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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
of all linear frames in the individual ¬bers of E, see 10.11. This is a principal
bundle with structure group GL(n), n = the ¬ber dimension of E. Clearly E
is identi¬ed with the ¬ber bundle associated to GL(Rn , E) with standard ¬ber
Rn . The construction of associated bundles establishes a natural equivalence
between the category PBm (GL(n)) and the category VBm,n := VB © FMm,n .
A linear connection D on a vector bundle E is usually de¬ned as a linear
morphism D : E ’ J 1 E splitting the target jet projection J 1 E ’ E, see sec-
tion 17. One ¬nds easily that there is a canonical bijection between the linear
connections on E and the principal connections on GL(Rn , E), see 11.11. That
is why we can say that Q(GL(Rn , E)) =: QE is the bundle of linear connections
on E. In the special case E = T BE this gives a well-known fact from the theory
of classical linear connections on a manifold.

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


An interesting geometrical problem is how a linear connection D on a vector
bundle E and a classical linear connection Λ on the base manifold BE can
induce a classical linear connection on the total space E. More precisely, we
are looking for operators which are natural on the category VBm,n . Taking into
account the natural equivalence between VBm,n and PBm (GL(n)), we see that
this is a problem on base-extending GL(n)-natural operators. But we ¬nd it
more instructive to apply the direct approach in this section. Thus, our problem
is to ¬nd all operators Q • QT B QT which are natural on VBm,n .
54.2. First we describe a concrete construction of such an operator. Let us
denote the covariant di¬erentiation with respect to a connection by the symbol
of the connection itself. Thus, if X is a vector ¬eld on BE and s is a section
of E, then DX s is a section of E. Further, let X D denote the horizontal lift
of vector ¬eld X with respect to D. Moreover, using the translations in the
individual ¬bers of E, we derive from every section s : BE ’ E a vertical vector
¬eld sV on E called the vertical lift of s.
Proposition. For every linear connection D on a vector bundle E and every
classical linear connection Λ on BE there exists a unique classical linear connec-
tion “ = “(D, Λ) on the total space E with the following properties

“X D Y D = (ΛX Y )D , “X D sV = (DX s)V ,
(1)
“sV X D = 0, “sV σ V = 0,

for all vector ¬elds X, Y on BE and all section s, σ of E.
Proof. We use direct evaluation, because we shall need the coordinate expres-
sions in the sequel. Let xi , y p be some local linear coordinates on E and
X i = dxi , Y p = dy p be the induced coordinates on T E. If
p
dy p = Dqi (x)y q dxi
(2)


are the equations of D and ξ i (x) ‚xi or sp (x) is the coordinate form of X or s,
respectively, then DX s is expressed by

‚sp i p
ξ ’ Dqi sq ξ i .
(3) i
‚x

The coordinate expression of X D is

‚ ‚
p
ξi + Dqi y q ξ i p
(4) i
‚x ‚y

and sV is given by


sp (x)
(5) .
‚y p

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
54. Induced linear connections on the total space of vector and principal bundles 411


Let
dX i = Λi X j dxk
(6) jk

be the coordinate expression of Λ and let
dX i = (“i X j + “i Y p )dxk + (“i X j + “i Y p )dy q ,
jk pk jq pq
(7)
dY p = (“p X i + “p Y q )dxj + (“p X i + “p Y q )dy r
qr
ij qj ir

be the coordinate expression of “. Evaluating (1), we obtain

“p = Dp ’ Drj Dqi + Dqk Λk y q ,
p p
“i = Λ i , r
jk jk ij
ij j qi
‚x
(8)
“j = “j = 0, “p = “p = Dqi ,
p
“i = 0, “p = 0.
pq qr
ip pi iq qi

This proves the existence and the uniqueness of “.
54.3. Since the di¬erence of two classical linear connections on E is a tensor
¬eld of T E — T — E — T — E, we shall heavily use the gauge natural di¬erence
tensors in characterizing all gauge natural operators Q • QT B QT .
The projection T π : T E ’ T BE de¬nes the dual inclusion E•T BE ’ T — E.


