introduce an appropriate geometric operation.

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

388 Chapter XI. General theory of Lie derivatives

De¬nition. Given a base-preserving morphism • : J q Y ’ Λk T — M , its formal

exterior di¬erential D• : J q+1 Y ’ Λk+1 T — M is de¬ned by

D•(jx s) = d(• —¦ j q s)(x)

q+1

for every local section s of Y , where d means the exterior di¬erential at x ∈ M

of the local exterior k-form • —¦ j q s on M .

If f : J q Y ’ R is a function, we have a coordinate decomposition

Df = (Di f )dxi

‚f p

‚f

where Di f = ‚xi + |±|¤q ‚y± y±+i : J q+1 Y ’ R is the so called formal (or total)

p

derivative of f , provided ±+i means the multi index arising from ± by increasing

its i-th component by 1. If the coordinate expression of • is ai1 ...ik dxi1 §· · ·§dxik ,

then

D• = Di ai1 ...ik dxi § dxi1 § · · · § dxik .

To determine the Euler morphism, it su¬ces to discuss the variation L· » with

‚

respect to the vertical vector ¬elds. If · p (x, y) ‚yp is the coordinate expression

of such a vector ¬eld, then the coordinate expression of J r · is

‚

(D± · p ) p

‚y±

|±|¤r

where D± means the iterated formal derivative with respect to the multi index

±. In the following assertion we do not indicate explicitly the pullback of L· »

to J 2r Y .

49.3. Proposition. For every r-th order Lagrangian » : J r Y ’ Λm T — M ,

there exists a morphism K(») : J 2r’1 Y ’ V — J r’1 Y — Λm’1 T — M and a unique

morphism E(») : J 2r Y ’ V — Y — Λm T — M satisfying

J r’1 ·, K(»)

L· » = D

(1) + ·, E(»)

for every vertical vector ¬eld · on Y .

Proof. Write ω = dx1 § · · · § dxm , ωi = i ±i p

kp dy± — ωi ,

ω, K(») =

‚ |±|¤r’1

‚xi

E(») = Ep dy p — ω. Since L· » = T » —¦ J r ·, the coordinate expression of L· » is

‚L p

(2) p D± · .

‚y±

|±|¤r

Comparing the coe¬cients of the individual expressions D± · p in (1), we ¬nd the

following relations

Lj1 ...jr = Kp 1 ...jr )

(j

p

.

.

.

Lj1 ...jq = Di Kp1 ...jq i + Kp 1 ...jq )

j (j

p

(3)

.

.

.

Lj = Di Kp + Kp

ji i

p

i

Lp = Di Kp + Ep

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49. Lie derivatives of morphisms of ¬bered manifolds 389

j ...j j ...j i

where Lp1 q = ±! ‚y± and Kp1 q = ±! kp , provided ± is the multi index

‚L ±i

p

q! q!

corresponding to j1 . . . jq , |±| = q. Evaluating Ep by a backward procedure, we

¬nd

‚L

(’1)|±| D±

(4) Ep = p

‚y±

|±|¤r

for any K™s, so that the Euler morphism is uniquely determined. The quantities

j ...j i

Kp1 q , which are not symmetric in the last two superscripts, are not uniquely

determined by virtue of the symmetrizations in (3). Nevertheless, the global

existence of a K(») can be deduced by a recurrent construction of some sections

of certain a¬ne bundles. This procedure is straightforward, but rather technical.

The reader is referred to [Kol´ˇ, 84b]

ar

We remark that one can prove easily by proposition 49.3 that a section s of

Y is an extremal of » if and only if E(») —¦ j 2r s = 0.

49.4. The construction of the Euler morphism can be viewed as an operator

transforming every base-preserving morphism » : J r Y ’ Λm T — M into a base-

preserving morphism E(») : J 2r Y ’ V — Y — Λm T — M . Analogously to L· », the

Lie derivative of E(») with respect to a projectable vector ¬eld · on Y over ξ

on M is de¬ned by

L· E(») := L(J 2r ·,V — ·—Λm T — ξ) E(»).

An important question is whether the Euler operator commutes with Lie

di¬erentiation. From the uniqueness assertion in proposition 49.3 it follows that

E is a natural operator and from 49.3.(4) we see that E is a linear operator.

49.5. We ¬rst deduce a general result of such a type. Consider two natural

bundles over m-manifolds F and H, a natural surjective submersion q : H ’ F

and two natural vector bundles over m-manifolds G and K.

Proposition. Every linear natural operator A : (F, G) (H, K) satis¬es

Lξ (Af ) = A(Lξ f )

for every base-preserving morphism f : F M ’ GM and every vector ¬eld ξ on

M.

Proof. By 47.8.(2) and an analogy of 47.4.(1), we have

G(Flξ ) —¦ f —¦ F (Flξ ) ’ f .

1

Lξ f = lim ’t t

t’0 t

Since A commutes with limits by 19.9, we obtain by linearity and naturality

K(Flξ ) —¦ Af —¦ H(Flξ ) ’ Af = Lξ (Af )

1

ALξ (f ) = lim ’t t

t’0 t

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

390 Chapter XI. General theory of Lie derivatives

49.6. Our original problem on the Euler morphism can be discussed in the same

way as in the proof of proposition 49.5, but the functors in question are de¬ned

on the local isomorphisms of ¬bered manifolds. Hence the answer to our problem

is a¬rmative.

Proposition. It holds

L· E(») = E(L· »)

for every r-th order Lagrangian » and every projectable vector ¬eld · on Y .

49.7. A projectable vector ¬eld · on Y is said to be a generalized in¬nitesimal

automorphism of an r-th order Lagrangian », if L· E(») = 0. By proposition

49.6 we obtain immediately the following interesting assertion.

Corollary. Higher order Noether-Bessel-Hagen theorem. A projectable

vector ¬eld · is a generalized in¬nitesimal automorphism of an r-th order La-

grangian » if and only if E(L· ») = 0.

49.8. An in¬nitesimal automorphism of » means a projectable vector ¬eld ·

satisfying L· » = 0. In particular, corollary 49.7 and 49.3.(4) imply that every

in¬nitesimal automorphism is a generalized in¬nitesimal automorphism.

50. The general bracket formula

50.1. The generalized Lie derivative of a section s of an arbitrary ¬bered man-

ifold Y ’ M with respect to a projectable vector ¬eld · on Y over ξ on M is

¯

˜

a section L· s : M ’ V Y . If · is another projectable vector ¬eld on Y over ξ

¯

on M , a general problem is whether there exists a reasonable formula for the

˜·

generalized Lie derivative L[·,¯] s of s with respect to the bracket [·, · ]. Since

¯

L· s is not a section of Y , we cannot construct the generalized Lie derivative of

˜

L· s with respect to · . However, in the case of a vector bundle E ’ M we have

¯

de¬ned L· L· s : M ’ E.

¯

Proposition. If · and · are two linear vector ¬elds on a vector bundle E ’ M ,

¯

then

L[·,¯] s = L· L· s ’ L· L· s

(1) ¯ ¯

·

for every section s of E.

At this moment, the reader can prove this by direct evaluation using 47.10.(2).

But we shall give a conceptual proof resulting from more general considerations

in 50.5. By direct evaluation, the reader can also verify that the above proposi-

tion does not hold for arbitrary projectable vector ¬elds · and · on E. However,

¯

if F M is a natural vector bundle, then Fξ is a linear vector ¬eld on F M for

every vector ¬eld ξ on M , so that we have

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

50. The general bracket formula 391

Corollary. If F M is a natural vector bundle, then

L[ξ,ξ] s = Lξ Lξ s ’ Lξ Lξ s

¯ ¯ ¯

¯

for every section s of F M and every vector ¬elds ξ, ξ on M .

