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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

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