The contracted curvature κ(D) of D is a tensor ¬eld of T — BE — T — BE. On the
other hand, the Liouville vector ¬eld L of E is a section of T E. Hence L — κ(D)
is one of the di¬erence tensors we need.
Let ∆ be the horizontal form of D in the sense of 31.5, so that ∆ is a tensor
ˆ
of T E — T — E. The contracted torsion tensor S of Λ is a section of T — BE and
ˆ ˆ
we construct two kinds of tensor product ∆ — S and S — ∆.
According to 28.13, all natural operators transforming Λ into a section of
T BE — T — BE form an 8-parameter family, which we denote by G(Λ). Hence


L — G(Λ) is an 8-parameter family of gauge natural di¬erence tensors. Finally,
let N (Λ) be the 3-parameter family de¬ned in 45.10.
Proposition. All gauge natural operators Q • QT B QT form the following
15-parameter family
¯
(1 ’ k1 )“(D, N (Λ)) + k1 “(D, N (Λ)) + k2 L — κ(D)+
(1)
ˆ ˆ
k3 ∆ — S + k4 S — ∆ + L — G(Λ)
where bar denotes the conjugate connection.
We remark that the list (1) is essentially simpli¬ed if we assume Λ to be
ˆ
without torsion. Then S vanishes, N (Λ) is reduced to Λ and the 8-parameter
family G(Λ) is reduced to a two-parameter family generated by the two di¬erent
contractions R1 and R2 of the curvature tensor of Λ. This yields the following
Corollary. All gauge natural operators transforming a linear connection D on
E and a linear symmetric connection Λ on T BE into a linear connection on T E
form the following 4-parameter family
¯
(1 ’ k1 )“(D, Λ) + k1 “(D, Λ) + L — (k2 κ(D) + k3 R1 + k4 R2 ).
(2)


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


54.4. To prove proposition 54.3, ¬rst we take into account that, analogously to
51.16 and 23.7, every gauge natural operator A : Q • QT B QT has a ¬nite
order. Let S = J0 Q(R — R ’ R ) be the ¬ber over 0 ∈ Rm of the r-th
r r m n m

jet prolongation of the connection bundle of the vector bundle Rm — Rn ’ Rm ,
let Z r = J0 T Rm be the ¬ber over 0 ∈ Rm of the r-th jet prolongation of the
r

connection bundle of T Rm and V be the ¬ber over 0 ∈ Rm of the connection
bundle of T (Rm — Rn ) with respect to the total projection QT (Rm — Rn ) ’
(Rm — Rn ) ’ Rm . Then all S r , Z r , Rn and V are Wm (GL(n)) =: Wm,n -
r+1 r+1

spaces. In fact, Wm,n acts on Z r by means of the base homomorphisms into
r+1

Gr+1 , on Rn by means of the canonical projection into GL(n) and on V by means
m
1
of the jet homomorphism into Wm,n . The r-th order gauge natural operators
r+1
A : Q • QT B QT are in bijection with Wm,n -equivariant maps (denoted by
the same symbol) A : S r —Z r —Rn ’ V satisfying q —¦A = pr3 , where q : V ’ Rn
is the canonical projection.
Formula 54.2.(2) induces on S r the jet coordinates
p
0 ¤ |±| ¤ r
(1) D± = (Dqi± ),

where ± is a multi index of range m. On Z r , 54.2.(6) induces analogously the
coordinates

Λβ = (Λi ), 0 ¤ |β| ¤ r.
(2) jkβ

On V , we consider the coordinates y = (y p ) and

“A ,
(3) A, B, C = 1, . . . , m + n
BC

given by 54.2.(7). Hence the coordinate expression of any smooth map f : S r —
Z r — Rn ’ V satisfying q —¦ f = pr3 is y p = y p and

“A = fBC (D± , Λβ , y).
A
(4) BC

The coordinate form of a linear isomorphism of vector bundle Rm —Rn ’ Rm
is

xi = f i (x), y p = fq (x)y q .
p
(5) ¯ ¯
r+1
The induced coordinates on Wm,n are

ap = ‚β fq (0),
ai = ‚± f i (0), p
0 < |±| ¤ r + 1, 0 ¤ |β| ¤ r + 1.
± qβ

The above-mentioned homomorphism Wm,n ’ Gr+1 consists in suppressing the
r+1
m
coordinates ap .