This result covers the classical cases of Lie di¬erentiation.

50.2. We are going to discuss the most general situation. Let M , N be two

¯

manifolds, f : M ’ N be a map, ξ, ξ be two vector ¬elds on M and ·, · be two

¯

vector ¬elds on N . Our problem is to ¬nd a reasonable expression for

˜¯·

L([ξ,ξ],[·,¯]) f : M ’ T N.

(1)

˜

Since L(ξ,·) f is a map of M into T N , we cannot construct its Lie derivative

¯¯

with respect to the pair (ξ, · ), since · is a vector ¬eld on N and not on T N .

¯

However, if we replace · by its ¬‚ow prolongation T · , we have de¬ned

¯ ¯

˜¯ ¯ ˜

L(ξ,T ·) L(ξ,·) f : M ’ T T N.

(2)

On the other hand, we can construct

˜ ˜ ¯·

L(ξ,T ·) L(ξ,¯) f : M ’ T T N.

(3)

Now we need an operation transforming certain special pairs of the elements

of the second tangent bundle T T Q of any manifold Q into the elements of T Q.

Consider A,B ∈ T Tz Q satisfying

(4) πT Q (A) = T πQ (B) and T πQ (A) = πT Q (B).

Since the canonical involution κ : T T Q ’ T T Q exchanges both projections, we

have πT Q (A) = πT Q (κB), T πQ (A) = T πQ (κB). Hence A and κB are in the

same ¬ber of T T Q with respect to projection πT Q and their di¬erence A ’ κB

satis¬es T πQ (A ’ κB) = 0. This implies that A ’ κB is a tangent vector to the

¬ber Tz Q of T Q and such a vector can be identi¬ed with an element of Tz Q,

which will be denoted by A · B.

50.3. De¬nition. A · B ∈ T Q is called the strong di¬erence of A, B ∈ T T Q

satisfying 50.2.(4).

In the case Q = Rm we have T T Rm = Rm — Rm — Rm — Rm . If A =

(x, a, b, c) ∈ T T Rm , then B satisfying 50.2.(4) is of the form B = (x, b, a, d) and

one ¬nds easily

A · B = (x, c ’ d)

(1)

From the geometrical de¬nition of the strong di¬erence it follows directly

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

392 Chapter XI. General theory of Lie derivatives

Lemma. If A, B ∈ T T Q satisfy 50.2.(4) and f : Q ’ P is any map, then

T T f (A), T T f (B) ∈ T T P satisfy the condition of the same type and it holds

T T f (A) · T T f (B) = T f (A · B) ∈ T P.

50.4. We are going to deduce the bracket formula for generalized Lie derivatives.

First we recall that lemma 6.13 reads

¯ ¯ ¯

[ζ, ζ] = T ζ —¦ ζ · T ζ —¦ ζ

(1)

for every two vector ¬elds on the same manifold.

The maps 50.2.(2) and 50.2.(3) satisfy the condition for the existence of

˜¯ ¯ ˜ ˜

the strong di¬erence. Indeed, we have πT N —¦ L(ξ,T ·) L(ξ,·) f = L(ξ,·) f since

˜ ˜

any generalized Lie derivative of L(ξ,·) f is a vector ¬eld along L(ξ,·) f . On

¯

the other hand, T πN —¦ (L(ξ,T ·) L(ξ,·) f ) = T πN ‚ T (Fl· ) —¦ L(ξ,·) f —¦ Flξ =

¯

˜¯ ¯ ˜ ˜

’t

t

‚t 0

¯

(Fl·

¯

Flξ ) ˜¯¯

‚

—¦f —¦ = L(ξ, · )f .

’t t

‚t 0

Proposition. It holds

˜¯· ˜ ˜ ¯· ˜¯ ¯ ˜

L([ξ,ξ],[·,¯]) f = L(ξ,T ·) L(ξ,¯) f · L(ξ,T ·) L(ξ,·) f

(2)

Proof. We ¬rst recall that the ¬‚ow prolongation of · satis¬es T · = κ —¦ T ·. By

¯

˜¯ ¯ ˜

47.1.(1) we obtain L(ξ,T ·) L(ξ,·) f = T (T f —¦ ξ ’ · —¦ f ) —¦ ξ ’ T · —¦ (T f —¦ ξ ’ · —¦ f ) =

¯

¯ ¯

T T f —¦ T ξ —¦ ξ ’ T · —¦ T f —¦ ξ ’ κ —¦ T · —¦ T f —¦ ξ + κ —¦ T · —¦ · —¦ f as well as a similar

¯ ¯

˜(ξ,T ·) L(ξ,¯) f . Using (1) we deduce that the right hand side of

˜ ¯·

expression for L

¯ ¯ ¯

(2) is equal to T f —¦(T ξ —¦ ξ ·T ξ —¦ξ)’(T · —¦ · ·T · —¦·)f = T f —¦[ξ, ξ]’[·, · ]—¦f .

¯ ¯ ¯

50.5. In the special case of a section s : M ’ Y of a ¬bered manifold Y ’ M

and of two projectable vector ¬elds · and · on Y , 50.4.(2) is specialized to

¯

˜· ˜ ˜¯ ˜ ¯˜

L[·,¯] s = LV· L· s · LV · L· s

(1)

where V· or V · is the restriction of T · or T · to the vertical tangent bundle

¯ ¯

V Y ‚ T Y . Furthermore, if we have a vector bundle E ’ M and a linear vector

¬eld on E, then V· is of the form V· = · • ·, since the tangent map of a linear

map coincides with the original map itself. Thus, if we separate the restricted Lie

derivatives in (1) in the case · and · are linear, we ¬nd L[·,¯] s = L· L· s’L· L· s.

¯ ¯ ¯

·

This proves proposition 50.1.

Remarks

The general concept of Lie derivative of a map f : M ’ N with respect to a

pair of vector ¬elds on M and N was introduced by [Trautman, 72]. The oper-

ations with linear vector ¬elds from the second half of section 47 were described

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

Remarks 393

in [Janyˇka, Kol´ˇ, 82]. In the theory of multilinear natural operators, the com-

s ar

mutativity with the Lie di¬erentiation is also used as the starting point, see

[Kirillov, 77, 80]. Proposition 48.4 was proved by [Cap, Slov´k, 92]. According

a

to [Janyˇka, Modugno, to appear], there is a link between the in¬nitesimally nat-

s

ural operators and certain systems in the sense of [Modugno, 87a]. The concept

of a sector r-form was introduced in [White, 82].

The Lie derivatives of morphisms of ¬bered manifolds were studied in [Kol´ˇ,ar

82a] in connection with the higher order variational calculus in ¬bered manifolds.

We remark that a further analysis of formula 49.3.(3) leads to an interesting fact

that a Lagrangian of order at least three with at least two independent variables

does not determine a unique Poincar´-Cartan form, but a family of such forms

e

only, see e.g. [Kol´ˇ, 84b], [Saunders, 89]. The general bracket formula from

ar

section 50 was deduced in [Kol´ˇ, 82c].

ar

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

394

CHAPTER XII.

GAUGE NATURAL BUNDLES

AND OPERATORS

In chapters IV and V we have explained that the natural bundles coincide

with the associated ¬ber bundles to higher order frame bundles on manifolds.

