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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

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

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
ˆy 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) =

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

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