E

xw T P —

TP — TW &

&& TW TE

x

TG TG

¯

¦

&& xxT p

Tp (

x

TM

and from it one easily sees that the induced connection is linear in both vector

bundle structures. We say that it is a linear connection on the associated bundle.

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110 Chapter III. Bundles and connections

Recall now from 6.11 the vertical lift vlE : E —M E ’ V E, which is an

isomorphism, pr1 “πE “¬berwise linear and also p“T p“¬berwise linear.

¯

Now we de¬ne the connector K of the linear connection ¦ by

¯

K := pr2 —¦ (vlE )’1 —¦ ¦ : T E ’ V E ’ E —M E ’ E.

Lemma. The connector K : T E ’ E is a vector bundle homomorphism for

both vector bundle structures on T E and satis¬es K —¦ vlE = pr2 : E —M E ’

T E ’ E.

So K is πE “p“¬berwise linear and T p“p“¬berwise linear.

Proof. This follows from the ¬berwise linearity of the components of K and from

its de¬nition.

11.11. Linear connections. If (E, p, M ) is a vector bundle, a connection

Ψ ∈ „¦1 (E; T E) such that Ψ : T E ’ V E ’ T E is also T p“T p“¬berwise linear

is called a linear connection. An easy check with 11.9 or a direct construction

shows that Ψ is then induced from a unique principal connection on the linear

frame bundle GL(Rn , E) of E (where n is the ¬ber dimension of E).

Equivalently a linear connection may be speci¬ed by a connector K : T E ’ E

with the three properties of lemma 11.10. For then HE := {ξu : K(ξu ) = 0p(u) }

is a complement to V E in T E which is T p“¬berwise linearly chosen.

11.12. Covariant derivative on vector bundles. Let (E, p, M ) be a vector

bundle with a linear connection, given by a connector K : T E ’ E with the

properties in lemma 11.10.

For any manifold N , smooth mapping s : N ’ E, and vector ¬eld X ∈ X(N )

we de¬ne the covariant derivative of s along X by

:= K —¦ T s —¦ X : N ’ T N ’ T E ’ E.

(1) Xs

∞

If f : N ’ M is a ¬xed smooth mapping, let us denote by Cf (N, E) the vector

space of all smooth mappings s : N ’ E with p —¦ s = f “ they are called sections

of E along f . From the universal property of the pullback it follows that the

vector space Cf (N, E) is canonically linearly isomorphic to the space C ∞ (f — E)

∞

of sections of the pullback bundle. Then the covariant derivative may be viewed

as a bilinear mapping

∞ ∞

: X(N ) — Cf (N, E) ’ Cf (N, E).

(2)

Lemma. This covariant derivative has the following properties:

(3) X s is C ∞ (N, R)-linear in X ∈ X(N ). So for a tangent vector Xx ∈

∞

Tx N the mapping Xx : Cf (N, E) ’ Ef (x) makes sense and we have

( X s)(x) = X(x) s.

∞

(4) X s is R-linear in s ∈ Cf (N, E).

(5) X (h.s) = dh(X).s + h. X s for h ∈ C ∞ (N, R), the derivation property

of X .

(6) For any manifold Q and smooth mapping g : Q ’ N and Yy ∈ Ty Q we

have T g.Yy s = Yy (s —¦ g). If Y ∈ X(Q) and X ∈ X(N ) are g-related,

then we have Y (s —¦ g) = ( X s) —¦ g.

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

11. Principal and induced connections 111

Proof. All these properties follow easily from the de¬nition (1).

For vector ¬elds X, Y ∈ X(M ) and a section s ∈ C ∞ (E) an easy computation

shows that

RE (X, Y )s : = s’ ’

Xs [X,Y ] s

X Y Y

]’

= ([ X, [X,Y ] )s

Y

is C ∞ (M, R)-linear in X, Y , and s. By the method of 7.3 it follows that RE is a 2-

form on M with values in the vector bundle L(E, E), i.e. RE ∈ „¦2 (M ; L(E, E)).

It is called the curvature of the covariant derivative.

