a¯

tion Tm G = G (g—Rm— ) with the multiplication (a, Z)(¯, Z) = (a¯, Ad(¯’1 )Z+

1

a a

¯ see 15.6. Let us now express the action of Wm G = (G1 —G) (g—Rm— ) on

1

Z), m

∼ j 1 •(0, e) ∈ W 1 G, and Y ∼ j 1 s ∈ J 1 (Rm —G),

m—

S = (g—R ). Given (A, a, Z) = =0

m 0

1

s(0) = (0, e), we have A = j0 •0 , a = pr2 —¦ •(0, e), Z = T »a’1 —¦ T0 • and ¯

1 1

Y = T0 s, where s = pr2 —¦ s. By de¬nition, Q•(j0 s) = j0 q and if we require

˜ ˜

’1

q (0) := pr2 —¦q(0) = e we have q = ρa —¦•—¦s—¦•’1 , where ρ denotes the principal

˜ 0

right action of G. Then we evaluate

’1

—¦ µ —¦ (•, s) —¦ •’1 = conj(a) —¦ µ —¦ (»a’1 —¦ •, s) —¦ •’1 .

q = ρa

˜ ¯˜ ¯˜

0 0

Hence by applying the tangent functor we get the action in form

(A, a, Z)(Y ) = Ad(a)(Y + Z) —¦ A’1 .

(1)

Proposition. For every principal bundle (P, p, M, G) the bundle of principal

connections QP is the associated ¬ber bundle W 1 P [g — Rm— , ] with the action

given by (1).

Since the standard ¬ber of QP is a Euclidean space, there are always global

sections of QP and so we have reproved in this way that every principal ¬ber

bundle admits principal connections.

17.6. The a¬ne structure on QP . In 17.2 and 17.3 we deduced that every

principal connection on P determines a bijection between principal connections

on P and the right invariant sections in C ∞ (V P — T — M ’ P ). For every

principal ¬ber bundle (P, p, M, G), let us denote by LP the associated vector

bundle P [g, Ad]. Since the fundamental ¬eld mapping (u, A) ’ ζA (u) ∈ Vu P

identi¬es V P with P — g and (ua, Ad(a’1 )(A)) ’ T Ra —¦ ζA (u), there is the

induced identi¬cation P [g, Ad] ∼ V P/G. Hence every element in LP can be

=

viewed as a right invariant vertical vector ¬eld on a ¬ber of P . Let us now

consider g — Rm— as a standard ¬ber of the vector bundle LP — T — M with the

left action of the product of Lie groups G — G1 given by

m

(a, A)(Y ) = Ad(a)(Y ) —¦ A’1 .

(1)

At the same time, we can view g — Rm— as the standard ¬ber of QP with the

1

action of Wm G given in 17.5.(1). Using the canonical a¬ne structure on the

vector space g — Rm— , we get for every two elements Y1 , Y2 ∈ g — Rm—

((A, a, Z), Y1 ) ’ ((A, a, Z), Y2 ) = Ad(a)(Y1 ’ Y2 ) —¦ A’1 ,

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

162 Chapter IV. Jets and natural bundles

cf. 15.6.(3). Hence QP is an associated a¬ne bundle to W 1 P with the modelling

vector bundle LP — T — M = W 1 P [g — Rm— ] corresponding to the action (1) of

the Lie subgroup G1 — G ‚ Wm G via the canonical homomorphism Wm G ’

1 1

m

G1 — G. Since the curvature R of a principal connection is a right invariant

m

section in C ∞ (V P — Λ2 T — M ’ P ), we can view the curvature as an operator

R : C ∞ (QP ’ M ) ’ C ∞ (LP — Λ2 T — M ’ M ). By the de¬nition, R commutes

with the action of the PBm (G)-morphisms, so that this is a typical example of

the so called gauge natural operators which will be treated in chapter XII.

17.7. Principal connections on higher order frame bundles. Let us con-

sider a frame bundle P r M and the bundle of principal connections QP r M . The

composition Q —¦ P r is a bundle functor on Mfm of order r + 1, so that there

is the canonical structure QP r M ∼ P r+1 M [gr — Rm— ], but there also is the

= m

∼ W 1 P r [gr — Rm— ; ] described in 17.6. It is an easy exer-

r

identi¬cation QP M = m

cise to verify that the former structure of an associated bundle is obtained from

the latter one by the natural reduction iM : P r+1 M ’ W 1 P r M , see proposition

15.7.

