s ∈ Tm S we have to show D(a.y) =

∞ ∞ ∞

m

a.D(y). Each a is a composition of a jet of a linear map f and of a jet from

∞ ∞

the kernel B1 of the jet projection π1 . If f is linear, then there are linear

maps gi , i = 1, 2, . . . , l, lying in the image of the exponential map of G1 such

m

∞

that f = g1 —¦ . . . —¦ gl . Since Tm S = ∪r Vr there must be an r ∈ N such that y

∞ ∞

and all elements (j0 gp —¦ . . . —¦ j0 gl ) · y are in Vr for all p ¤ l. Thus, D(a.y) =

r+k r+k

r r

Dr (j0 f.j0 s) = j0 f.Dr (j0 s) = a.D(y), for Dr preserves all the fundamental

¬elds.

r

Since the whole kernel B1 lies in the image of the exponential map for each

∞ ∞

r < ∞, an analogous consideration for j0 f ∈ B1 completes the proof of the

lemma.

+

48.8. Lemma. The natural operator A on Mfm which is determined by the

G∞ + -equivariant map D coincides on Rm with the operator D.

m

Proof. There is the associated map A : J ∞ F Rm ’ GRm to the operator ARm .

Let us write A0 for its restriction (J ∞ F )0 Rm ’ G0 Rm and similarly for the

map D corresponding to the original operator D. Now let tx : Rm ’ Rm be

the translation by x. Then the map A (and thus the operator A) is uniquely

determined by A0 since by naturality of A we have (t’x )— —¦ ARm —¦ (tx )— = ARm .

But tx is the ¬‚ow at time 1 of the constant vector ¬eld X. For every vector ¬eld

X and section s we have

˜ ˜

LX ((FlX )— s) = LX (F (FlX ) —¦ s —¦ FlX ) = ‚t (F (FlX ) —¦ s —¦ FlX )

‚

’t ’t

t t t

˜ X— ˜

X X

= T (F (Fl’t )) —¦ LX s —¦ Flt = (Flt ) (LX s)

and so using in¬nitesimal naturality, for every complete vector ¬eld X we com-

pute

(FlX )— (D(FlX )— s) =

‚

’t t

‚t

˜ ˜

= ’(FlX )— LX (D(FlX )— s) + (FlX )— (V D)((FlX )— LX s) =

’t ’t

t t

˜ ˜

= (FlX )— ’LX (D(FlX )— s) + (V D)(LX ((FlX )— s)) = 0.

’t t t

Thus (t’x )— —¦ D —¦ (tx )— = D and since A0 = D0 this completes the proof.

Lemmas 48.7 and 48.8 imply the assertion of theorem 48.4. Indeed, if M = Rm

we get the result immediately and it follows for general M by locality of the

operators in question.

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

48. Commuting with natural operators 385

48.9. As we have seen, if F is a natural vector bundle, then V F is naturally

equivalent to F • F and the second component of our general Lie derivative is

just the usual Lie derivative. Thus, the condition of the in¬nitesimal naturality

becomes the usual form D —¦ LX = LX —¦ D if D : C ∞ (F M ) ’ C ∞ (GM ) is linear.

More generally, if F is a sum F = E1 • · · · • Ek of k natural vector bundles,

G is a natural vector bundle and D is k-linear, then we have

˜ D F (FlX ) —¦ (s1 , . . . , sk ) —¦ FlX

‚

pr2 —¦ V D(LX (s1 , . . . , sk )) = ’t t

‚t 0

k

D(s1 , . . . , LX si , . . . , sk ).

=

i=1

Hence for the k-linear operators we have

Corollary. Every natural k-linear operator A : E1 • · · · • Ek F satis¬es

k

LX AM (s1 , . . . , sk ) = AM (s1 , . . . , LX si , . . . , sk )

(1) i=1

for all s1 ∈ C ∞ E1 M ,. . . ,sk ∈ C ∞ Ek M , X ∈ C ∞ T M .

Formula (1) covers, among others, the cases of the Fr¨licher-Nijenhuis bracket

o

and the Schouten bracket discussed in 30.10 and 8.5.

48.10. The converse implication follows immediately for vector bundle functors

+

on Mfm . But we can prove more.

Let E1 , . . . , Ek be r-th order natural vector bundles corresponding to actions

»i of the jet group Gr on standard ¬bers Si , and assume that with the re-

m

1

stricted actions »i |Gm the spaces Si are invariant subspaces in spaces of the

form •j (—pj Rm — —qj Rm— ). In particular this applies to all natural vector bun-

dles which are constructed from the tangent bundle. Given any natural vector

bundle F we have

Theorem. Every local regular k-linear operator

AM : C ∞ (E1 M ) • · · · • C ∞ (Ek M ) ’ C ∞ (F M )

over an m-dimensional manifold M which satis¬es 48.9.1 is a value of a unique

natural operator A on Mfm .

The theorem follows from the theorem 48.4 and the next lemma

+

Lemma. Every k-linear natural operator A : E1 • · · · • Ek F on Mfm

extends to a unique natural operator on Mfm .

Let us remark, the proper sense of this lemma is that every operator in ques-

tion obeys the necessary commutativity properties with respect to all local di¬eo-

morhpisms between oriented m-manifolds and hence determines a unique natural

operator over the whole Mfm .