∞

For f : N ’ M , vector ¬elds X, Y ∈ X(N ) and a section s ∈ Cf (N, E)

along f one may prove that

= (f — RE )(X, Y )s = RE (T f.X, T f.Y )s.

s’ ’

Xs [X,Y ] s

X Y Y

We will do this in 37.15.(2).

11.13. Covariant exterior derivative. Let (E, p, M ) be a vector bundle with

a linear connection, given by a connector K : T E ’ E.

For a smooth mapping f : N ’ M let „¦(N ; f — E) be the vector space of all

forms on N with values in the vector bundle f — E. We can also view them as

forms on N with values along f in E, but we do not introduce an extra notation

for this.

The graded space „¦(N ; f — E) is a graded „¦(N )-module via

(• § ¦)(X1 , . . . , Xp+q ) =

1

= sign(σ) •(Xσ1 , . . . , Xσp )¦(Xσ(p+1) , . . . , Xσ(p+q) ).

p! q!

σ

It can be shown that the graded module homomorphisms H : „¦(N ; f — E) ’

„¦(N ; f — E) (so that H(• § ¦) = (’1)deg H. deg • • § H(¦)) are exactly the map-

pings µ(A) for A ∈ „¦p (N ; f — L(E, E)), which are given by

(µ(A)¦)(X1 , . . . , Xp+q ) =

1

= sign(σ) A(Xσ1 , . . . , Xσp )(¦(Xσ(p+1) , . . . , Xσ(p+q) )).

p! q!

σ

The covariant exterior derivative d : „¦p (N ; f — E) ’ „¦p+1 (N ; f — E) is de¬ned

by (where the Xi are vector ¬elds on N )

p

(’1)i

(d ¦)(X0 , . . . , Xp ) = Xi ¦(X0 , . . . , Xi , . . . , Xp )

i=0

(’1)i+j ¦([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xp ).

+

0¤i<j¤p

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112 Chapter III. Bundles and connections

Lemma. The covariant exterior derivative is well de¬ned and has the following

properties.

(1) For s ∈ C ∞ (f — E) = „¦0 (N ; f — E) we have (d s)(X) = X s.

(2) d (• § ¦) = d• § ¦ + (’1)deg • • § d ¦.

(3) For smooth g : Q ’ N and ¦ ∈ „¦(N ; f — E) we have d (g — ¦) = g — (d ¦).

(4) d d ¦ = µ(f — RE )¦.

Proof. It su¬ces to investigate decomposable forms ¦ = • — s for • ∈ „¦p (N )

and s ∈ C ∞ (f — E). Then from the de¬nition we have d (• — s) = d• — s +

(’1)p • § d s. Since by 11.12.(3) d s ∈ „¦1 (N ; f — E), the mapping d is well

de¬ned. This formula also implies (2) immediately. (3) follows from 11.12.(6).

(4) is checked as follows:

d d (• — s) = d (d• — s + (’1)p • § d s) by (2)

= 0 + (’1)2p • § d d s

= • § µ(f — RE )s by the de¬nition of RE

= µ(f — RE )(• — s).

11.14. Let (P, p, M, G) be a principal ¬ber bundle and let ρ : G ’ GL(W ) be

a representation of the structure group G on a ¬nite dimensional vector space

W.

Theorem. There is a canonical isomorphism from the space of P [W, ρ]-valued

di¬erential forms on M onto the space of horizontal G-equivariant W -valued

di¬erential forms on P :

q : „¦(M ; P [W, ρ]) ’ „¦hor (P ; W )G := {• ∈ „¦(P ; W ) : iX • = 0

for all X ∈ V P, (rg )— • = ρ(g ’1 ) —¦ • for all g ∈ G}.

In particular for W = R with trivial representation we see that

p— : „¦(M ) ’ „¦hor (P )G = {• ∈ „¦hor (P ) : (rg )— • = •}

is also an isomorphism. The isomorphism

q : „¦0 (M ; P [W ]) = C ∞ (P [W ]) ’ „¦0 (P ; W )G = C ∞ (P, W )G

hor

is a special case of the one from 10.12.

Proof. Recall the smooth mapping „ G : P —M P ’ G from 10.2, which satis¬es

r(ux , „ G (ux , vx )) = vx , „ G (ux .g, ux .g ) = g ’1 .„ G (ux , ux ).g , and „ G (ux , ux ) = e.