The standard action of G2 on Z 0 is given by 25.2.(3). The action of Wm,n
1
m
on S 0 is a special case of 52.1.(5) for the group G = GL(n). This yields

Dqi = ap “r as aj + ap ar aj .
¯p
(6) r sj ˜q ˜i rj ˜q ˜i


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
54. Induced linear connections on the total space of vector and principal bundles 413


The canonical action of GL(n) on Rn is

y p = ap y q .
(7) ¯ q

1
Using standard evaluation, we deduce from 54.2.(7) that the action of Wm,n on
V is (7) and

ai + ai “l = “i al am + “i ap y q al + “i al ap y q + “i ap y r aq y s
¯ lm j k ¯ pl ¯ lp j ¯ pq
(8) jk l jk k
qj rj
qk sk
¯ ¯
il i qk i qr s
(9) al “pj = “qk ap aj + “qr ap asj y
ai “l = “i ak aq + “i aq y s ar
¯ kq j p ¯ qr
(10) l jp p
sj
¯ rs p q
ai “l = “i ar as
(11) l pq

ap y q + ap y q “k + ap “q = “p ak al + “p aq y r al +
¯ ij ¯
(12) ij q ij j
qij ql ri
qk kl

“p ak aq y r + “p aq y s ar y t
¯i ¯ qr tj
rj si
kq

ap y r “k + ap + ap “r = “p ar ak + “p ar as y t
¯ q i ¯ rs q ti
(13) qi r qi
qi
rk rk

ap y r “k + ap + ap “r = “p aj ar + “p ar y t as
¯ ¯ rs ti q
(14) iq r iq jr i q
qi
rk
ap y s “j + ap “s = “p as at
¯ st
(15) qr s qr qr
sj

r+1
54.5. Let H ‚ Wm,n be the subgroup determined by the (r + 1)-th jets of the
products of linear isomorphisms on both Rm and Rn , which is canonically isomor-
phic to GL(m)—GL(n). The standard prolongation procedure and 54.5.(8)“(15)
p
imply that the actions of H on Dqi± , Λi and “A are tensorial.
BC
jkβ
i
Consider the equivariance of fpq with respect to the ¬ber homotheties. This
yields
k ’2 fpq = fpq (D± , Λβ , ky).
i i


Multiplying by k 2 and letting k ’ 0, we obtain

“i = fpq = 0.
i
(1) pq

i
The equivariance of fjp with respect to the ¬ber homotheties gives

k ’1 fjp = fjp (D± , Λβ , ky).
i i


This implies in the same way

“i = 0.
(2) jp

i p
For fpj and fqr we ¬nd quite similarly

“i = 0, “p = 0.
(3) pj qr

p
For fqi the ¬ber homotheties give
p p
fqi = fqi (D± , Λβ , ky).

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

p
Letting k ’ 0 we ¬nd fqi are independent of y p . Then the base homotheties
yield
p p
kfqi = fqi (k 1+|±| D± , k 1+|β| Λβ ).
p p
By the homogeneous function theorem, fqi are linear in Dqi , Λi and independent
jk
of D± , Λβ with |±| > 0, |β| > 0. By the generalized invariant tensor theorem,
we obtain

fqi = aDqi + bδq Dri + cδq Λj + dδq Λj .
p p pr p p
(4) ji ij


Let K ‚ Wm,n be the subgroup characterized by ai = δj , ap = δq . By 25.2.(3),
r+1 i p
q
j
54.4.(6) and 54.4.(13), the equivariance of (4) on K reads

ap = aap + bδq ar + cδq aj + dδq ai .
p p p
ri ij
qi qi ji

This implies a = 1, b = 0, c + d = 0, i.e.

“p = Dqi + c1 δq (Λj ’ Λj ),
p p
c1 ∈ R.
(5) qi ji ij

p
For fiq we deduce in the same way

“p = Dqi + c2 δq (Λj ’ Λj ),
p p
c2 ∈ R.
(6) iq ji ij


The ¬ber homotheties yield that fjk is independent of y p . Then the base
i
p p
homotheties imply that fjk is linear in Dqi , Λi and independent of Dqi± , Λi
i
jk jkβ
with |±| > 0, |β| > 0. By the generalized invariant tensor theorem, we obtain

fjk = aΛi + bΛi + cδj Λl + dδj Λl +
i i i
jk kj lk kl
(7) ip ip
eδk Λl + f δk Λl + gδj Dpk + hδk Dpj .
i i
lj jl


By 25.2.(3), 54.4.(6) and 54.4.(8), the equivariance of (7) on K reads

ai = (a + b)ai + (c + d)δj al + (e + f )δk al + gδj ap + hδk ap .
i i i i
jk jk lk lj pj
pk

This implies a + b = 1, c + d = e + f = g = h = 0, i.e.

