1

β —¦ ρA = ρa —¦ β

¯

(¯A )— X = j0 (¯A —¦ c) ∈ TuA W 1 P

1

ρ ρ

θP —¦ (¯A )— X = j0 (•’1 —¦ ψ ’1 —¦ β —¦ ρA —¦ c) = j0 (ρa —¦ •’1 —¦ ψ ’1 —¦ β —¦ c).

1 1

ρ ¯

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156 Chapter IV. Jets and natural bundles

Hence

’1

1

—¦ β —¦ c) = θP —¦ (¯A )— (X).

—¦ θP (X) = A’1 (j0 (ψ ρ

A’1

Unfortunately, θP is not horizontal since the principal bundle projection on

1

W P is p —¦ β.

16.4. Lemma. Let (P, p, M, G) be a principal bundle and q : W 1 P = P 1 M —M

J 1 P ’ P 1 M be the projection onto the ¬rst factor. Then the following diagram

u

commutes

θP

Rm • g T W 1P

u u

pr1 Tq

u θM

Rm T P 1M

Proof. Consider X = j0 c ∈ Tu W 1 P , u = j 1 ψ(0, e). Then T q(X) = j 1 (q —¦ c) and

1

1

q(u) = j0 ψ0 . It holds

’1

pr1 —¦ θP (X) = pr1 (j0 (ψ ’1 —¦ β —¦ c)) = j0 (ψ0 —¦ p —¦ β —¦ c)

1 1

’1

1

= j0 (ψ0 —¦ p —¦ q —¦ c) = θM —¦ T q(X)

¯

where p : P 1 M ’ M is the canonical projection.

¯

16.5. Canonical forms on frame bundles. Let us consider a frame bun-

dle P r M and the ¬rst order principal prolongation W 1 (P r’1 M ). We know

1 r’1

the canonical form θ ∈ „¦1 (W 1 (P r’1 M ); Rm • gr’1 )Wm Gm and the reduction

m

iM : P r M ’ W 1 (P r’1 M ) to the structure group Gr , see 15.7. So we can de¬ne

m

the canonical form θr on P r M to be the pullback i— θ ∈ „¦1 (P r M, Rm • gr’1 ).

m

M

By virtue of 16.3 there is the linear action ¯ = —¦ κ where κ is the group ho-

momorphism corresponding to iM , see 15.7, and θr is a pseudo tensorial form

of type ¯. The form θr can also be described directly. Given X ∈ Tu P r M ,

¯

we set u = πr’1 u, X = T πr’1 (X) ∈ Tu P r’1 M . Since every u = j0 f ∈ P r M

r r r

¯ ¯

determines a linear map u = T(0,e) P r’1 f : Rm • gr’1 ’ Tj r’1 f P r’1 M we get

˜ m 0

’1 ¯

r

θ (X) = u (X).

˜

16.6. Coordinate functions of sections of associated bundles. Let us

¬x an associated bundle E = P [S; ] to a principal bundle (P, p, M, G). The

canonical map „E : P —M E ’ S determines the so called frame form σ : P ’ S

of a section s : M ’ E, σ(u) = „E (u, s(p(u))). As we proved in 15.5, J r E =

W r P [Tm S; r ], m = dimM , and so for every ¬xed section s : M ’ E the frame

r

form σ r of its r-th prolongation j r s is a map σ r : W r P ’ Tm S. If we choose

r

some local coordinates (U, •), • = (y p ), on S, then there are the induced local

coordinates y± on (π0 )’1 (U ) ‚ Tm S, 0 ¤ |±| ¤ r, and for every section s : M ’

p r r

E the compositions y± —¦σ r de¬ne (on the corresponding preimages) the coordinate

p

functions ap of j r s induced by the local chart (U, •). We deduced in 15.5 that

±

for every u = j r ψ(0, e) = (j0 ψ0 , j r ψ1 (ψ0 (0))) ∈ W r P

r

σ r (u) = j0 „E (ψ1 —¦ ψ0 , s —¦ ψ0 ).

r

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16. Canonical di¬erential forms 157

In particular, for the ¬rst order case we get

ap (u) = y p —¦ σ(u)

ap (u) = dy p (j0 „E (ψ1 —¦ ψ0 , s —¦ ψ0 ) —¦ c)

1

i

where c : R ’ Rm is the curve t ’ txi .