However, in both di¬erential geometry and mathematical physics one can meet

¬ber bundles associated to an ˜abstract™ principal bundle with an arbitrary struc-

ture group G. If we modify the idea of bundle functor to such a situation, we

obtain the concept of gauge natural bundle. This is a functor on principal ¬ber

bundles with structure group G and their local isomorphisms with values in ¬ber

bundles, but with ¬bration over the original base manifold. The most important

examples of gauge natural bundles and of natural operators between them are

related with principal connections. In this chapter we ¬rst develop a description

of all gauge natural bundles analogous to that in chapter V. In particular, we

prove that the regularity condition is a consequence of functoriality and locality

and that any gauge natural bundle is of ¬nite order. We also present sharp

estimates of the order depending on the dimensions of the standard ¬bers. So

the r-th order gauge natural bundles coincide with the ¬ber bundles associated

to r-th principal prolongations of principal G-bundles (see 15.3), which are in

r

bijection with the actions of the group Wm G on manifolds.

Then we discuss a few concrete problems on ¬nding gauge natural opera-

tors. The geometrical results of section 52 are based on a generalization of the

Utiyama theorem on gauge natural Lagrangians. First we determine all gauge

natural operators of the curvature type. In contradistinction to the essential

uniqueness of the curvature operator on general connections, this result depends

on the structure group in a simple way. Then we study the di¬erential forms

of Chern-Weil type with values in an arbitrary associated vector bundle. We

¬nd it interesting that the full list of all gauge natural operators leads to a new

geometric result in this case. Next we determine all ¬rst order gauge natural

operators transforming principal connections to the tangent bundle. In the last

section we ¬nd all gauge natural operators transforming a linear connection on

a vector bundle and a classical linear connection on the base manifold into a

classical linear connection on the total space.

51. Gauge natural bundles

We are going to generalize the description of all natural bundles F : Mfm ’

FM derived in sections 14 and 22 to the gauge natural case. Since the concepts

and considerations are very similar to some previous ones, we shall proceed in a

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

51. Gauge natural bundles 395

rather brief style.

51.1. Let B : FM ’ Mf be the base functor. Fix a Lie group G and recall

the category PBm (G), whose objects are principal G-bundles over m-manifolds

¯

and whose morphisms are the morphisms of principal G-bundles f : P ’ P with

¯

the base map Bf : BP ’ B P lying in Mfm .

De¬nition. A gauge natural bundle over m-dimensional manifolds is a functor

F : PBm (G) ’ FM such that

(a) every PBm (G)-object π : P ’ BP is transformed into a ¬bered manifold

qP : F P ’ BP over BP ,

¯

(b) every PBm (G)-morphism f : P ’ P is transformed into a ¬bered mor-

¯

phism F f : F P ’ F P over Bf ,

(c) for every open subset U ‚ BP , the inclusion i : π ’1 (U ) ’ P is trans-

’1

formed into the inclusion F i : qP (U ) ’ F P .

If we intend to point out the structure group G, we say that F is a G-natural

bundle.

¯ r r

51.2. If two PBm (G)-morphisms f , g : P ’ P satisfy jy f = jy g at a point

y ∈ Px of the ¬ber of P over x ∈ BP , then the fact that the right translations

r r

of principal bundles are di¬eomorphisms implies jz f = jz g for every z ∈ Px . In

this case we write jr f = jr g.

x x

De¬nition. A gauge natural bundle F is said to be of order r, if jr f = jr g

x x

implies F f |Fx P = F g|Fx P .

51.3. De¬nition. A G-natural bundle F is said to be regular if every smoothly

parameterized family of PBm (G)-morphisms is transformed into a smoothly pa-

rameterized family of ¬bered maps.

51.4. Remark. By de¬nition, a G-natural bundle F : PBm (G) ’ FM satis¬es

B —¦ F = B and the projections qP : F P ’ BP form a natural transformation

q : F ’ B.

In general, we can consider a category C over ¬bered manifolds, i.e. C is

endowed with a faithful functor m : C ’ FM. If C admits localization of objects

and morphisms with respect to the preimages of open subsets on the bases with

analogous properties to 18.2, we can de¬ne the gauge natural bundles on C as

functors F : C ’ FM satisfying B—¦F = B—¦m and the locality condition 51.1.(c).

Let us mention the categories of vector bundles as examples. The di¬erent way

of localization is the source of a crucial di¬erence between the bundle functors

on categories over manifolds and the (general) gauge natural bundles. For any

¯

two ¬bered maps f , g : Y ’ Y we write jr f = jr g, x ∈ BY , if jy f = jy g for

r r

x x

all y ∈ Yx . Then we say that f and g have the same ¬ber r-jet at x. The space

¯ ¯

of ¬ber r-jets between C-objects Y and Y is denoted by Jr (Y, Y ). For a general

category C over ¬bered manifolds the ¬niteness of the order of gauge natural

bundles is expressed with the help of the ¬ber jets. The description of ¬nite

order bundle functors as explained in section 18 could be generalized now, but

there appear di¬culties connected with the (generally) in¬nite dimension of the

corresponding jet groups. Since we will need only the gauge natural bundles

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

396 Chapter XII. Gauge natural bundles and operators

on PBm (G) in the sequel, we will restrict ourselves to this category. Then the

description will be quite analogous to that of classical natural bundles. Some

basic steps towards the description in the general case were done in [Slov´k, 86]

a

where the in¬nite dimensional constructions are performed with the help of the

smooth spaces in the sense of [Fr¨licher, 81].

o

51.5. Examples.

(1) The choice G = {e} reproduces the natural bundles on Mfm

(2) The functors Qr : PBm (G) ’ FM of r-th order principal connections

mentioned in 17.4 are examples of r-th order regular gauge natural bundles.

(3) The gauge natural bundles W r : PBm (G) ’ PB m (Wm G) of r-th principal

r

prolongation de¬ned in 15.3 play the same role as the frame bundles P r : Mfm ’

FM did in the description of natural bundles.

r

(4) For every manifold S with a smooth left action of Wm G, the construction

of associated bundles to the principal bundles W r P yields a regular gauge natural

bundle L : PBm (G) ’ FM. We shall see that all gauge natural bundles are of

this type.

51.6. Proposition. Every r-th order regular gauge natural bundle is a ¬ber

bundle associated to W r .

Proof. Analogously to the case of natural bundles, an r-th order regular gauge

natural bundle F is determined by the system of smooth associated maps

¯ ¯

FP,P : Jr (P, P ) —BP F P ’ F P

¯

and the restriction of FRm —G,Rm —G to the ¬ber jets at 0 ∈ Rm yields an action

of Wm G = Jr (Rm — G, Rm — G)0 on the ¬ber S = F0 (Rm — G). The same

r

0

considerations as in 14.6 complete now the proof.

51.7. Theorem. Let F : PBm (G) ’ Mf be a functor endowed with a natu-

ral transformation q : F ’ B such that the locality condition 51.1.(c) holds.

Then S := (qRm —G )’1 (0) is a manifold of dimension s ≥ 0 and for every

P ∈ ObPBm (G), the mapping qP : F P ’ BP is a locally trivial ¬ber bun-

dle with standard ¬ber S, i.e. F : PBm (G) ’ FM. The functor F is a regular

gauge natural bundle of a ¬nite order r ¤ 2s + 1. If moreover m > 1, then

s s

r ¤ max{

(1) , + 1}.

m’1 m

All these estimates are sharp.

Brie¬‚y, every gauge natural bundle on PB m (G) with s-dimensional ¬bers is

one of the functors de¬ned in example 51.5.(4) with r bounded by the estimates

from the theorem depending on m and s but not on G. The proof is based on

the considerations from chapter V and it will require several steps.

