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 .

qβ

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