We shall describe the ¬rst order prolongation in more details. Let us denote ei ,

i = 1, . . . , m, the canonical basis in Rm and let e± , ± = m + 1, . . . , m + dimG, be

a linear basis of the Lie algebra g. So the canonical form θ on W 1 P decomposes

into θ = θi ei + θ± e± . Let us further write Y± for the fundamental vector ¬elds

on S determined by e± and let ω ± be the dual basis to that induced from e±

‚

p

on V P . Hence if the coordinate formulas for Y± are Y± = ·± (y) ‚yp , then for

z ∈ Ex , u ∈ Px , X ∈ Vu P , y = „E (u, z) we get

„E ( , z)— X = ’Y± (y)ω ± (X) = ’·± (y)ω ± (X) ‚yp .

p ‚

The next proposition describes the coordinate functions of j 1 s on W 1 P by

means of the canonical form θ and the coordinate functions ap of s on P .

Proposition. Let ap be the coordinate functions of a geometric object ¬eld

¯

s : M ’ E and let ap , ap be the coordinate functions of j 1 s. Then ap = ap —¦ β,

¯

i

1

where β : W P ’ P is the target projection, and

dap + ·± (aq )θ± = ap θi .

p

i

Proof. The equality ap = ap —¦ β follows directly from the de¬nition. We shall

¯

evaluate da (X) with arbitrary X ∈ Tu W 1 P , where u ∈ W 1 P , u = j 1 ψ(0, e) =

p

(j0 ψ0 , j 1 ψ1 (ψ0 (0))). The frame u determines the linear isomorphism

1

u = T(0,e) ψ : Rm • g ’ Tu P,

˜ ¯

u = β(u). We shall denote θi (X) = ξ i , θ± (X) = ξ ± , so that θ(X) = u’1 (β— X) =

¯ ˜

¯ ¯ ¯ ¯ ¯

ξ ei + ξ e± . Let us write X = β— X = X1 + X2 with X1 = u(ξ ei ), X2 = u(ξ ± e± )

i ± i

˜ ˜

i m

and let c be the curve t ’ tξ ei on R . We have

a¯

d¯p (X1 ) = dy p (j0 (σ —¦ ψ1 —¦ ψ0 —¦ c))

1

= dy p (j0 („E (ψ1 —¦ ψ0 , s —¦ ψ0 ) —¦ c)) = ap (u)ξ i

1

i

p¯ ¯ 2 ) = ’· p (aq (¯))ξ ± ‚ p .

p

d¯ (X2 ) = dy („E ( , s(p(¯)))— X

a u u ± ‚y

Hence

dap (X) = d¯p (β— X) = ap (u)θi (X) ’ ·± (aq (u))θ± (X).

p

a i

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

158 Chapter IV. Jets and natural bundles

17. Connections and the absolute di¬erentiation

17.1. Jet approach to general connections. The (general) connections on

any ¬ber bundle (Y, p, M, S) were introduced in 9.3 as the vector valued 1-forms

¦ ∈ „¦1 (Y ; V Y ) with ¦ —¦ ¦ = ¦ and Im¦ = V Y . Equivalently, any connection

is determined by the horizontal projection χ = idT Y ’ ¦, or by the horizontal

subspaces χ(Ty Y ) ‚ Ty Y in the individual tangent spaces, i.e. by the horizontal

distribution. But every horizontal subspace χ(Ty Y ) is complementary to the

vertical subspace Vy Y and therefore it is canonically identi¬ed with a unique

1 1 1 1

element jy s ∈ Jy Y . On the other hand, each jy s ∈ Jy Y determines a subspace in

Ty Y complementary to Vy Y . This leads us to the following equivalent de¬nition.

De¬nition. A (general) connection “ on a ¬ber bundle (Y, p, M ) is a section

“ : Y ’ J 1 Y of the ¬rst jet prolongation β : J 1 Y ’ Y .

Now, the horizontal lifting γ : T M —M Y ’ T Y corresponding to a connection

1

“ is given by the composition of jets, i.e. for every ξx = j0 c ∈ Tx M and y ∈ Y ,

p(y) = x, we have γ(ξx , y) = “(y) —¦ ξx . Given a vector ¬eld ξ, we get the “-

lift “ξ ∈ X(Y ), “ξ(y) = “(y) —¦ ξ(p(y)) which is a projectable vector ¬eld on

Y ’ M . Note that for every connection “ on p : Y ’ M and ξ ∈ Ty Y it holds

χ(ξ) = γ(T p(ξ), y) and ¦ = idT Y ’ χ.

Since the ¬rst jet prolongations carry a natural a¬ne structure, we can con-

sider J 1 as an a¬ne bundle functor on the category FMm,n of ¬bered manifolds

with m-dimensional bases and n-dimensional ¬bers and their local ¬bered mani-

fold isomorphisms. The corresponding vector bundle functor is V — T — B, where

B : FMm,n ’ Mfm is the base functor, see 12.11. The choice of a (general)

connection “ on p : Y ’ M yields an identi¬cation of J 1 Y ’ Y with V Y —T — M .