51.8. Let us point out that the restriction of any gauge natural bundle F to

trivial principal bundles M — G and to morphisms of the form f — id : M — G ’

N — G can be viewed as a natural bundle Mfm ’ FM. Hence the action „ of

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

51. Gauge natural bundles 397

the abelian group of ¬ber translations tx : Rm — G ’ Rm — G, (y, a) ’ (x + y, a),

i.e. „x = F tx , is a smooth action by 20.3. This implies immediately the assertion

on ¬ber bundle structure in 51.7, cf. 20.3. Further, analogously to 20.5.(1) we

¬nd that the regularity of F follows if we verify the smoothness of the induced

action of the morphisms keeping the ¬ber over 0 ∈ Rm on the standard ¬ber

S = F0 (Rm — G).

51.9. Lemma. Let U ‚ S be a relatively compact open set and write

F •(U ) ‚ S

QU =

•

where the union goes through all • ∈ PB m (G)(Rm — G, Rm — G) with •0 (0) =

(0). Then there is r ∈ N such that for all z ∈ QU and all PB m (G)-morphisms

•, ψ : Rm — G ’ Rm — G, •0 (0) = ψ0 (0) = 0, the condition jr • = jr ψ implies

0 0

F •(z) = F ψ(z).

Proof. Every morphism • : Rm — G ’ Rm — G is identi¬ed with the couple •0 ∈

˜

C ∞ (Rm , Rm ), • ∈ C ∞ (Rm , G). So F induces an operator F : C ∞ (Rm , Rm —

¯

G) ’ C ∞ (F (Rm — G), F (Rm — G)) which is qRm —G -local and the map qRm —G

is locally non-constant. Consider the constant map e : Rm ’ G, x ’ e, and the

ˆ

m m

map idRm — e : R ’ R — G corresponding to idRm —G . By corollary 19.8, there

ˆ

˜

r r

is r ∈ N such that j0 f = j0 (idRm — e) implies F f (z) = z for all z ∈ U . Hence if

ˆ

jr • = jr idRm —G , then F •(z) = z for all z ∈ U and the easy rest of the proof is

0 0

quite analogous to 20.4.

51.10. Proposition. Every gauge natural bundle is regular.

Proof. The whole proof of 20.5 goes through for gauge natural bundles if we

∞

choose local coordinates near to the unit in G and replace the elements j0 fn ∈

G∞ by the couples (j0 fn , j0 •n ) ∈ G∞ Tm G and idRm by idRm — e. Let us

∞ ∞ ∞

¯ ˆ

m m

remark that also „x gets the new meaning of F (tx ).

51.11. Since every natural bundle F : Mf ’ FM can be viewed as the gauge

¯

natural bundle F = F —¦ B : PBm (G) ’ FM, the estimates from theorem 51.7

must be sharp if they are correct, see 22.1. Further, the considerations from 22.1

applied to our situation show that we complete the proof of 51.7 if we deduce

r

that every smooth action of Wm G on a smooth manifold S factorizes to an action

k

of Wm G, k ¤ r, with k satisfying the estimates from 51.7.

r

So let us consider a continuous action ρ : Wm G ’ Di¬(S) and write H for its

kernel. Hence H is a closed normal Lie subgroup and the kernel H0 ‚ Gr of m

the restriction ρ0 = ρ|Gr always contains the normal Lie subgroup Bk ‚ Gr r

m m

with k = 2dimS + 1 if m = 1 and k = max{ dimS , dimS + 1} if m > 1. Let us

m’1 m

r r k

denote Kk the kernel of the jet projection Wm G ’ Wm G.

Lemma. For every Lie group G and all r, k ∈ N, r > k ≥ 1, the normal closed

r r r

Lie subgroup in Wm G generated by Bk {e} equals to Kk .

Proof. The Lie group Wm G can be viewed as the space of ¬ber jets Jr (Rm —

r

0

G, Rm — G)0 and so its Lie algebra wr g coincides with the space of ¬ber jets at

m

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398 Chapter XII. Gauge natural bundles and operators

0 ∈ Rm of (projectable) right invariant vector ¬elds with projections vanishing

at the origin. If we repeat the consideration from the proof of 13.2 with jets

replaced by ¬ber jets, we get the formula for Lie bracket in wr g, [jr X, jr Y ] =

m 0 0

’jr [X, Y ]. Since every polynomial vector ¬eld in wr g decomposes into a sum

m

0

r r r

of X1 ∈ gm and a vertical vector ¬eld X2 from the Lie algebra Tm g of Tm G, we

get immediately the action of gr on Tm g, [j0 X1 + 0, 0 + jr X2 ] = ’jr LX1 X2 .

r r

m 0 0

Now let us ¬x a base ei of g and elements Yi ∈ Tm g, Yi = jr x1 ei . Taking any

r

0

functions fi on Rm with j0 fi = 0, the r-jets of the ¬elds Xi = fi ‚/‚x1 lie in the

k

kernel br ‚ gr and we get

m

k

[j0 Xi , jr Yi ] = ’jr fi ei ∈ Tm g.

r r

0 0

i

Hence [br , Tm g] contains the whole Lie algebra of the kernel Kk and so the latter

r r

k

algebra must coincide with the ideal in wr g generated by br {0}. Since the

m k

r

kernel K1 is connected this completes the proof.

51.12. Corollary. Let G be a Lie group and S be a manifold with a continuous

r

left action of Wm G, dimS = s ≥ 0. Then the action factorizes to an action of

s s

k

Wm G with k ¤ 2s + 1. If m > 1, then k ¤ max{ m’1 , m + 1}. These estimates

are sharp.

The corollary concludes the proof of theorem 51.7.

51.13. Given two G-natural bundles F , E : PBm (G) ’ FM, every natural

transformation T : F ’ E is formed be a system of base preserving FM-

morphisms, cf. 14.11 and 51.8. In the same way as in 14.12 one deduces

Proposition. Natural transformations F ’ E between two r-th order G-

natural bundles over m-dimensional manifolds are in a canonical bijection with

the Wm G-equivariant maps F0 ’ E0 between the standard ¬bers F0 = F0 (Rm —

r

G), E0 = E0 (Rm — G).

51.14. De¬nition. Let F and E be two G-natural bundles over m-dimensional

manifolds. A gauge natural operator D : F E is a system of regular operators

∞ ∞

DP : C F P ’ C EP for all PBm (G)-objects π : P ’ BP such that

(a) DP (F f —¦ s —¦ Bf ’1 ) = F f —¦ DP s —¦ Bf ’1 for every s ∈ C ∞ F P and every

¯

¯

PBm (G)-isomorphism f : P ’ P ,

(b) Dπ’1 (U ) (s|U ) = (DP s)|U for every s ∈ C ∞ F P and every open subset

U ‚ BP .

51.15. For every k ∈ N and every gauge natural bundle F of order r its com-

position J k —¦ F with the k-th jet prolongation de¬nes a gauge natural bundle

functor of order k + r, cf. 14.16. In the same way as in 14.17 one deduces

Proposition. The k-th order gauge natural operators F E are in a canonical

bijection with the natural transformations J k F ’ E.

In particular, this proposition implies that the k-th order G-natural operators

s k

E are in a canonical bijection with the Wm G-equivariant maps J0 F ’ E0 ,

F

where s is the maximum of the orders of J k F and E and J0 F = J0 F (Rm — G).

k k

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52. The Utiyama theorem 399

51.16. Consider the G-natural connection bundle Q and an arbitrary G-natural

bundle E.

Proposition. Every gauge natural operator A : Q E has ¬nite order.

