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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



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

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

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