Let • ∈ „¦k (P ; W )G , X1 , . . . , Xk ∈ Tu P , and X1 , . . . , Xk ∈ Tu P such that

hor

Tu p.Xi = Tu p.Xi for each i. Then we have for g = „ G (u, u ), so that ug = u :

q(u, •u (X1 , . . . , Xk )) = q(ug, ρ(g ’1 )•u (X1 , . . . , Xk ))

= q(u , ((rg )— •)u (X1 , . . . , Xk ))

= q(u , •ug (Tu (rg ).X1 , . . . , Tu (rg ).Xk ))

= q(u , •u (X1 , . . . , Xk )), since Tu (rg )Xi ’ Xi ∈ Vu P.

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

11. Principal and induced connections 113

By this prescription a vector bundle valued form ¦ ∈ „¦k (M ; P [W ]) is uniquely

determined.

For the converse recall the smooth mapping „ W : P —M P [W, ρ] ’ W

from 10.7, which satis¬es „ W (u, q(u, w)) = w, q(ux , „ W (ux , vx )) = vx , and

„ W (ux g, vx ) = ρ(g ’1 )„ W (ux , vx ).

For ¦ ∈ „¦k (M ; P [W ]) we de¬ne q ¦ ∈ „¦k (P ; W ) as follows. For Xi ∈ Tu P

we put

(q ¦)u (X1 , . . . , Xk ) := „ W (u, ¦p(u) (Tu p.X1 , . . . , Tu p.Xk )).

Then q ¦ is smooth and horizontal. For g ∈ G we have

((rg )— (q ¦))u (X1 , . . . , Xk ) = (q ¦)ug (Tu (rg ).X1 , . . . , Tu (rg ).Xk )

= „ W (ug, ¦p(ug) (Tug p.Tu (rg ).X1 , . . . , Tug p.Tu (rg ).Xk ))

= ρ(g ’1 )„ W (u, ¦p(u) (Tu p.X1 , . . . , Tu p.Xk ))

= ρ(g ’1 )(q ¦)u (X1 , . . . , Xk ).

Clearly the two constructions are inverse to each other.

11.15. Let (P, p, M, G) be a principal ¬ber bundle with a principal connection

¦ = ζ —¦ ω, and let ρ : G ’ GL(W ) be a representation of the structure group G

on a ¬nite dimensional vector space W . We consider the associated vector bundle

¯

(E := P [W, ρ], p, M, W ), the induced connection ¦ on it and the corresponding

covariant derivative.

Theorem. The covariant exterior derivative dω from 11.5 on P and the co-

variant exterior derivative for P [W ]-valued forms on M are connected by the

mapping q from 11.14, as follows:

q —¦ d = dω —¦ q : „¦(M ; P [W ]) ’ „¦hor (P ; W )G .

Proof. Let ¬rst f ∈ „¦0 (P ; W )G = C ∞ (P, W )G , then we have f = q s for s ∈

hor

∞

C (P [W ]). Moreover we have f (u) = „ W (u, s(p(u))) and s(p(u)) = q(u, f (u))

by 11.14 and 10.12. Therefore T s.T p.Xu = T q(Xu , T f.Xu ), where T f.Xu =

(f (u), df (Xu )) ∈ T W = W — W . If χ : T P ’ HP is the horizontal projection

as in 11.5, we have T s.T p.Xu = T s.T p.χ.Xu = T q(χ.Xu , T f.χ.Xu ). So we get

(q d s)(Xu ) = „ W (u, (d s)(T p.Xu ))

= „ W (u, T p.Xu s) by 11.13.(1)

= „ W (u, K.T s.T p.Xu ) by 11.12.(1)

= „ W (u, K.T q(χ.Xu , T f.χ.Xu )) from above

= „ W (u, pr2 .vl’1 .¦.T q(χ.Xu , T f.χ.Xu ))

¯ by 11.10

P [W ]

’1

„ W (u, pr2 .vlP [W ] .T q.(¦ — Id)(χ.Xu , T f.χ.Xu )))

= by 11.8

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

114 Chapter III. Bundles and connections

’1

= „ W (u, pr2 .vlP [W ] .T q(0u , T f.χ.Xu ))) since ¦.χ = 0

’1

= „ W (u, q.pr2 .vlP —W .(0u , T f.χ.Xu ))) since q is ¬ber linear

= „ W (u, q(u, df.χ.Xu )) = (χ— df )(Xu )

= (dω q s)(Xu ).