The most important case is r = 1, since the functor QP 1 associates to each

manifold M the bundle of linear connections on M . Let us deduce the coordinate

expressions of the actions of Wm G1 and G2 on (g1 — Rm— ) = Hom(Rm , gl(m)).

1

m m m

Given (A, B, Z) ∈ Wm Gm , A = (aj ) ∈ Gm , B = (bi ) ∈ G1 , Z = (zjk ) ∈

11 i 1 i

m

j

(g1 — Rm— ), “ = (“i ) ∈ (g1 — Rm— ), we have Ad(B)(Z) = (bi znj ˜n ), so that

m

bk

m m m

jk

17.5.(1) implies

(A, B, Z)(“i ) = (bi (“m + znl )˜l ˜n ).

m

ak bj

jk m nl

The coordinate expression of the homomorphism i0 : G2 ’ Wm G1 deduced in

1

m m

15.8 yields the formula

(ai , ai )(“i ) = (ai “m al an + ai al an ).

m nl ˜k ˜j nl ˜k ˜j

j jk jk

We remark that the “i introduced in this way di¬er from the classical Christo¬el

jk

symbols, [Kobayashi, Nomizu, 69], by sign and by the order of subscripts, see

17.15.

Let us mention brie¬‚y the second order case. We have to deal with (A, B, Z) ∈

Wm Gm , A = (ai ) ∈ G1 , B = (bi , bi ) ∈ G2 , Z = (zjk , zjkl ) ∈ (g2 — Rm— ). We

12 i i

m m m

j j jk

compute

Ad(B)(Z) —¦ A’1 = (bi zsm am˜s , bi zsm am˜s

p

˜k bj p p ˜l bjk

p

+ bi zmnq aq ˜n˜m + bi zmn an˜m˜s + bi zmn an˜p˜m )

p p s

˜ l b j bk ˜ l bj bk ˜ l b j bk

p ps ps

and we have to compose this action with the homomorphism i0 : G3 ’ Wm G2 .

1

m m

For every a = (ai , ai , ai ) ∈ G3 , the formula derived in 15.8 implies

m

j jk jkl

a.(“i , “i ) = ai “m al an + ai al an ,

m nl ˜k ˜j nl ˜k ˜j

jk jkl

ai “p aq an am + ai “p am as + ai “p an am as

p mnq ˜l ˜k ˜j p sm ˜l ˜jk ps mn ˜l ˜j ˜k

+ ai “s am ap an + ai aq am an + ai as am .

ps nm ˜l ˜j ˜k mnq ˜l ˜k ˜j sm ˜kj ˜l

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

17. Connections and the absolute di¬erentiation 163

17.8. The absolute di¬erential. Let us consider a ¬xed principal connection

“ : P ’ J 1 P on a principal ¬ber bundle (P, p, M, G) and an associated ¬ber

bundle E = P [S; ]. We recall the maps q : P — S ’ E and „ : P —M E ’ S, see

10.7, and we denote u : = q(u, ) : S ’ Ep(u) . Hence given local sections σ : M ’

˜

P and s : M ’ E with a common domain U ‚ M and a point x ∈ U , there is

’1

the map •σ,s : U y ’ σ(x) —¦ σ(y) —¦ s(y) ∈ Ex , i.e. •σ,s = q(σ(x), ) —¦ „ —¦ (σ, s).

In fact we use the local trivialization of E induced by σ to describe the local

behavior of s in a single ¬ber. If P and (consequently) also E are trivial bundles

and σ(x) = (0, e), then we get just the projection onto the standard ¬ber. Since

1

the principal connection “ associates to every u ∈ Px a 1-jet “(u) = jx σ of a

section σ, for every local section s : M ’ E and point x in its domain the one

jet of •σ,s at x describes the local behavior of s at x up to the ¬rst order. Our

construction does not depend on the choice of u ∈ Px , for “ is right invariant.

So we de¬ne the absolute (or covariant) di¬erential s(x) of s at x with respect

to the principal connection “ by

s(x) = jx •σ,s ∈ Jx (M, Ex )s(x) ∼ Hom(Tx M, Vs(x) E).

1 1

=

If E is an associated vector bundle, then there is the canonical identi¬cation

Vs(x) E = Ex . Then we have s(x) ∈ Hom(Tx M, Ex ) and we shall see that this

coincides with the values of the covariant derivative as de¬ned in section 11.