Proof. By the multilinear Peetre theorem A is of some ¬nite order . Thus A is

determined by the associated k-linear (Gr+ )+ -equivariant map A : Tm S1 — . . . —

m

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

386 Chapter XI. General theory of Lie derivatives

Tm Sk ’ Q. Recall that the jet group Gr+ is the semidirect product of GL(m)

m

r+

and the kernel B1 , while (Gr+ )+ is the semidirect product of the connected

m

r+

+

component GL (m) of the unit and the same kernel B1 . Thus, in particular

the map A : Tm S1 — . . . — Tm Sk ’ Q is k-linear and GL+ (m)-equivariant. By

the descriptions of (Gr+ )+ and Gr+ above we only have to show that any such

m m

map is GL(m) equivariant, too. Using the standard polarization technique we

can express the map A by means of a GL+ (m) invariant tensor. But looking

at the proof of the Invariant tensor theorem one concludes that the spaces of

GL+ (m) invariant and of GL(m) invariant tensors coincide, so the map A is

GL(m) equivariant.

48.11. Lie derivatives of sector forms. At the end of this section we present

an original application of proposition 48.1. This is related with the di¬erentiation

of a certain kind of r-th order forms on a manifold M . The simplest case is

the ˜ordinary™ di¬erential of a classical 1-form on M . Such a 1-form ω can be

considered as a map ω : T M ’ R linear on each ¬ber. Beside its exterior

di¬erential dω : §2 T M ’ R, E. Cartan and some other classical geometers used

another kind of di¬erentiating ω in certain concrete geometric problems. This

was called the ordinary di¬erential of ω to be contrasted from the exterior one.

We can de¬ne it by constructing the tangent map T ω : T T M ’ T R = R — R,

which is of the form T ω = (ω, δω). The second component δω : T T M ’ R is

said to be the (ordinary) di¬erential of ω . In an arbitrary order r we consider

the r-th iterated tangent bundle T r M = T (· · · T (T M ) · · · ) (r times) of M .

The elements of T r M are called the r-sectors on M. Analogously to the case

r = 2, in which we have two well-known vector bundle structures pT M and

T pM on T T M over T M , on T r M there are r vector bundle structures pT r’1 M ,

T pT r’2 M , . . . , T · · · T pM (r ’ 1 times) over T r’1 M .

De¬nition. A sector r-form on M is a map σ : T r M ’ R linear with respect

to all r vector bundle structures on T r M over T r’1 M .

A sector r-form on M at a point x is the restriction of a sector r-form an

M to the ¬ber (T r M )x . Denote by T— M ’ M the ¬ber bundle of all sector

r

r-forms at the individual points on M , so that a sector r-form on M is a section

r r

of T— M . Obviously, T— M ’ M has a vector bundle structure induced by the

linear combinations of R-valued maps. If f : M ’ N is a local di¬eomorphism

and A : (T r M )x ’ R is an element of (T— M )x , we de¬ne (T— f )(A) = A —¦

r r

(T r f ’1 )f (x) : (T r N )f (x) ’ R, where f ’1 is constructed locally. Since T r f is a

r

linear morphism for all r vector bundle structures, (T— f )(A) is an element of

r r

(T— N )f (x) . Hence T— is a natural bundle. In particular, for every vector ¬eld X

on M and every sector r-form σ on M we have de¬ned the Lie derivative

r

LX σ = LT—r X σ : M ’ T— M.

For every sector r-form σ : T r ’ R we can construct its tangent map T σ : T T r M

’ T R = R — R, which is of the form (σ, δσ). Since the tangent functor preserves

vector bundle structures,

δσ : T r+1 M ’ R

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

49. Lie derivatives of morphisms of ¬bered manifolds 387

is linear with respect to all r + 1 vector bundle structures on T r+1 M over T r M ,

so that this is a sector (r + 1)-form on M .

48.12. De¬nition. The operator δ : C ∞ T— M ’ C ∞ T— M will be called the

r r+1

di¬erential of sector forms.

By de¬nition, δ is a natural operator. Obviously, δ is a linear operator as

well. Applying proposition 48.1, we obtain

48.13. Corollary. δ commutes with the Lie di¬erentiation, i.e.

δ(LX σ) = LX (δσ)

for every sector r-form σ and every vector ¬eld X.

49. Lie derivatives of morphisms of ¬bered manifolds

We are going to show a deeper application of the geometrical approach to

Lie di¬erentiation in the higher order variational calculus in ¬bered manifolds.

For the sake of simplicity we restrict ourselves to the geometrical aspects of the

problem.

49.1. By an r-th order Lagrangian on a ¬bered manifold p : Y ’ M we mean

a base-preserving morphism

» : J r Y ’ Λm T — M, m = dim M.

For every section s : M ’ Y , we obtain the induced m-form » —¦ j r s on M .

We underline that from the geometrical point of view the Lagrangian is not a

function on J r Y , since m-forms (and not functions) are the proper geometric

objects for integration on X. If xi , y p are local ¬ber coordinates on Y , the in-

duced coordinates on J r Y are xi , y± for all multi indices |±| ¤ r. The coordinate

p

expression of » is

L(xi , y± )dxi § · · · § dxm

p

but such a decomposition of » into a function on J r Y and a volume element on

M has no geometric meaning.

If · is a projectable vector ¬eld on Y over ξ on M , we can construct, similarly

to 47.8.(2), the Lie derivative L· » of » with respect to ·

L· » := L(J r ·,Λm T — ξ) » : J r Y ’ Λm T — M

which coincides with the classical variation of » with respect to ·.

49.2. The geometrical form of the Euler equations for the extremals of » is

the so-called Euler morphism E(») : J 2r Y ’ V — Y — Λm T — M . Its geometric