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By the homogeneous function theorem, fij is linear in y p , i.e.
p p
fij = Fijq (D± , Λβ )y q .

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
54. Induced linear connections on the total space of vector and principal bundles 415

The base homotheties then imply
p p
k 2 Fijq = Fijq (k 1+|±| D± , k 1+|β| Λβ ).

p p
By the homogeneous function theorem, Fijq is linear in Dqij , Λi , bilinear in
p p
i i
Dqi , Λjk and independent of Dqi± , Λjkβ with |±| > 1, |β| > 1. Using the gen-
eralized invariant tensor theorem, we obtain Fijq in the form of a 40-parameter
family. The equivariance of fij with respect to K then yields

“p = p p p r r
(1 ’ c6 )Dqij + c6 Dqji + c7 δq (Drij ’ Drji )’
(1) ij

p p p p
c6 Dri Dqj + (c6 ’ 1)Drj Dqi + (1 ’ c3 )Dqk Λk + c3 Dqk Λk +
r r
ij ji
p p
(c4 ’ c1 )Dqi (Λl ’ Λl ) + (c5 ’ c2 )Dqj (Λl ’ Λl )+
lj jl li il

δq Gij (Λ) y q

where Gij (Λ) is the coordinate form of G(Λ).
One veri¬es directly that (1) and 54.5.(1)“(3), (5), (6), (8) is the coordinate
expression of 54.3.(1).
54.7. The case of principal bundles. An analogous problem is to study the
gauge natural operators transforming a connection D on a principal G-bundle
π : P ’ BP and a classical linear connection Λ on the base manifold BP into a
classical linear connection on the total space P . First we present a geometrical
construction of such an operator.
Let vA be the vertical component of a vector A ∈ Ty P and bA be its projection
to the base manifold. Consider a vector ¬eld X on BP such that jx X = Λ(bA),
x = π(y). Construct the lift X D of X and the fundamental vector ¬eld •(vA)
determined by vA. An easy calculation shows that the rule

A ’ jy (X D + •(vA))

determines a classical linear connection NP (D, Λ) : T P ’ J 1 (T P ’ P ) on P .
54.8. We are going to determine all gauge natural operators of the above type.
The result of 54.3 suggests us that the case Λ is without torsion is much simpler
than the general case. That is why we restrict ourselves to a symmetric Λ. Since
the di¬erence of two classical linear connections on P is a tensor ¬eld of type
T P — T — P — T — P , we characterize all gauge natural operators in question as
a sum of the operator N from 54.7 and of the gauge natural di¬erence tensor
¬elds. We construct geometrically the following 3 systems of di¬erence tensor
I. The connection form of D is a linear map ω : T P ’ g. Take any bilinear
map f1 : g — g ’ g and compose ω • ω with f1 . This de¬nes an n3 -parameter
system of di¬erence tensor ¬elds T P — T P ’ V P , n = dimG.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
416 Chapter XII. Gauge natural bundles and operators

II. The curvature form Dω of ω is a bilinear map T P • T P ’ g. Take any
linear map f2 : g ’ g and compose Dω with f2 . This yields an n2 -parameter
system of di¬erence tensor ¬elds.
III. By 28.7, all natural operators transforming a linear symmetric connection
Λ on BP into a tensor ¬eld of T — BP — T — BP form a 2-parameter family linearly
generated by both di¬erent contractions R1 and R2 of the curvature tensor of Λ.
The tangent map of the bundle projection P ’ BP de¬nes the dual injection
P • T — BP ’ T — P . Taking any fundamental vector ¬eld Y determined by a
vector Y ∈ g, we obtain a 2n-parameter system of di¬erence tensor ¬elds linearly
¯ ¯
generated by Y — R1 and Y — R2 .
54.9. Proposition. All gauge natural operators transforming a connection on
P and a classical linear symmetric connection of the base manifold BP into
a classical linear connection on P form the (n3 + n2 + 2n)-parameter family
generated by operator N and by the above families I, II, and III of the di¬erence
tensor ¬elds.
The proof consists in straightforward application of our techniques, but it is
too long to be performed here. We refer the reader to [Kol´ˇ, to appear a].

Our approach to gauge natural bundles and operators generalizes directly the
theory of natural bundles. So we also prove the regularity originally assumed
in [Eck, 81]. Let us mention that, analogously to chapter XI, we can de¬ne
the Lie derivative of sections of gauge natural bundles with respect to the right
invariant vector ¬elds on the corresponding principal ¬ber bundles and then
the in¬nitesimally gauge natural operators. The relation between the gauge
naturality and in¬nitesimal gauge naturality is similar to the case of natural
bundles if the gauge group is connected; more information can be found in [Cap,
Slov´k, 92].
The ¬rst application of our methods for ¬nding gauge natural operators was
presented in [Kol´ˇ, 87b]. The considerations in that paper are restricted to
the case the structure group is the general linear group GL(n) in an arbitrary
dimension (independent of the dimension of the base manifold), for in such a
case one can apply directly the results from chapter VI. [Kol´ˇ, 87b] has also
determined all GL(n)-natural operators transforming a principal connection on
a principal bundle P and a classical linear connection on the base manifold into
a principal connection on W 1 P . The curvature-like operators were found in the
special case G = GL(n) in [Kol´ˇ, 87b] and the general problem was solved
in [Kol´ˇ, to appear a]. The greater part of the results from section 52 was
deduced in [Kol´ˇ, to appear b]. Proposition 53.3 was proved for the special
case G = GL(n) in [Kol´ˇ, 91], the general result is ¬rst presented in this book.
Section 54 is based on [Gancarzewicz, Kol´ˇ, 91].

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

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