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CHAPTER IV. JETS AND NATURAL BUNDLES

Back

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

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

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 coordinates,

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 globally

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 important

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 operators.

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

m on manifolds. Moreover, natural transformations between

the r-th order natural bundles bijectively correspond to Gr

m-equivariant maps.

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

m-equivariant maps, which will be discussed

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 prolongations

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 prolongation

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 geometry.

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 2 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 ; _ : R ! M have the r-th contact at zero, if for every

smooth function ' on M the di_erence ' _ 􀀀 ' _ _ vanishes to r-th order at

0 2 R.

In this case we write _r _. Obviously, _r is an equivalence relation. For

r = 0 this relation means (0) = _(0).

Lemma. If _r _, then f _ _r f _ _ for every map f : M ! N.

Proof. If ' is a function on N, then ' _ f is a function on M. Hence ' _ f _ 􀀀

' _ f _ _ 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 2 M, if for every curve : R ! M with (0) = x the curves f _ and

g _ have the r-th order contact at zero.

In such a case we write jrx

f = jrx

g or jrf(x) = jrg(x).

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

jrx

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

denoted by Jr(M;N). For X = jrx

f 2 Jr(M;N), the point x =: _X is the

source of X and the point f(x) =: _X is the target of X. We denote by _r

s ,

0 _ s _ r, the projection jrx

f 7! jsx

f of r-jets into s-jets. By Jr

x(M;N) or

Jr(M;N)y we mean the set of all r-jets of M into N with source x 2 M or

target y 2 N, respectively, and we write Jr

x(M;N)y = Jr

x(M;N) \ Jr(M;N)y.

The map jrf : M ! Jr(M;N) is called the r-th jet prolongation of f : M ! N.

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

118 Chapter IV. Jets and natural bundles

12.3. Proposition. If two pairs of maps f, _ f : M ! N and g, _g : N ! Q

satisfy jrx

f = jrx

_ f and jr

yg = jr

y _g, f(x) = y = _ f(x), then jrx

(g _ f) = jrx

(_g _ _ f).

Proof. Take a curve onM with (0) = x. Then jrx

f = jrx

_ f implies f _ _r _ f _,

lemma 12.1 gives _g _ f _ _r _g _ _ f _ and jr

yg = jr

y _g yields g _ f _ _r _g _ f _ .

Hence g _ f _ _r _g _ _ f _ . _

In other words, r-th order contact of maps is preserved under composition. If

X 2 Jr

x(M;N)y and Y 2 Jr

y (N;Q)z are of the form X = jrx

f and Y = jr

yg, we

can de_ne the composition Y _ X 2 Jr

x(M;Q)z by

Y _ X = jrx

(g _ f):

By the above proposition, Y _X does not depend on the choice of f and g. We

remark that we _nd it useful to denote the composition of r-jets by the same

symbol as the composition of maps. Since the composition of maps is associative,

the same holds for r-jets. Hence all r-jets form a category, the units of which

are the r-jets of the identity maps of manifolds. An element X 2 Jr

x(M;N)y

is invertible, if there exists X􀀀1 2 Jr

y (N;M)x such that X􀀀1 _ X = jrx

(idM)

and X _ X􀀀1 = jr

y(idN). By the implicit function theorem, X 2 Jr(M;N) is

invertible if and only if the underlying 1-jet _r

1X is invertible. The existence of

such a jet implies dimM = dimN. We denote by invJr(M;N) the set of all

invertible r-jets of M into N.

12.4. Let f : M ! _M be a local di_eomorphism and g : N ! _N be a map.

Then there is an induced map Jr(f; g) : Jr(M;N) ! Jr( _M ; _N ) de_ned by

Jr(f; g)(X) = (jr

yg) _ X _ (jrx

f)􀀀1

where x = _X and y = _X are the source and target of X 2 Jr(M;N). Since

the jet composition is associative, Jr is a functor de_ned on the product category

Mfm_Mf. (We shall see in 12.6 that the values of Jr lie in the category FM.)

12.5. We are going to describe the coordinate expression of r-jets. We recall

that a multiindex of range m is a m-tuple _ = (_1; : : : ; _m) of non-negative

integers. We write j_j = _1 + _ _ _ + _m, _! = _1! _ _ _ _m! (with 0! = 1), x_ =

(x1)_1 : : : (xm)_m for x = (x1; : : : ; xm) 2 Rm. We denote by

D_f = @j_jf

(@x1)_1 : : : (@xm)_m

the partial derivative with respect to the multiindex _ of a function f : U _

Rm ! R.

