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