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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

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