Proposition. Given a local coordinate system xi on M in a neighborhood of x

and a local coordinate system yp on N in a neighborhood of f(x), two maps f,

g : M ! N satisfy jrx

f = jrx

g if and only if all the partial derivatives up to order

r of the components fp and gp of their coordinate expressions coincide at x.

Proof. We _rst deduce that two curves (t), _(t) : R ! N satisfy _r _ if and

only if

(1) dk(yp _ )(0)

dtk = dk(yp _ _)(0)

dtk

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

12. Jets 119

k = 0; 1; : : : ; r, for all coordinate functions yp. On one hand, if _r _, then

yp _ 􀀀 yp _ _ vanishes to order r, i.e. (1) is true. On the other hand, let (1)

hold. Given a function ' on N with coordinate expression '(y1; : : : ; yn), we _nd

by the chain rule that all derivatives up to order r of ' _ _ depend only on the

partial derivatives up to order r of ' at (0) and on (1). Hence ' _ 􀀀 ' _ _

vanishes to order r at 0.

If the partial derivatives up to the order r of fp and gp coincide at x, then

the chain rule implies f _ _r g _ by (1). This means jrx

f = jrx

g. Conversely,

assume jrx

f = jrx

g. Consider the curves xi = ait with arbitrary ai. Then the

coordinate condition for f _ _r g _ reads

(2)

j_j=k

(D_fp(x))a_ =

j_j=k

(D_gp(x))a_

k = 0; 1; : : : ; r. Since ai are arbitrary, (2) implies that all partial derivatives up

to order r of fp and gp coincide at x. _

Now we can easily prove that the auxiliary relation _r _ can be expressed

in terms of r-jets.

Corollary. Two curves , _ : R ! M satisfy _r _ if and only if jr

0 = jr

0_.

Proof. Since xi _ and xi _ _ are the coordinate expressions of and _, (1) is

equivalent to jr

0 = jr

0_. _

12.6. Write Lr

m;n = Jr

0 (Rm;Rn)0. By proposition 12.5, the elements of Lr

m;n

can be identi_ed with the r-th order Taylor expansions of the generating maps,

i.e. with the n-tuples of polynomials of degree r in m variables without absolute

term. Such an expression X

1_j_j_r

_x_

will be called the polynomial representative of an r-jet. Hence Lr

m;n is a numerical

space of the variables ap

_. Standard combinatorics yields dimLr

m;n =

_􀀀m+r

􀀀 1

. The coordinates on Lr

m;n will sometimes be denoted more explicitly

by ap

i , ap

ij ; : : : ; ap

i1:::ir

, symmetric in all subscripts. The projection _r

s : Lr

m;n

! Ls

m;n consists in suppressing all terms of degree > s.

The jet composition Lr

m;n

_Lr

n;q

! Lr

m;q is evaluated by taking the composition

of the polynomial representatives and suppressing all terms of degree higher

than r. Some authors call it the truncated polynomial composition. Hence the

jet composition Lr

m;n

_Lr

n;q

! Lr

m;q is a polynomial map of the numerical spaces

in question. The sets Lr

m;n can be viewed as the sets of morphisms of a category

Lr over non-negative integers, the composition in which is the jet composition.

The set of all invertible elements of Lr

m;m with the jet composition is a Lie

group Gr

m called the r-th di_erential group or the r-th jet group in dimension m.

For r = 1 the group G1

m is identi_ed with GL(m;R). That is why some authors

use GLr(m;R) for Gr

In the case M = Rm, we can identify every X 2 Jr(Rm;Rn) with a triple

(_X; (jr_Xt􀀀1

_X) _ X _ (jr

0 t_X); _X) 2 Rm _ Lr

m;n

_ Rn, where tx means the

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

120 Chapter IV. Jets and natural bundles

translation on Rm transforming 0 into x. This product decomposition de_nes

the structure of a smooth manifold on Jr(Rm;Rn) as well as the structure of

a _bered manifold _r

0 : Jr(Rm;Rn) ! Rm _ Rn. Since the jet composition in

Lr is polynomial, the induced map Jr(f; g) of every pair of di_eomorphisms

f : Rm ! Rm and g : Rn ! Rn is a _bered manifold isomorphism over (f; g).