We can de¬ne a structure of an associated bundle on the union of the man-

1

ifolds Jx (M, Ex ), x ∈ M , where the mappings s take their values. Let

us consider the principal ¬ber bundle P 1 M —M P with the principal action

r(a1 ,a2 ) (u1 , u2 ) = (u1 .a1 , u2 .a2 ) of the Lie group G1 — G (here the dots mean

m

the obvious principal actions). We de¬ne

„ : (P 1 M —M P ) —M (∪x∈M Jx (M, Ex )) ’ Tm S

1 1

„ ((j0 f, u), jx •) = j0 (˜’1 —¦ • —¦ f ).

1 1 1

u

Let us further de¬ne a left action ¯ of G1 —G on Tm S by (remember E = P [S; ])

1

m

¯((j 1 h, a2 ), j 1 q) = j 1 ( —¦ q) —¦ j0 h’1 .

1

a2

0 0 0

One veri¬es easily that „ determines the structure of the associated bundle

E1 = (P 1 M —M P )[Tm S; ¯] and that for every section s : M ’ E its absolute

1

di¬erential s with respect to a ¬xed principal connection “ on P is a smooth

section of E1 . Hence can be viewed as an operator

: C ∞ (E) ’ C ∞ ((P 1 M —M P )[Tm S; ¯]).

1

17.9. Absolute di¬erentiation along vector ¬elds. Let E, P , “ be as in

17.8. Given a tangent vector Xx ∈ Tx M , we de¬ne the absolute di¬erentiation

in the direction Xx of a section s : M ’ E to be the value s(x)(Xx ) ∈ Vs(x) E.

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

164 Chapter IV. Jets and natural bundles

Applying this procedure to a vector ¬eld X ∈ X(M ) we get a map M’

Xs:

V E with the following properties

πE —¦

(1) Xs =s

(2) s=f Xs +g s

f X+gY Y

for all vector ¬elds X, Y ∈ X(M ) and smooth functions f , g on M , πE : V E ‚

T E ’ E being the canonical projection.

So every X ∈ X(M ) determines an operator X : C ∞ (E) ’ C ∞ (V E) and

the whole procedure of the absolute di¬erentiation can be viewed as an operator

: C ∞ (T M —M E) ’ C ∞ (V E).

By the de¬nition of the connection form ¦E of the induced connection “E , it

holds

= ¦E —¦ T s —¦ X

(3) Xs

= T s —¦ X ’ (“E X) —¦ s.

(4) Xs

17.10. The frame forms. For every vector ¬eld X ∈ X(M ) and every map

s : P ’ S we de¬ne

¯

P ’ T S, T s —¦ “E X

Xs:

¯ Xs =

¯ ¯

s : P 1 M —M P ’ Tm S,

1

s(v, u) = T s —¦ T σ —¦ v,

¯ ¯ ¯

1

where “(u) = jx σ, x = p(u). We call s the absolute di¬erential of s while

¯ ¯ Xs

¯

is called the absolute di¬erential along X.

Proposition. Let s : P ’ S be the frame form of a section s : M ’ E. Then

¯

s is the frame form of s and for every X ∈ X(M ), X s is the frame form of

¯ ¯

X s.

Proof. The map X s is a section of V E = P [T S] and s(u) = „E (u, s —¦ p(u)),

¯

1 1

u ∈ P . Further, for every u ∈ Px with “(u) = jx σ, we have s(x) = jx (˜—¦¯—¦σ) ∈

us

Hom(Tx M, Vs(x) E). Hence for every X ∈ X(M ) we get X s = T u —¦ T (¯ —¦ σ) —¦ X

˜ s

and since the di¬eomorphism T S ’ (V E)x determined by u ∈ P is just T u, the

˜

frame form of X s is X s. ¯

In order to prove the other equality, let us evaluate

s(x) = {(v, u), (jx (˜’1 —¦ •)) —¦ v}.

1

u

1

Since • = u —¦ s —¦ σ, where “(u) = jx σ, the frame form of

˜¯ s is s.

