¯

˜· ˜ ˜¯ ˜ ¯˜

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

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

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

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

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