Having two manifolds M and N, every local charts ': U ! Rm and : V ! Rn

determine an identi_cation (_r

0)􀀀1(U_V ) _= Jr(Rm;Rn). Since the chart changings

are smooth maps, this de_nes the structure of a smooth _bered manifold on

0 : Jr(M;N) ! M _N. Now we see that Jr is a functor Mfm _Mf ! FM.

Obviously, all jet projections _r

s are surjective submersions.

12.7. Remark. In de_nition 12.2 we underlined the geometrical approach to

the concept of r-jets. We remark that there exists a simple algebraic approach

as well. Consider the ring C1

x (M;R) of all germs of smooth functions on a

manifold M at a point x and its subset M(M; x) of all germs with zero value

at x, which is the unique maximal ideal of C1

x (M;R). Let M(M; x)k be the

k-th power of the ideal M(M; x) in the algebraic sense. Using coordinates one

veri_es easily that two maps f, g : M ! N, f(x) = y = g(x), determine the

same r-jet if and only if ' _ f 􀀀 ' _ g 2M(M; x)r+1 for every ' 2 C1

y (N;R).

12.8. Velocities and covelocities. The elements of the manifold Tr

kM :=

0 (Rk;M) are said to be the k-dimensional velocities of order r on M, in short

(k; r)-velocities. The inclusion Tr

kM _ Jr(Rm;M) de_nes the structure of a

smooth _ber bundle on Tr

kM ! M. Every smooth map f : M ! N is extended

into an FM-morphism Tr

k f : Tr

kM ! Tr

kN de_ned by Tr

k f(jr

0g) = jr

0 (f _ g).

Hence Tr

k is a functor Mf ! FM. Since every map Rk ! M1 _M2 coincides

with a pair of maps Rk ! M1 and Rk ! M2, functor Tr

k preserves products.

For k = r = 1 we obtain another de_nition of the tangent functor T = T1

1 .

We remark that we can now express the contents of de_nition 12.2 by saying

that jrx

f = jrx

g holds if and only if the restrictions of both Tr

1 f and Tr

1 g to

(Tr

1M)x coincide.

The space Tr_

k M = Jr(M;Rk)0 is called the space of all (k; r)-covelocities on

M. In the most important case k = 1 we write in short Tr_

1 = Tr_. Since Rk is a

vector space, Tr_

k M ! M is a vector bundle with jrx

'(u) + jrx

(u) = jrx('(u) +

(u)), u 2 M, and kjrx

'(u) = jrx

k'(u), k 2 R. Every local di_eomorphism

f : M ! N is extended to a vector bundle morphism Tr_

k f : Tr_

k M ! Tr_

k N,

jrx

' 7! jr

f(x)(' _ f􀀀1), where f􀀀1 is constructed locally. In this sense Tr_

k is a

functor on Mfm. For k = r = 1 we obtain the construction of the cotangent

bundles as a functor T1_

1 = T_ onMfm. We remark that the behavior of Tr_

k on

arbitrary smooth maps will be reected in the concept of star bundle functors

we shall introduce in 41.2.

12.9. Jets as algebra homomorphisms. The multiplication of reals induces

a multiplication in every vector space Tr_

x M by

(jrx

'(u))(jrx

(u)) = jrx

('(u) (u));

which turns Tr_

x M into an algebra. Every jrx

f 2 Jr

x(M;N)y de_nes an algebra

homomorphism hom(jrx

f) : Tr_

y N ! Tr_

x M by jr

y' 7! jrx

(' _ f). To deduce

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

12. Jets 121

the converse assertion, consider some local coordinates xi on M and yp on N

centered at x and y. The algebra Tr_

y N is generated by jr

0yp. If we prescribe

quite arbitrarily the images _(jr

0yp) in Tr_

x M, this is extended into a unique

algebra homomorphism _: Tr_

y N ! Tr_

x M. The n-tuple _(jr

0yp) represents

the coordinate expression of a jet X 2 Jr

x(M;N)y and one veri_es easily _ =

hom(X). Thus we have proved

Proposition. There is a canonical bijection between Jr

x(M;N)y and the set of

all algebra homomorphisms Hom(Tr_

y N; Tr_

x M).

For r = 1 the product of any two elements in T_

xM is zero. Hence the algebra

homomorphisms coincide with the linear maps T_

yN ! T_

xM. This gives an

identi_cation J1(M;N) = TN T_M (which can be deduced by several other

ways as well).