¯

17.11. If E = P [S; ] is an associated vector bundle, then we can use the canon-

ical identi¬cation S ∼ Ty S for each point y ∈ S. Consider a section s : M ’ E

=

1

and its frame form s : P ’ S. Then s(x) ∈ Jx (M, Ex ) can be viewed as a

¯

value of a form Ds ∈ „¦1 (M ; E). The corresponding S-valued tensorial 1-form

D¯ : T P ’ S is de¬ned by D¯ = d¯ —¦ χ = (χ— d)(¯), where χ is the horizontal

s s s s

projection of “E . Of course, this formula de¬nes the absolute di¬erentiation

D : „¦k (P ; S) ’ „¦k+1 (P ; S) for all k ≥ 0, cf. section 11. The absolute di¬er-

entials of higher order can also be de¬ned in the nonlinear case. However, this

requires an inductive procedure and we refer the reader to [Kol´ˇ, 73 b].

ar

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

17. Connections and the absolute di¬erentiation 165

17.12. We are going to deduce a general coordinate formula for the absolute

di¬erentiation of sections of an arbitrary associated ¬ber bundle. We shall do it

in a geometric way, which reduces the problem to the proposition 16.6. For every

principal connection “ : P ’ J 1 P the image of the map “ de¬nes a reduction

“

R(“) : P 1 M —M P ’ P 1 M —M “(P ) ’ P 1 M —M J 1 P = W 1 P

’

of the principal bundle W 1 P to the structure group

G1 — G ’ G1 T m G = G1

1

(G (g — Rm— )).

m m m

˜

Let us write θ for the restriction of the canonical form θ on W 1 P to P 1 M —M

“(P ), let ω be the connection form of “ and θM will denote the canonical form

θM ∈ „¦1 (P 1 M ; Rm ).

Lemma. The following diagram is commutative

u w TP M

β— pr1

T (P 1 M —M “(P )) 1

TP

ω

u u u

θM

˜

θ

u wR

pr2 pr1

m m

R •g

g

Proof. For every u ∈ W 1 P , u = j 1 ψ(0, e), β(u) = u, we have the isomorphism

¯

u : R • g ’ Tu P and for every X ∈ Tu W P , θ(X) = u’1 (β— X). If X ∈

m 1

˜ ˜

¯

1 m

T (P M —M “(P )), we denote θ(X) = Y1 + Y2 ∈ R • g. Then u(Y1 ) = T (ψ1 —¦

˜

ψ0 )Y1 = χ(β— X) and u(Y2 ) = β— X ’ u(Y1 ) = ¦(β— X), where ¦ and χ are the

˜ ˜

vertical and horizontal projections determined by “. Since the restriction of u to

˜

m

the second factor in R •g coincides with the fundamental vector ¬eld mapping,

the commutativity of the left-hand square follows.

The commutativity of the right-hand one was proved in 16.4.

17.13. Lemma. Let s : M ’ E be a section, s : P ’ S its frame form and

¯

let s : W P ’ Tm S be the frame form of j s. Then for all u ∈ P 1 M —M P ∼

1 1 1 1

¯ =

1 1 1

P M —M “(P ) ‚ W P it holds s (u) = s(u).

¯ ¯

Proof. If u = j 1 ψ(0, e), u = β(u), then “(¯) = j 1 ψ1 (ψ0 (0)). Since we know

¯ u

s (u) = j0 („E (ψ1 —¦ ψ0 , s —¦ ψ0 )), we get s(u) = j0 (¯ —¦ ψ1 —¦ ψ0 ) = s1 (u).

1 1 1

¯ ¯ s ¯

17.14. Proposition. Let E, S, P , “, ω be as before and consider a local chart

(U, •), • = (y p ), on S. Let ei , i = 1, . . . , m be the canonical basis in Rm and e± ,

i

± = m + 1, . . . , m + dimG be a base of Lie algebra g. Let us denote θM = θM ei

the canonical form on P 1 M , ω = ω ± e± , j1 and j2 be the canonical projections

¯i

— —i ‚

on P 1 M —M P . Further, let us write ω ± = j2 ω ± , θM = j1 θM and let ·± (y) ‚yp

p

¯

be the fundamental vector ¬elds corresponding to e± . For a section s : M ’ E

let ap , ap be the coordinate functions of s on P 1 M —M P while ap be those of

¯

i

s. Then it holds

dap + ·± (aq )¯ ± = ap θM . ¯i

p

ω i

Proof. In 16.6 we described the coordinate functions bp , bp of j 1 s de¬ned on

i

W 1 P , bp = β — ap , dbp + ·± (bq )θ± = bp θi . According to 17.13, the functions ap ,

p

¯ i

ap are restrictions of bp , bp to P 1 M —M P . But then the proposition follows

i i

from lemma 17.12.

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

166 Chapter IV. Jets and natural bundles

17.15. Example. We ¬nd it instructive to apply this general formula to the

simplest case of the absolute di¬erential of a vector ¬eld ξ on a manifold M

with respect to a classical linear connection “ on M . Since we consider the

standard action y i = ai y j of GL(m) on Rm , the fundamental vector ¬elds ·j oni