Chosen any ¬bered atlas •± : (Rm+n ’ Rm ) ’ (Y ’ M ) with •± (Rm+n ) = U± ,

we can use the canonical ¬‚at connection on Rm+n to get such identi¬cations on

J 1 U± . In this way we obtain the local sections γ± : U± ’ (V — T — B)(U± ) which

correspond to the Christo¬el forms introduced in 9.7. More explicitly, if we pull

back the sections γ± to Rm+n ’ Rm and use the product structure, then we

obtain exactly the Christo¬el forms.

In 9.4 we de¬ned the curvature R of a (general) connection “ by means of the

Fr¨licher-Nijenhuis bracket, 2R = [¦, ¦]. It holds R[X1 , X2 ] = ¦([χX1 , χX2 ])

o

for all vector ¬elds X1 , X2 on Y . In other words, given two vectors A1 , A2 ∈

Ty Y , we extend them to arbitrary vector ¬elds X1 and X2 on Y and we have

R(A1 , A2 ) = ¦([χX1 , χX2 ](y)). Clearly, we can take for X1 and X2 projectable

vector ¬elds over some vector ¬elds ξ1 , ξ2 on M . Then χXi = γξi , i = 1, 2. This

implies that R can be interpreted as a map R(y, ξ1 , ξ2 ) = ¦([γξ1 , γξ2 ](y)). Such

a map is identi¬ed with a section Y ’ V Y — Λ2 T — M . Obviously, the latter

formula can be rewritten as

R(y, ξ1 , ξ2 ) = [γξ1 , γξ2 ](y) ’ γ([ξ1 , ξ2 ])(y).

This relation is usually expressed by saying that the curvature is the obstruction

against lifting the bracket of vector ¬elds.

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

17. Connections and the absolute di¬erentiation 159

17.2. Principal connections. Consider a principal ¬ber bundle (P, p, M, G)

with the principal action r : P — G ’ P . We shall also denote by r the canonical

right action r : J 1 P — G ’ J 1 P given by rg (jx s) = jx (rg —¦ s) for all g ∈ G

1 1

and jx s ∈ J 1 P . In accordance with 11.1 we de¬ne a principal connection “ on

1

a principal ¬ber bundle P with a principal action r as an r-equivariant section

“ : P ’ J 1 P of the ¬rst jet prolongation J 1 P ’ P .

Let us recall that for every principal bundle, there are the canonical right

actions of the structure group on its tangent bundle and vertical tangent bundle.

By de¬nition, for every vector ¬eld ξ ∈ X(M ) and principal connection “ the “-

lift “ξ is a right invariant projectable vector ¬eld on P . Furthermore, a principal

connection induces an identi¬cation J 1 P ∼ V P — T — M which maps principal

=

connections into right invariant sections.

17.3. Induced connections on associated ¬ber bundles. Let us consider

an associated ¬ber bundle E = P [S; ]. Every local section σ of P determines a

local trivialization of E. Hence the idea of the de¬nition of induced connections

used in 11.8 gets the following simple form. For any principal connection “ on

P we de¬ne the section “E : E ’ J 1 E by “E {u, s} = jx {σ, s}, where u ∈ Px

1

ˆ

1

and s ∈ S are arbitrary, “(u) = jx σ and s means the constant map M ’ S

ˆ

with value s. It follows immediately that the parallel transport PtE (c, {u, s}) of

an element {u, s} ∈ E along a curve c : R ’ M is the curve t ’ {Pt(c, u, t), s}

where Pt is the G-equivariant parallel transport with respect to the principal

connection on P .

We recall the canonical principal bundle structure (T P, T p, T M, T G) on T P

and T E = T P [T S, T ], see 10.18. The horizontal lifting determined by the

induced connection “E is given for every ξ ∈ X(M ) by

“E ξ({u, s}) = {“ξ(u), 0s } ∈ (T E)ξ(p(u)) ,

(1)

where 0s ∈ Ts S is the zero tangent vector. Let us now consider an arbi-

trary general connection “E on E. Chosen an auxiliary principal connection

“P on P , we can express the horizontal lifting γE in the form “E ξ({u, s}) =

{“P ξ(u), γ (ξ(p(u)), s)}. The map γ is uniquely determined if the action is in-

¯ ¯

¬nitesimally e¬ective, i.e. the fundamental ¬eld mapping g ’ X(S) is injective.

Then it is not di¬cult to check that the horizontal lifting γE can be expressed

in the form (1) with certain principal connection “ on P if and only if the map

γ takes values in the fundamental ¬elds on S. This is equivalent to 11.9.