12.10. Kernel descriptions. The projection _r

r􀀀1 : Tr_M ! Tr􀀀1_M is a

linear morphism of vector bundles. Its kernel is described by the following exact

sequence of vector bundles over M

(1) 0 􀀀! SrT_M 􀀀! Tr_M

r􀀀1 􀀀􀀀􀀀! Tr􀀀1_M 􀀀! 0

where Sr indicates the r-th symmetric tensor power. To prove it, we _rst construct

a map p: r

_ T_M ! Tr_M. Take r functions f1; : : : ; fr on M with

values zero at x and construct the r-jet at x of their product. One sees directly

that jrx

(f1 : : : fr) depends on j1

xf1; : : : ; j1

xfr only and lies in ker(_r

r􀀀1). We have

jrx

(f1 : : : fr) = j1

xf1 _ _ _ j1

xfr, where means the symmetric tensor product,

so that p is uniquely extended into a linear isomorphism of SrT_M into

ker(_r

r􀀀1).

Next we shall use a similar idea for a geometrical construction of an identi

_cation, which is usually justi_ed by the coordinate evaluations only. Let ^y

denote the constant map of M into y 2 N.

Proposition. The subspace (_r

r􀀀1)􀀀1(jr􀀀1

x ^y) _ Jr

x(M;N)y is canonically identi

_ed with TyN SrT_

xM.

Proof. Let B 2 TyN and j1

xfp 2 T_

xM, p = 1; : : : ; r. For every jr

y' 2 Tr_

y N,

take the value B' 2 R of the derivative of ' in direction B and construct a

function (B')f1(u) : : : fr(u) on M. It is easy to see that jr

y' 7! jrx((B')f1 : : : fr)

is an algebra homomorphism Tr_

y N ! Tr_

x M. This de_nes a map p: TyN _

r-times

_: : :_T_

xM ! Jr

x(M;N)y. Using coordinates one veri_es that p generates

linearly the required identi_cation. _

For r = 1 we have a distinguished element j1

x^y in every _ber of J1(M;N) !

M _ N. This identi_es J1(M;N) with TN T_M.

In particular, if we apply the above proposition to the projection

r􀀀1 : (Tr

kM)x ! (Tr􀀀1

k M)x, x 2 M, we _nd

(2) (_r

r􀀀1)􀀀1(jr􀀀1

0 ^x) = TxM SrRk_:

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

122 Chapter IV. Jets and natural bundles

12.11. Proposition. _r

r􀀀1 : Jr(M;N) ! Jr􀀀1(M;N) is an a_ne bundle,

the modelling vector bundle of which is the pullback of TN SrT_M over

Jr􀀀1(M;N).

Proof. Interpret X 2 Jr

x(M;N)y and A 2 TyN SrT_

xM _ Jr

x(M;N)y as algebra

homomorphisms Tr_

y N ! Tr_

x M. For every _ 2 Tr_

y N we have _r

r􀀀1(A(_))

= 0 and _r

0(X(_)) = 0. This implies X(_)A( ) = 0 and A(_)A( ) = 0

for any other 2 Tr_

y N. Hence X(_ ) + A(_ ) = X(_)X( ) = (X(_) +

A(_))(X( )+A( )), so that X +A is also an algebra homomorphism Tr_

y N !

Tr_

x M. Using coordinates we _nd easily that the map (X;A) 7! X + A gives

rise to the required a_ne bundle structure. _

Since the tangent space to an a_ne space is the modelling vector space, we obtain

immediately the following property of the tangent map T_r

r􀀀1 : TJr(M;N)

! TJr􀀀1(M;N).

Corollary. For every X 2 Jr

x(M;N)y, the kernel of the restriction of T_r

r􀀀1 to

TXJr(M;N) is TyN SrT_

xM.

12.12. The frame bundle of order r. The set PrM of all r-jets with source

0 of the local di_eomorphisms of Rm into M is called the r-th order frame

bundle of M. Obviously, PrM = invTrm

(M) is an open subset of Trm

(M),

which de_nes a structure of a smooth _ber bundle on PrM ! M. The group

m acts smoothly on PrM on the right by the jet composition. Since for

every jr

0', jr

0 2 Pr

xM there is a unique element jr

0 ('􀀀1 _ ) 2 Gr

m satisfying

(jr

0')_(jr

0 ('􀀀1_ )) = jr

0 , PrM is a principal _ber bundle with structure group

m. For r = 1, the elements of invJ1

0 (Rm;M)x are identi_ed with the linear

isomorphisms Rm ! TxM and G1

m = GL(m), so that P1M coincides with the

bundle of all linear frames in TM, i.e. with the classical frame bundle of M.

