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.

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

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

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

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

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

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