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).

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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

Ei = Ap (a)Ej aj + Ap (a)cq ar X j Ci

¯p q s

(1) ˜i

q q rs j

Di = Ap (a)Dj aj + Ap (a)cq ar X j Bi ’ Ej aj X k .

¯p q ¯p

s

(2) ˜i

q q rs j ki

p q

By 53.6.(7), the ¬rst equation implies Ei = µp Rij X j , (µp ) ∈ Z, in the same way

q q

as in 53.6. Using ap in the second equation, we ¬nd that the D™s are independent

ij

of S. Then the use of ai implies

jk

p

(3) Ei = 0.

The equivariance of D™s with a = e, ai = δj now reads

i

j

Di (X, “q + aq , R) = Di (X, “, R) + cp aq X j γs Rik X k .

p p rs

qr j

j j

Di¬erentiating with respect to aq and setting aq = 0, we ¬nd that the D™s are

j j

of the form

Di = cp “q X j γs Rik X k + Fip (X j , Rkl X l ).

p q

rs

qr j

The ˜absolute terms™ Fip can be determined by lemma 53.6. This yields

Di = cp “q X j γs Rik X k + kq Rij X j ,

p pq

rs p

(kq ) ∈ Z.

(4) qr j

One veri¬es easily that (3), (4) together with 53.6.(7)“(8) and 53.4.(1)“(2) is

the coordinate form of proposition 53.3.

54. Induced linear connections on the total space

of vector and principal bundles

54.1. Gauge natural operators Q • QT B QT . Given a vector bundle

π : E ’ BE of ¬ber dimension n, we denote by GL(Rn , E) ’ BE the bundle