Every velocities space Tr

kM is a _ber bundle associated with PrM with standard

_ber Lr

k;m. The basic idea consists in the fact that for every jr

0f 2 (Tr

kM)x

and jr

0' 2 Pr

xM we have jr

0 ('􀀀1 _ f) 2 Lr

k;m, and conversely, every jr

0g 2 Lr

k;m

and jr

0' 2 Pr

xM determine jr

0 ('_g) 2 (Tr

kM)x. Thus, if we formally de_ne a left

action Gr

_ Lr

k;m

! Lr

k;m by (jr

0h; jr

0g) 7! jr

0 (h _ g), then Tr

kM is canonically

identi_ed with the associated _ber bundle PrM[Lr

k;m].

Quite similarly, every covelocities space Tr_

k M is a _ber bundle associated

with PrM with standard _ber Lr

m;k with respect to the left action Gr

_Lr

m;k

m;k, (jr

0h; jr

0g) 7! jr

0 (g _ h􀀀1). Furthermore, PrM _ PrN is a principal _ber

bundle over M _ N with structure group Gr

_ Gr

n. The space Jr(M;N) is a

_ber bundle associated with PrM _PrN with standard _ber Lr

m;n with respect

to the left action (Gr

_Gr

n)_Lr

m;n

! Lr

m;n, ((jr

0'; jr

0 ); jr

0f) 7! jr

0 ( _f _'􀀀1).

Every local di_eomorphism f : M ! N induces a map Prf : PrM ! PrN

by Prf(jr

0') = jr

0 (f _ '). Since Gr

m acts on the right on both PrM and PrN,

Prf is a local principal _ber bundle isomorphism. Hence Pr is a functor from

Mfm into the category PB(Gr

m).

Given a left action of Gr

m on a manifold S, we have an induced map

fPrf; idSg: PrM[S] ! PrN[S]

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

12. Jets 123

between the associated _ber bundles with standard _ber S, see 10.9. The rule

M 7! PrM[S], f 7! fPrf; idSg is a bundle functor onMfm as de_ned in 14.1. A

very interesting result is that every bundle functor onMfm is of this type. This

will be proved in section 22, but the proof involves some rather hard analytical

results.

12.13. For every Lie group G, Tr

kG is also a Lie group with multiplication

(jr

0f(u))(jr

0g(u)) = jr

0 (f(u)g(u)), u 2 Rk, where f(u)g(u) is the product in

G. Clearly, if we consider the multiplication map _: G _ G ! G, then the

multiplication map of Tr

kG is Tr

k _: Tr

kG _ Tr

kG ! Tr

kG. The jet projections

s : Tr

kG ! Ts

kG are group homomorphisms. For s = 0, there is a splitting

_ : G ! Tr

kG of _r

0 = _ : Tr

kG ! G de_ned by _(g) = jr

0 ^g, where ^g means the

constant map of Rk into g 2 G. Hence Tr

kG is a semidirect product of G and of

the kernel of _ : Tr

kG ! G.

If G acts on the left on a manifold M, then Tr

kG acts on Tr

kM by

(jr

0f(u))(jr

0g(u)) = jr

􀀀

f(u)(g(u))

;

where f(u)(g(u)) means the action of f(u) 2 G on g(u) 2 M. If we consider

the action map ` : G _ M ! M, then the action map of the induced action is

k ` : Tr

kG _ Tr

kM ! Tr

kM. The same is true for right actions.

12.14. r-th order tangent vectors. In general, consider the dual vector

bundle Tr_

k M = (Tr_

k M)_ of the (k; r)-covelocities bundle on M. For every map

f : M ! N the jet composition A 7! A _ (jrx

f), x 2 M, A 2 (Tr_

k N)f(x) de_nes

a linear map _(jrx

f) : (Tr_

k N)f(x)

! (Tr_

k M)x. The dual map (_(jrx

f))_ =:

(Tr_

k f)x : (Tr_

k M)x ! (Tr_

k N)f(x) determines a functor Tr_

k onMf with values

in the category of vector bundles. For r > 1 these functors do not preserve

products by the dimension argument. In the most important case k = 1 we shall

write Tr_

1 = T(r) (in order to distinguish from the r-th iteration of T). The

elements of T(r)M are called r-th order tangent vectors on M. We remark that

for r = 1 the formula TM = (T_M)_ can be used for introducing the vector

bundle structure on TM.