Proof. By 51.8, every G-natural bundle F determines a classical natural bundle

N F by N F (M ) = F (M —G), N F (f ) = F (f —idG ). Given a G-natural operator

D: F E, we denote by N D its restriction to N F , i.e. N DM = DM —G . Clearly,

N D is a classical natural operator N F ’ N E.

Since our operator A is determined locally, we may restrict ourselves to the

product bundle M — G. Then we have a classical natural operator N A. In this

situation the standard ¬ber g — Rm— of Q coincides with the direct product of

dimG copies of Rm— . Hence we can apply proposition 23.5.

52. The Utiyama theorem

52.1. The connection bundle. First we write the equations of a connection

“ on Rm — G in a suitable form. Let ep be a basis of g and let ω p be the

corresponding (left) Maurer-Cartan forms given by p ω p (Xg )ep = T (»g’1 )(X).

Let

(ω p )e = “p (x)dxi

(1) i

be the equations of “(x, e), x ∈ Rm , e = the unit of G. Since “ is right-invariant,

its equations on the whole space Rm — G are

ω p = “p (x)dxi .

(2) i

The connection bundle QP = J 1 P/G is a ¬rst order gauge natural bundle

with standard ¬ber g — Rm— . Having a PB m (G)-isomorphism ¦ of Rm — G into

itself

y = •(x) · y,

(3) x = f (x),

¯ ¯ f (0) = 0

with • : Rm ’ G, its 1-jet j1 ¦ ∈ Wm G is characterized by

1

0

(ap ) = j0 (a’1 · •(x)) ∈ g — Rm— ,

1

(ai ) = j0 f ∈ G1 .

1

a = •(0) ∈ G,

(4) j m

i

Let Ap (a) be the coordinate expression of the adjoint representation of G. In

q

15.6 we deduced the following equations of the action of Wm G on g — Rm—

1

“p = Ap (a)(“q + aq )˜j .

¯

(5) a

q i j j i

The ¬rst jet prolongation J 1 QP of the connection bundle is a second order

gauge natural bundle, so that its standard ¬ber S1 = J0 Q(Rm — G), with the

1

coordinates “p , “p = ‚“p /‚xj , is a Wm G-space. The second order partial

2

i ij i

derivatives ap of the map a’1 · •(x) together with ai = ‚jk f i (0) are the addi-

2

ij jk

2

tional coordinates on Wm G. Using 15.5, we deduce from (5) that the action of

2

Wm G on S1 has the form (5) and

“p = Ap (a)“q ak al + Ap (a)aq ak al +

¯

(6) ˜i ˜j ˜i ˜j

q q

ij kl kl

+ Dqr (a)“q ar ak al + Eqr (a)aq ar ak al + Ap (a)(“q + aq )˜k

p p

k l ˜i ˜j k l ˜i ˜j k aij

q k

where the D™s and E™s are some functions on G, which we shall not need.

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400 Chapter XII. Gauge natural bundles and operators

52.2. The curvature. To deduce the coordinate expression of the curvature

tensor, we shall use the structure equations of “. By 52.1.(1), the components

•p of the connection form of “ are

•p = ω p ’ “p (x)dxi .

(1) i

The structure equations of “ reads

p

d•p = cp •q § •r + Rij dxi § dxj

(2) qr

p

where cp are the structure constants of G and Rij is the curvature tensor. Since

qr

ω p are the Maurer-Cartan forms of G, we have dω p = cp ω q § ω r . Hence the

qr

exterior di¬erentiation of (1) yields

d•p = cp (•q + “q dxi ) § (•r + “r dxj ) + “p (x)dxi § dxj .

(3) qr j

i ij

Comparing (2) with (3), we obtain

Rij = “p + cp “q “r .

p

(4) qr i j

[ij]

52.3. Generalization of the Utiyama theorem. The curvature of a connec-

tion “ on P can be considered as a section CP “ : BP ’ LP — Λ2 T — BP , where

LP = P [g, Ad] is the so-called adjoint bundle of P , see 17.6. Using the language

of the theory of gauge natural bundles, D. J. Eck reformulated a classical result

by Utiyama in the following form: All ¬rst order gauge natural Lagrangians on

the connection bundle are of the form A —¦ C, where A is a zero order gauge

natural Lagrangian on the curvature bundle and C is the curvature operator,

[Eck, 81]. By 49.1, a ¬rst order Lagrangian on a connection bundle QP is a

morphism J 1 QP ’ Λm T — BP , so that the Utiyama theorem deals with ¬rst

Λm T — B. We are going to generalize this

order gauge natural operators Q

result. Since the proof will be based on the orbit reduction, we shall directly

discuss the standard ¬bers in question.

Denote by γ : S1 ’ g — Λ2 Rm— the formal curvature map 52.2.(4). One

sees easily that γ is a surjective submersion. The semi-direct decomposition

W m G = G2

2 2 2

Tm G together with the target jet projection Tm G ’ G de¬nes a

m

group homomorphism p : Wm G ’ G2 —G. Let Z be a G2 —G-space, which can

2

m m

be considered as a Wm G-space by means of p. The standard ¬ber g — Λ2 Rm— of

2

the curvature bundle is a G1 — G-space, which can be interpreted as G2 — G-

m m

space by means of the jet homomorphism π1 : G2 ’ G1 .

2

m m

Proposition. For every Wm G-map f : S1 ’ Z there exists a unique G2 — G-

2

m

map g : g — Λ2 Rm— ’ Z satisfying f = g —¦ γ.

Proof. On the kernel K of p : Wm G ’ G2 — G we have the coordinates ap ,

2

m i

ap = ap introduced in 52.1. Let us replace the coordinates “p on S1 by

ij ji ij

Rij = “p + cp “q “r ,

p

Sij = “p ,

p

(1) qr i j

[ij] (ij)

while “p remain unchanged. Hence the coordinate form of γ is (“p , Rij , Sij ) ’

p p

i i

(Rij ). From 52.1.(5) and 52.1.(6) we can evaluate ap and ap in such a way that

p

i ij

¯ p = 0 and S p = 0. This implies that each ¬ber of γ is a K-orbit. Then we

¯

“i ij

apply 28.1.

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

52. The Utiyama theorem 401

52.4. To interpret the proposition 52.3 in terms of operators, it is useful to

introduce a more subtle notion of principal prolongation W s,r P of order (s, r),

s ≥ r, of a principal ¬ber bundle P (M, G). Formally we can construct the ¬ber

product over M

W s,r P = P s M —M J r P

(1)

and the semi-direct product of Lie groups

W m G = Gs

s,r r

(2) Tm G

m

with respect to the right action (A, B) ’ B —¦ πr (A) of Gs on Tm G. The right

s r

m

action of Wm G on W s,r P is given by a formula analogous to 15.4

s,r

(u, v)(A, B) = (u —¦ A, v.(B —¦ πr (A’1 —¦ u’1 ))),

s

u ∈ P s M , v ∈ J r P , A ∈ Gs , B ∈ Tm G. In the case r = 0 we have a

r

m

direct product of Lie groups Wm G = Gs — G and the usual ¬bered product

s,0

m

s,0 s

W P = P M —M P of principal ¬ber bundles.

To clarify the geometric substance of the previous construction, we have to use

the concept of (r, s, q)-jet of a ¬bered manifold morphism introduced in 12.19.

