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m

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

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