Now we turn to the general case. It su¬ces to check the formula for a decom-

posable P [W ]-valued form Ψ = ψ — s ∈ „¦k (M, P [W ]), where ψ ∈ „¦k (M ) and

s ∈ C ∞ (P [W ]). Then we have

dω q (ψ — s) = dω (p— ψ · q s)

= dω (p— ψ) · q s + (’1)k χ— p— ψ § dω q s by 11.5.(1)

= χ— p— dψ · q s + (’1)k p— ψ § q d s from above and 11.5.(4)

= p— dψ · q s + (’1)k p— ψ § q d s

= q (dψ — s + (’1)k ψ § d s)

= q d (ψ — s).

11.16. Corollary. In the situation of theorem 11.15 above we have for the cur-

vature form „¦ ∈ „¦2 (P ; g) and the curvature RP [W ] ∈ „¦2 (M ; L(P [W ], P [W ]))

hor

the relation

qL(P [W ],P [W ]) RP [W ] = ρ —¦ „¦,

where ρ = Te ρ : g ’ L(W, W ) is the derivative of the representation ρ.

Proof. We use the notation of the proof of theorem 11.15. By this theorem we

have for X, Y ∈ Tu P

(dω dω qP [W ] s)u (X, Y ) = (q d d s)u (X, Y )

= (q RP [W ] s)u (X, Y )

= „ W (u, RP [W ] (Tu p.X, Tu p.Y )s(p(u)))

= (qL(P [W ],P [W ]) RP [W ] )u (X, Y )(qP [W ] s)(u).

On the other hand we have by theorem 11.5.(8)

(dω dω q s)u (X, Y ) = (χ— iR dq s)u (X, Y )

= (dq s)u (R(X, Y )) since R is horizontal

= (dq s)(’ζ„¦(X,Y ) (u)) by 11.2

ζ

‚ „¦(X,Y )

= (q s)(Fl’t (u))

‚t 0

„ W (u. exp(’t„¦(X, Y )), s(p(u. exp(’t„¦(X, Y )))))

‚

= ‚t 0

„ W (u. exp(’t„¦(X, Y )), s(p(u)))

‚

= ‚t 0

ρ(exp t„¦(X, Y ))„ W (u, s(p(u)))

‚

= by 10.7

‚t 0

= ρ („¦(X, Y ))(q s)(u).

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

Remarks 115

Remarks

The concept of connections on general ¬ber bundles was formulated at about

1970, see e. g. [Libermann, 73]. The theorem 9.9 that each ¬ber bundle admits

a complete connection is contained in [Wolf, 67], with an incorrect proof. It is

an exercise in [Greub-Halperin-Vanstone I, 72, p 314]. The proof given here and

the generalization 9.11 of the Ambrose Singer theorem are from [Michor, 88], see

also [Michor, 91], which are also the source for 11.8 and 11.9. The results 11.15

and 11.16 appear here for the ¬rst time.

¦

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

116

CHAPTER IV.

JETS AND NATURAL BUNDLES

In this chapter we start our systematic treatment of geometric objects and

operators. It has become commonplace to think of geometric objects on a man-

ifold M as forming ¬ber bundles over the base M and to work with sections

of these bundles. The concrete objects were frequently described in coordinates

by their behavior under the coordinate changes. Stressing the change of coor-

dinates, we can say that local di¬eomorphisms on the base manifold operate on

the bundles of geometric objects. Since a further usual assumption is that the

resulting transformations depend only on germs of the underlying morphisms,

we actually deal with functors de¬ned on all open submanifolds of M and local

di¬eomorphisms between them (let us recall that local di¬eomorphisms are glob-

ally de¬ned locally invertible maps), see the preface. This is the point of view

introduced by [Nijenhuis, 72] and worked out later by [Terng, 78], [Palais, Terng,

77], [Epstein, Thurston, 79] and others. These functors are fully determined by

their restriction to any open submanifold and therefore they extend to the whole

category Mfm of m-dimensional manifolds and local di¬eomorphisms. An im-

portant advantage of such a de¬nition of bundles of geometric objects is that we

get a precise de¬nition of geometric operators in the concept of natural opera-

tors. These are rules transforming sections of one natural bundle into sections of

another one and commuting with the induced actions of local di¬eomorphisms

between the base manifolds.