Then W s,r P can be de¬ned as the space of all (r, r, s)-jets at (0, e) of the local

principal bundle isomorphisms Rm — G ’ P and the group Wm G is the ¬ber

s,r

of W s,r (Rm — G) over 0 ∈ Rm endowed with the jet composition. The proof is

left to the reader as an easy exercise. Furthermore, in the same way as in 51.2

¯ r,r,s r,r,s

we deduce that if two PBm (G)-morphisms f, g : P ’ P satisfy jy f = jy g

at a point y ∈ Px , x ∈ BP , then this equality holds at every point of the ¬ber

Px . In this case we write jr,r,s f = jr,r,s g.

x x

Now we can say that natural bundle F is of order (s, r), s ≥ r, if jr,r,s f =

x

jr,r,s g implies F f |Fx P = F g|Fx P . Using the proposition 51.10 we deduce quite

x

similarly to 51.6 that every gauge natural bundle of order (s,r) is a ¬ber bundle

associated to W s,r .

Then the proposition 52.3 is equivalent to the following assertion.

General Utiyama theorem. Let F be a gauge natural bundle of order (2, 0).

Then for every ¬rst order gauge natural operator A : Q F there exists a

¯ : L — Λ2 T — B ’ F satisfying A = A —¦ C, where

¯

unique natural transformation A

L — Λ2 T — B is the curvature operator.

C: Q

In all concrete problems in this chapter the result will be applied to gauge

natural bundles of order (1,0). By de¬nition, every such a bundle has the order

(2,0) as well.

L—Λ2 T — B

52.5. Curvature-like operators. The curvature operator C : Q

is a gauge natural operator because of the geometric de¬nition of the curva-

L — —2 T — B.

ture. We are going to determine all gauge natural operators Q

(We shall see that the values of all of them lie in L — Λ2 T — B. But this is an

interesting geometric result that the antisymmetry of such operators is a con-

sequence of their gauge naturality.) Let Z ‚ L(g, g) be the subspace of all

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402 Chapter XII. Gauge natural bundles and operators

linear maps commuting with the adjoint action of G. Since every z ∈ Z is an

equivariant linear map between the standard ¬bers, it induces a vector bundle

morphism zP : LP ’ LP . Hence we can construct a modi¬ed curvature operator

¯

C(z)P : (¯P — Λ2 T — idBP ) —¦ CP .

z

L — —2 T — B are the modi¬ed

Proposition. All gauge natural operators Q

curvature operators C(z) for all z ∈ Z.

Proof. By 51.16, every gauge natural operator A on the connection bundle has

r+1

¬nite order. The r-th order gauge natural operators correspond to the Wm G-

p

equivariant maps J0 Q ’ g — —2 Rm— . Let “i± be the induced coordinates on

r

J0 Q, where ± is a multi index of range m with |±| ¤ r. On g — —2 Rm— we have

r

p

the canonical coordinates Rij and the action

¯p q

Rij = Ap (a)Rkl ak al .

(1) ˜i ˜j

q

Hence the coordinate components of the map associated to A are some func-

tions fij (“q ). If we consider the canonical inclusion of G1 into Wm G, then

p r+1

m

k±

analogously to 14.20 the transformation laws of all quantities “p are tensorial.

i±

The equivariance with respect to the homotheties in G1 gives a homogeneity

m

condition

c2 fij (“q ) = fij (c1+|±| “q )

p p

0 = c ∈ R.

(2) k± k±

By the homogeneous function theorem, fij is independent of “p with |±| ≥ 2.

p

i±

Hence A is a ¬rst order operator and we can apply the general Utiyama theorem.

The associated map

g : g — Λ2 Rm— ’ g — —2 Rm—

(3)

of the induced natural transformation L — Λ2 T — B ’ L — —2 T — B is of the form

p q

gij (Rkl ). Using the homotheties in G1 we ¬nd that g is linear. If we ¬x one

m

coordinate in g on the right-hand side of the arrow (3), we obtain a linear G1 -

m

map —n Λ2 Rm— ’ —2 Rm— . By 24.8.(5), this map is a linear combination of the

individual inclusions Λ2 Rm— ’ —2 Rm— , i.e.

p pq

(4) gij = zq Rij .

2

Using the equivariance with respect to the canonical inclusion of G into Wm G,

p

we ¬nd that the linear map (zq ) : g ’ g commutes with the adjoint action.

52.6. Remark. In the case that the structure group is the general linear group

GL(n) of an arbitrary dimension n, the invariant tensor theorem implies directly

that the Ad-invariant linear maps gl(n) ’ gl(n) are generated by the identity

and the map X ’ (traceX)id. Then the proposition 52.5 gives a two-parameter

L — —2 T — B, which the ¬rst author

family of all GL(n)-natural operators Q

deduced by direct evaluation in [Kol´ˇ, 87b]. In general it is remarkable that

ar

the study of the case of the special structure group GL(n), to which we can

apply the generalized invariant tensor theorem, plays a useful heuristic role in

the theory of gauge natural operators.

L — —2 T — B

Further we remark that all gauge natural operators Q • Q

transforming pairs of connections on an arbitrary principal ¬ber bundle P into

sections of LP — —2 T — BP are determined in [Kurek, to appear a].

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

52. The Utiyama theorem 403

52.7. Generalized Chern-Weil forms. We recall that for every vector bun-

dle E ’ M , a section of E — Λr T — M is called an E-valued r-form, see 7.11.

For E = M — R we obtain the usual exterior forms on M . Consider a linear

˜

action ρ of a Lie group G on a vector space V and denote by V the G-natural

0

bundle over m-manifolds determined by this action of G = Wm G. We are going

to construct some gauge natural operators transforming every connection “ on a

˜

principal bundle P (M, G) into a V (P )-valued exterior form. In the special case

of the identity action of G on R, i.e. ρ(g) = idR for all g ∈ G, we obtain the

classical Chern-Weil forms of “, [Kobayashi,Nomizu, 69].

Let h : S r g ’ V be a linear G-map. We have S r (g — Λ2 Rm— ) = S r g —

¯

S r Λ2 Rm— , so that we can de¬ne h : g — Λ2 Rm— ’ V — Λ2r Rm— by

¯ A ∈ g — Λ2 Rm— ,

h(A) = (h — Alt)(A — · · · — A),

(1)

where Alt : S r Λ2 Rm— ’ Λ2r Rm— is the tensor alternation. Since g — Λ2 Rm—

˜

or V — Λ2r Rm— is the standard ¬ber of the curvature bundle or of V (P ) —

¯ ¯

Λ2r T — M , respectively, h induces a bundle morphism hP : L(P ) — Λ2 T — M ’

˜

V (P ) — Λ2r T — M . For every connection “ : M ’ QP , we ¬rst construct its

˜

curvature CP “ and then a V (P )-valued 2r-form

˜ ¯

(2) hP (“) = hP (CP “).

Such forms will be called generalized Chern-Weil forms.

Let I(g, V ) denote the space of all polynomial G-maps of g into V . Every

H ∈ I(g, V ) is determined by a ¬nite sequence of linear G-maps hri : S ri g ’ V ,

i = 1, . . . , n. Then

˜ ˜

˜

HP (“) = hr1 (“) + · · · + hrn (“)

P P

˜ ˜

is a section of V (P ) — ΛT — M for every connection “ on P . By de¬nition, H is

˜

V — ΛT — B.

a gauge natural operator Q

˜ ˜

V — ΛT — B are of the form H

52.8. Theorem. All G-natural operators Q

for all H ∈ I(g, V ).

Proof. Consider some linear coordinates y p on g and z a on V and the induced

p

coordinates yij on g — Λ2 Rm— and zi1 ...is on V — Λs Rm— .

a

˜

V — Λs T — B has a ¬nite order k.