¯

17.4. The bundle of (principal) connections. We intend to treat principal

connections as sections of an appropriate bundle. We have de¬ned them as right

invariant sections of the ¬rst jet prolongation of principal bundles, so that given

a principal connection “ on (P, p, M, G) and a point x ∈ M , its value on the

whole ¬ber Px is determined by the value in any point from Px . We de¬ne QP

to be the set of orbits J 1 P/G. Since the source projection ± : J 1 P ’ M is G-

invariant, we have the projection QP ’ M , also denoted by ±. Furthermore, for

¯¯¯ ¯

every morphism of principal ¬ber bundles (•, •1 ) : (P, p, M, G) ’ (P , p, M , G)

¯

over •1 : G ’ G it holds

J 1 •(jx (ra —¦ s)) = j•0 (x) (r•1 (a) —¦ • —¦ s —¦ •’1 )

1 1

0

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160 Chapter IV. Jets and natural bundles

¯

for all jx s ∈ J 1 P , a ∈ G. Hence the map J 1 • : J 1 P ’ J 1 P factors to a map

1

¯

Q• : QP ’ QP and Q becomes a functor with values in ¬bered sets. More

¯

explicitly, for every jx s in an orbit A ∈ QP the value Q•(A) is the orbit in J 1 P

1

going through J 1 •(jx s). By the construction, we have a bijective correspondence

1

between the sections of the ¬bered set QP ’ M and the G-equivariant sections

of J 1 P ’ P which are smooth along the individual ¬bers of P . It remains to

de¬ne a suitable smooth structure on QP .

Let us ¬rst assume P = Rm — G. Then there is a canonical representative

in each orbit J 1 (Rm — G)/G, namely jx s with s(x) = (x, e), e ∈ G being

1

the unit. Moreover, J 1 (Rm — G) is identi¬ed with Rm — J0 (Rm , G), jx s ’

1 1

(x, j0 (pr2 —¦ s —¦ tx )). Hence there is the induced smooth structure Q(Rm — G) ∼

1

=

m 1 m 1 m m

R — J0 (R , G)e and the canonical projection J (R — G) ’ Q(R — G)

becomes a surjective submersion. Let PBm be the category of principal ¬ber

bundles over m-manifolds and their morphisms covering local di¬eomorphisms

¯

on the base manifolds. For every PBm -morphism • : Rm — G ’ Rm — G and

element jx s ∈ A ∈ Q(Rm — G) with s(x) = (x, e), the orbit Q•(A) is determined

1

by J 1 •(jx s). This means that

1

’1

—¦ • —¦ s —¦ •’1 )

Q•(jx s) = j•0 (x) (ra

1 1

0

where a = pr2 —¦ •(x, e) and consequently Q• is smooth.

Now for every principal ¬ber bundle atlas (U± , •± ) on a principal ¬ber bundle

P the maps Q•± form a ¬ber bundle atlas (U± , Q•± ) on QP ’ M . Let us

summarize.

Proposition. The functor Q : PBm ’ FMm associates with each principal

¬ber bundle (P, p, M, G) the ¬ber bundle QP over the base M with standard

¬ber J0 (Rm , G)e . The smooth sections of QP are in bijection with the principle

1

connections on P .

The functor Q is a typical example of the so called gauge natural bundles

which will be studied in detail in chapter XII. On replacing the ¬rst jets by

k-jets in the above construction, we get the functor Qk : PBm ’ FMm of k-th

order (principal) connections.

17.5. The structure of an associated bundle on QP . Let us consider a

principal ¬ber bundle (P, p, M, G) and a local trivialization ψ : Rm — G ’ P .

By the de¬nition, the restriction of Qψ to the ¬ber S := (Q(Rm — G))0 is a

di¬eomorphism onto the ¬ber QPψ0 (0) . Since the functor Q is of order one, this

di¬eomorphism is determined by j 1 ψ(0, e) ∈ W 1 P , cf. 15.3. For the same reason,

every element j 1 •(0, e) ∈ Wm G determines a di¬eomorphism Q•|S : S ’ S. By

1

1

the de¬nition of the Lie group structure on Wm G, this de¬nes a left action of

Wm G on S. We de¬ne a mapping q : W 1 P — S ’ QP by

1

q(j 1 ψ(0, e), A) = Qψ(A).

Since q(j 1 (ψ —¦•)(0, e), Q•’1 (A)) = Qψ —¦Q•—¦Q•’1 (A), the map q identi¬es QP

with W 1 P [S; ]. We shall see in chapter XII that the map q is an analogy to our

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

17. Connections and the absolute di¬erentiation 161

identi¬cations of the values of bundle functors on Mfm with associated bundles

to frame bundles and that this construction goes through for every gauge natural

bundle.

We are going to describe the action in more details. We know that

S = J0 (Rm — G)/G ∼ (Rm — Tm G)0 /G ∼ J0 (Rm , G)e ∼ g — Rm— ,

1 1

=1

= =

see 17.4, and Wm G = G1

1 1

Tm G. Moreover, we have introduced the identi¬ca-