In the theory of natural bundles and operators, a prominent role is played

by jets. Roughly speaking, jets are certain equivalence classes of smooth maps

between manifolds, which are represented by Taylor polynomials. We start this

chapter with a thorough treatment of jets and jet bundles, and we investigate the

so called jet groups. Then we give the de¬nition of natural bundles and deduce

that the r-th order natural bundles coincide with the associated ¬bre bundles to

r-th order frame bundles. So they are in bijection with the actions of the r-th

order jet group Gr on manifolds. Moreover, natural transformations between

m

the r-th order natural bundles bijectively correspond to Gr -equivariant maps.

m

Let us note that in chapter V we deduce a rather general theory of functors on

categories over manifolds and we prove that both the ¬niteness of the order and

the regularity of natural bundles are consequences of the other axioms, so that

actually we describe all natural bundles here. Next we treat the basic properties

of natural operators. In particular, we show that k-th order natural operators

are described by natural transformations of the k-th order jet prolongations of

the bundles in question. This reduces even the problem of ¬nding ¬nite order

natural operators to determining Gr -equivariant maps, which will be discussed

m

in chapter VI.

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

12. Jets 117

Further we present the procedure of principal prolongation of principal ¬ber

bundles based on an idea of [Ehresmann, 55] and we show that the jet prolonga-

tions of associated bundles are associated bundles to the principal prolongations

of the corresponding principal bundles. This fact is of basic importance for the

theory of gauge natural bundles and operators, the foundations of which will be

presented in chapter XII. The canonical form on ¬rst order principal prolonga-

tion of a principal bundle generalizes the well known canonical form on an r-th

order frame bundle. These canonical forms are used in a formula for the ¬rst jet

prolongation of sections of arbitrary associated ¬ber bundles, which represents a

common basis for several algorithms in di¬erent branches of di¬erential geome-

try. At the end of the chapter, we reformulate a part of the theory of connections

from the point of view of jets, natural bundles and natural operators. This is

necessary for our investigation of natural operations with connections, but we

believe that this also demonstrates the power of the jet approach to give a clear

picture of geometric concepts.

12. Jets

12.1. Roughly speaking, two maps f , g : M ’ N are said to determine the

same r-jet at x ∈ M , if they have the r-th order contact at x. To make this idea

precise, we ¬rst de¬ne the r-th order contact of two curves on a manifold. We

recall that a smooth function R ’ R is said to vanish to r-th order at a point,

if all its derivatives up to order r vanish at this point.

De¬nition. Two curves γ, δ : R ’ M have the r-th contact at zero, if for every

smooth function • on M the di¬erence • —¦ γ ’ • —¦ δ vanishes to r-th order at

0 ∈ R.

In this case we write γ ∼r δ. Obviously, ∼r is an equivalence relation. For

r = 0 this relation means γ(0) = δ(0).

Lemma. If γ ∼r δ, then f —¦ γ ∼r f —¦ δ for every map f : M ’ N .

Proof. If • is a function on N , then • —¦ f is a function on M . Hence • —¦ f —¦ γ ’

• —¦ f —¦ δ has r-th order zero at 0.

12.2. De¬nition. Two maps f , g : M ’ N are said to determine the same

r-jet at x ∈ M , if for every curve γ : R ’ M with γ(0) = x the curves f —¦ γ and

g —¦ γ have the r-th order contact at zero.

In such a case we write jx f = jx g or j r f (x) = j r g(x).

r r

An equivalence class of this relation is called an r-jet of M into N . Obviously,

r

jx f depends on the germ of f at x only. The set of all r-jets of M into N is

denoted by J r (M, N ). For X = jx f ∈ J r (M, N ), the point x =: ±X is the