By 51.16 every G-natural operator A : Q

Hence its associated map f : J0 Q ’ V — Λs Rm— is of the form

k

zi1 ...is = fia ...is (“p ),

a

0 ¤ |±| ¤ k.

i±

1

The homotheties in G1 give a homogeneity condition

m

k s fia ...is (“p ) = fia ...is (k 1+|±| “p ).

i± i±

1 1

This implies that f is a polynomial map in “p . Fix a, p1 , |±1 |, . . . , pr , |±r | with

i±

|±1 | ≥ 2 and consider the subpolynomial of the a-th component of f which is

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404 Chapter XII. Gauge natural bundles and operators

formed by the linear combinations of “p1 ±1 . . . “pr ±r . It represents a GL(m)-map

1 r

i i

Rm— — S |±1 | Rm— — . . . — Rm— — S |±r | Rm— ’ Λp Rm— . Analogously to 24.8 we

deduce that this is the zero map because of the symmetric component S |±1 | Rm— .

Hence A is a ¬rst order operator.

Applying the general Utiyama theorem, we obtain f = g —¦ γ, where g is a

Gm — G-map g — Λ2 Rm— ’ V — Λs Rm— . The coordinate form of g is

1

p

a a

zi1 ...is = gi1 ...is (yij ).

Using the homotheties in G1 we ¬nd that s = 2r and g is a polynomial of degree

m

p

r in yij . Its total polarization is a linear map S r (g — Λ2 Rm— ) ’ V — Λ2r Rm— .

If we ¬x one coordinate in V and any r-tuple of coordinates in g, we obtain

an underlying problem of ¬nding all linear G1 -maps —r Λ2 Rm— ’ Λ2r Rm— . By

m

p pr

24.8.(5) each this map is a constant multiple of y[i1 i2 . . . yi2r’1 i2r ] . Hence g is of

1

the form

p pr

ca1 ...pr y[i1 i2 . . . yi2r’1 i2r ] .

p 1

2

The equivariance with respect to the canonical inclusion of G into Wm G implies

that (ca1 ...pr ) : S r g ’ V is a G-map.

p

52.9. Consider the special case of the identity action of G on R. Then every

linear G-map S r g ’ R is identi¬ed with a G-invariant element of S r g— and

the (M — R)-valued forms are the classical di¬erential forms on M . Hence

52.7.(2) gives the classical Chern-Weil forms of a connection. In this case the

theorem 52.8 reads that all gauge natural di¬erential forms on connections are

the classical Chern-Weil forms. All of them are of even degree. The exterior

di¬erential of a Chern-Weil form is a gauge natural form of odd degree. By the

theorem 52.8 it must be a zero form. This gives an interesting application of

gauge naturality for proving the following classical result.

Corollary. All classical Chern-Weil forms are closed.

52.10. In general, if one has a vector bundle E ’ M , an E-valued r-form

ω : Λr T M ’ E and a linear connection ∆ on E, one introduces the covariant

exterior derivative d∆ ω : Λr+1 T M ’ E, see 11.14. Consider the situation from

52.7. For every H ∈ I(g, V ) and every connection “ on P we have constructed

˜ ˜

a V (P )-valued form HP (“), which is of even degree. According to 11.11, “

˜ ˜

induces a linear connection “V on V (P ). Then d“V HP (“) is a gauge natural

˜

V (P )-valued form of odd degree. By the theorem 52.8 it is a zero form. Thus,

we have proved the following interesting geometric result.

Proposition. For every H ∈ I(g, V ) and every connection “ on P , it holds

˜

d“V HP (“) = 0.

52.11. Remark. We remark that another generalization of Chern-Weil forms

is studied in [Lecomte, 85].

52.12. Gauge natural approach to the Bianchi identity. It is remarkable

that the Bianchi identity for a principal connection “ : BP ’ QP can be deduced

in a similar way. Using the notation from 52.5, we ¬rst prove an auxiliary result.

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

53. Base-extending gauge natural operators 405

L—Λ3 T — B is the zero operator.

Lemma. The only gauge natural operator Q

Proof. By 51.16, every such operator A has ¬nite order. Let

fijk (“q ),

p

0 ¤ |±| ¤ r

l±

be its associated map. The homotheties in G1 yield a homogeneity condition

m

c3 fijk (“q ) = fijk (c1+|±| “q ),

p p

c ∈ R \ {0}.

(1) l± l±

Hence f is polynomial in “p , “p and “p of degrees d0 , d1 and d2 satisfying

i ij ijk

3 = d0 + 2d1 + 3d2 .

This implies f is linear in “p . But “p represent a linear GL(m)-map Rm— —

ijk ijk

S 2 Rm— ’ Λ3 Rm— for each p = 1, . . . , n. By 24.8 the only possibility is the

zero map. Hence A is a ¬rst order operator. By the general Utiyama theorem,

f factorizes through a map g : g — Λ2 Rm— ’ g — Λ3 Rm— . The equivariance

of g with respect to the homotheties in G1 yields a homogeneity condition

m

c3 g(y) = g(c2 y), y ∈ g — Λ2 Rm— . Since there is no integer satisfying 3 = 2d, g is

the zero map.

The curvature of “ is a section CP “ : BP ’ LP — Λ2 T — BP . According to

˜

the general theory, “ induces a linear connection “ on the adjoint bundle LP .

Hence we can construct the covariant exterior di¬erential

BP ’ LP — Λ3 T — BP.

(2) “ CP “ :

˜

By the geometric character of this construction, (2) determines a gauge natural

operator. Then our lemma implies

(3) “ CP (“) = 0.

˜

By 11.15, this is the Bianchi identity for “.

53. Base extending gauge natural operators

53.1. Analogously to 18.17, we now formulate the concept of gauge natural

operators in more general situation. Let F , E and H be three G-natural bundles

over m-manifolds.

De¬nition. A gauge natural operator D : F (E, H) is a system of regular

∞ ∞

operators DP : C F P ’ CBP (EP, HP ) for every PBm (G)-object P satisfying

DP (F f —¦ s —¦ Bf ’1 ) = Hf —¦ DP (s) —¦ Ef ’1 for every s ∈ C ∞ F P and every

¯

¯

PBm (G)-isomorphism f : P ’ P , as well as a localization condition analogous

to 51.14.(b).

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

406 Chapter XII. Gauge natural bundles and operators

53.2. Quite similarly to 18.19, one deduces

Proposition. k-th order gauge natural operators F (E, H) are in a canonical

r

bijection with the natural transformations J F • E ’ H.

If we have a natural transformation q : H ’ E such that every qP : HP ’ EP

is a surjective submersion and we require every DP (s) to be a section of qP , we

(H ’ E). Then we ¬nd in the same way as in 51.15 that the

write D : F

s

(H ’ E) are in bijection with the Wm -equivariant

G-natural operators F

k

maps f : J0 F — E0 ’ H0 , satisfying q0 —¦ f = pr2 , where q0 : H0 ’ E0 is the

restriction of qRm —G and s is the maximum of the orders in question.

(QT ’ T B). In 46.3 we deduced that

53.3. Gauge natural operators Q

every connection “ on principal bundle P ’ M with structure group G induces

a connection T “ on the principal bundle T P ’ T M with structure group T G.

Hence T is a (¬rst-order) G-natural operator Q (QT ’ T B). Now we are

(QT ’ T B). Since

going to determine all ¬rst-order G-natural operators Q

the di¬erence of two connections on T P ’ T BP is a section of L(T P )—T — T BP ,

(LT — T — T B ’

it su¬ces to determine all ¬rst-order G-natural operators Q

T B). The ¬ber of the total projection L(T (Rm — G)) — T — T Rm ’ T Rm ’ Rm

—

over 0 ∈ Rm is the product of Rm with tg — T0 T Rm , 0 ∈ T Rm = R2m . By 53.2

our operators are in bijection with the Wm G-equivariant maps J0 Q(Rm — G) —

2 1

—

Rm ’ Rm — tg — T0 T Rm over the identity of Rm .