“i = (1 ’ c3 )Λi + c3 Λi + c4 δj (Λl ’ Λl ) + c5 δk (Λl ’ Λl ).
i i
(8) jk jk kj lk kl lj jl

p
54.6. The study of fij is quite analogous to 54.5, but it leads to more extended
evaluations. That is why we do not perform all of them in detail here. The ¬ber
homotheties yield
p p
kfij = fij (D± , Λβ , ky).
p
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
jkl
p p
i i
Dqi , Λjk and independent of Dqi± , Λjkβ with |±| > 1, |β| > 1. Using the gen-
p
eralized invariant tensor theorem, we obtain Fijq in the form of a 40-parameter
p
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
p



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
1
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))
1
(1)

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
¬elds.
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].
ar


Remarks
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].
a
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
ar
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
ar
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
ar
in [Kol´ˇ, to appear a]. The greater part of the results from section 52 was
ar
deduced in [Kol´ˇ, to appear b]. Proposition 53.3 was proved for the special
ar
case G = GL(n) in [Kol´ˇ, 91], the general result is ¬rst presented in this book.
ar
Section 54 is based on [Gancarzewicz, Kol´ˇ, 91].
ar




Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
417


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


List of symbols
1j the multi index with j-th component one and all others zero, 13.2
r
± : J (M, N ) ’ M the source mapping of jets, 12.2
β : J r (M, N ) ’ N the target mapping of jets, 12.2
B : FM ’ Mf the base functor, 2.20
C ∞ E, also C ∞ (E ’ M ) the space of smooth sections of a ¬ber bundle
C ∞ (M, N ) the space of smooth maps of M into N

Cx (M, N ) the space of germs at x ∈ M , 1.4
conja : G ’ G the conjugation in a Lie group G by a ∈ G, 4.24
d usually the exterior derivative, 7.8
the algebra of dual numbers, 37.1
D
Dn = J0 (Rn , R) the algebra of r-jets of functions, 40.5
r r

(E, p, M, S), also simply E usually a ¬ber bundle with total space E, base M ,
and standard ¬ber S, 9.1
F usually the ¬‚ow operator of a natural bundle F , 6.19, 42.1
X
Flt , also Fl(t, X) the ¬‚ow of a vector ¬eld X, 3.7
FM the category of ¬bered manifolds and ¬ber respecting mappings, 2.20
FMm the category of ¬bered manifolds with m-dimensional bases and ¬ber
respecting mappings with local di¬eomorphisms as base maps, 12.16
FMm,n the category of ¬bered manifolds with m-dimensional bases and n-
dimensional ¬bers and locally invertible ¬ber respecting mappings,
17.1

FM the category of star bundles, 41.1
usually a general Lie group with multiplication µ : G — G ’ G, left
G
translation », and right translation ρ
the Lie algebra of a Lie group G
g
r
Gm the jet group (di¬erential group) of order r in dimension m, 12.6
r
the jet group of order r of the category FMm,n , 18.8
Gm,n
GL(n) the general linear group in dimension n with real coe¬cients, 4.30
GL(Rn , E) the linear frame bundle of a vector bundle E, 10.11
short for the k — k-identity matrix IdRk
Ik
r
invJ (M, N ) the bundle of invertible r-jets of M into N , 12.3
J rE the bundle of r-jets of local sections of a ¬ber bundle E ’ M , 12.16
r
J (M, N ) the bundle of r-jets of smooth functions from M to N , 12.2
j r f (x), also jx f the r-jet of a mapping f at x, 12.2
r
r
Kn the functor of (n, r)-contact elements, 12.15
L the Lie derivative, 6.15, 47.4
: G — S ’ S usually a left action of a Lie group
L(V, W ) the space of all linear maps of vector space V into a vector space W
LP = P [g, Ad] the adjoint bundle of principal bundle P (M, G), 17.6
Lr the r-th order skeleton of the category Mf , 12.6
M usually a (base) manifold

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