We know from 10.17 that T G coincides with the semidirect product G g

with the following multiplication

’1

(1) (g1 , X1 )(g2 , X2 ) = (g1 g2 , Ad(g2 )(X1 ) + X2 )

where Ad means the adjoint action of G. This identi¬es the Lie algebra tg of T G

with g — g and a direct calculation yields the following formula for the adjoint

action AdT G of T G

(2) AdT G (g, X)(Y, V ) = (Ad(g)(Y ), Ad(g)([X, Y ] + V )).

Hence the subspace 0 — g ‚ tg is AdT G -invariant, so that it de¬nes a subbundle

K(T P ) ‚ L(T P ). The injection V ’ (0, V ) induces a map IP : LP ’ K(T P ).

Every modi¬ed curvature C(z)P (“) of a connection “ on P , see 52.5, can be

interpreted as a linear morphism Λ2 T BP ’ LP . Then we can de¬ne a linear

map µ(C(z)P (“)) : T T BP ’ L(T P ) by

µ(C(z)P (“))(A) = IP (C(z)P (“)(π1 A § π2 A)), A ∈ T T BP

(3)

where π1 : T T BP ’ T BP is the bundle projection and π2 : T T BP ’ T BP is

the tangent map of the bundle projection T BP ’ BP . This determines one

(LT — T — T B ’ T B).

series µ(C(z)), z ∈ Z, of G-natural operators Q

Moreover, if we consider a modi¬ed curvature C(z)P (“) as a map C(z) : P •

2

Λ T BP ’ g, we can construct its vertical prolongation with respect to the ¬rst

factor

V1 C(z)P (“) : V P • Λ2 T BP ’ T g = tg.

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

53. Base-extending gauge natural operators 407

Then we add the vertical projection ν : T P ’ V P of the connection “ and we

use the projections π1 and π2 from (3). This yields a map

„ (C(z)P (“)) : T P • T T BP ’ tg

(4)

„ (C(z)P (“))(U, A) = V1 C(z)P (“)(νU, π1 A § π2 A), U ∈ T P, A ∈ T T BP.

The latter map can be interpreted as a section of L(T P ) — T — T BP , which gives

(LT —T — T B ’ T B).

another series „ (C(z)), z ∈ Z, of G-natural operators Q

(QT ’ T B) form

Proposition. All ¬rst-order gauge natural operators Q

the following 2dimZ-parameter family

T + µ(C(z)) + „ (C(¯)), z, z ∈ Z.

(5) z ¯

The proof will occupy the rest of this section.

53.4. Let “ be a connection on Rm — G with equations

ω p = “p (x)dxi .

(1) i

Let (µp ) be the second component of the Maurer-Cartan form of T G (the ¬rst

one is (ω p )) and let X i be the induced coordinates on T0 Rm . Applying the

description of the Maurer-Cartan form of T G from 37.16 to (1), we ¬nd the

equation of T “ is of the form (1) and

‚“p j i

X dx + “p dX i .

p i

(2) µ= i

j

‚x

d

x(t) ∈ T0 Rm de¬nes a map

53.5. Remark ¬rst that every dt 0

1 1 d

Tm G ’ T G, j0 • ’ (• —¦ x)(t).

(1) dt 0

Consider an isomorphism x = f (x), y = •(x) · y of Rm — G and an element of

¯ ¯

m m

V (T (R — G) ’ T R ). Clearly, such an element can be generated by a map

(x(t), y(t, u)) : R2 ’ Rm — G, t, u ∈ R. This map is transformed into

y = •(x(t)) · y(t, u).

(2) x = f (x(t)),

¯ ¯

Di¬erentiating with respect to t, we ¬nd

d¯

y d•(x(0)) dy(0, u)

(3) = T µ( , )

dt dt dt

where µ : G — G ’ G is the group composition. This implies that the next

di¬erentiation with respect to u yields the adjoint action of T G with respect to

(1). Thus, if (Y p , V p ) are the coordinates in tg given by our basis in g, then

we deduce by the latter observation that the action of Wm G on Rm — tg is

2

¯

X i = ai X j and

j

¯ ¯

Y p = Ap (a)Y q , V p = Ap (a)(cq ar X j Y s + V q ).

(4) q q rs j

On the other hand, the action of Wm G on T0 T Rm goes through the projection

2

into G2 and has the standard form

m

¯

d¯i = ai dxj , dX i = ai X j dxk + ai dX j .

(5) x j jk j

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

408 Chapter XII. Gauge natural bundles and operators

53.6. Our problem is to ¬nd all Wm G-equivariant maps f : Rm — J0 Q ’ Rm —

2 1

p p p

—

tg — T0 T Rm over idRm . On J0 Q, we replace “ij by Rij and Sij as in 52.3. The

1

—

coordinates on tg — T0 T Rm are given by

p p

Y p = Bi dxi + Ci dX i

(1)

p p

V p = Di dxi + Ei dX i .

(2)

Hence all components of f are smooth functions of X = (X i ), “ = (“p ), R =

i

p p

(Rij ), S = (Sij ). Using 53.5.(4)“(5), we deduce from (1) the transformation

laws

Ci = Ap (a)Cj aj

¯p q

(3) ˜i

q

Bi = Ap (a)Bj aj ’ Ap (a)Ck ak aj X l .

¯p q q

(4) ˜i ˜j li

q q

Let us start with the component Ci (X, “, R, S) of f . Using ap and ap , we

p

ij i

deduce that C™s are independent of “ and S. Then we have the situation of the

following lemma.

Lemma. All AdG — GL(m, R)-equivariant maps Rm — g — Λ2 Rm— ’ g — Rm—

q

have the form µp Rij X j with (µp ) ∈ Z.

q q

Proof. First we determine all GL(m, R)-maps h : Rm — —n —2 Rm— ’ —n Rm— ,

h = (hp (bq , X l )). If we consider the contraction h, v of h with v = (v i ) ∈ Rm ,

i jk

we can apply the tensor evaluation theorem to each component of h, v . This

yields

hp v i = •p (bq X i X j , br v i X j , bs X i v j , bt v i v j ).

ij ij ij

i ij

Di¬erentiating with respect to v i and setting v i = 0, we obtain

hp = •p (br X k X l )bq X j + ψq (br X k X l )bq X j

p

(5) q kl kl

i ij ji

with arbitrary smooth functions •p , ψq of n variables. If bp = Rij are antisym-

p

p

q ij

p p p

metric, we have Rij X i X j = 0 and Rij X j = ’Rji X j , so that

hp = µp Rij X j ,

q

µp ∈ R.

(6) q q

i

The equivariance with respect to G then yields Ap (a)µq = µp Aq (a), i.e. (µp ) ∈

q r qr q

Z.

p q p

Thus our lemma implies Ci = µp Rij X j , (µp ) ∈ Z. For the components Bi

q q

of f , the use of ap and ap gives that B™s are independent of “ and S. Then the

i ij

equivariance with respect to ai yields

jk

p

(7) Ci = 0.

Using our lemma again, we obtain

p pq

Bi = γq Rij X j , p

(γq ) ∈ Z.

(8)

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

54. Induced linear connections on the total space of vector and principal bundles 409

53.7. From 53.6.(2) we deduce the transformation laws