Dualizing the exact sequence 12.10.(1), we obtain

(1) 0 􀀀! T(r􀀀1)M 􀀀! T(r)M 􀀀! SrTM 􀀀! 0:

This shows that there is a natural injection of the (r􀀀1)-st order tangent vectors

into the r-th order ones. Analyzing the proof of 12.10.(1), one _nds easily that

(1) has functorial character, i.e. for every map f : M ! N the following diagram

commutes

(2)

0 wT(r􀀀1)M w

T(r􀀀1)f

T(r)M w

T(r)f

SrTM w

SrTf

0 wT(r􀀀1)N wT(r)N wSrTN w 0

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

124 Chapter IV. Jets and natural bundles

12.15. Contact elements. Let N be an n-dimensional submanifold of a manifold

M. For every local chart ': N ! Rn, the rule x 7! '􀀀1(x) considered as a

map Rn ! M is called a local parametrization of N. The concept of the contact

of submanifolds of the same dimension can be reduced to the concept of r-jets.

De_nition. Two n-dimensional submanifolds N and _N of M are said to have

r-th order contact at a common point x, if there exist local parametrizations

: Rn ! M of N and _ : Rn ! M of _N , (0) = x = _ (0), such that jr

0 = jr

_ .

An equivalence class of n-dimensional submanifolds of M will be called an

n-dimensional contact element of order r on M, in short a contact (n; r)-element

on M. We denote by Krn

M the set of all contact (n; r)-elements on M. We have

a canonical projection `point of contact' Krn

M ! M.

An (n; r)-velocity A 2 (Tr

nM)x is called regular, if its underlying 1-jet corresponds

to a linear map Rn ! TxM of rank n. For every local parametrization

of an n-dimensional submanifold, jr

0 is a regular (n; r)-velocity. Since in

the above de_nition we can reparametrize and _ in the same way (i.e. we

compose them with the same origin preserving di_eomorphism of Rm), every

contact (n; r)-element on M can be identi_ed with a class A _ Gr

n, where A is

a regular (n; r)-velocity on M. There is a unique structure of a smooth _bered

manifold on Krn

M ! M with the property that the factor projection from the

subbundle regTr

nM _ Tr

nM of all regular (n; r)-velocities into Krn

M is a surjective

submersion. (The simplest way how to check it is to use the identi_cation

of an open subset in Krn

Rm with the r-th jet prolongation of _bered manifold

Rn _ Rm􀀀n ! Rn, which will be described in the end of 12.16.)

Every local di_eomorphism f : M ! _M preserves the contact of submanifolds.

This induces a map Krn

f : Krn

M ! Krn

, which is a _bered manifold morphism

over f. Hence Krn

is a bundle functor on Mfm. For r = 1 each _ber (K1n

M)x

coincides with the Grassmann manifold of n-planes in TxM, see 10.5. That is

why K1n

M is also called the Grassmannian n-bundle of M.

12.16. Jet prolongations of _bered manifolds. Let p: Y ! M be a _bered

manifold, dimM = m, dim Y = m+n. The set JrY (also written as Jr(Y ! M)

or Jr(p: Y ! M), if we intend to stress the base or the bundle projection) of

all r-jets of the local sections of Y will be called the r-th jet prolongation of Y .

Using polynomial representatives we _nd easily that an element X 2 Jr

x(M; Y )

belongs to JrY if and only if (jr_Xp)_X = jrx

(idM). Hence JrY _ Jr(M; Y ) is a

closed submanifold. For every section s of Y ! M, jrs is a section of JrY ! M.

Let xi or yp be the canonical coordinates on Rm or Rn, respectively. Every

local _ber chart ': U ! Rm+n on Y identi_es (_r

0)􀀀1(U) with Jr(Rm;Rn). This

de_nes the induced local coordinates yp_ on JrY , 1 _ j_j _ r, where _ is any

multi index of range m.

Let q : Z ! N be another _bered manifold and f : Y ! Z be an FMmorphism

with the property that the base map f0 : M ! N is a local diffeomorphism.

Then the map Jr(f; f0) : Jr(M; Y ) ! Jr(N;Z) constructed in

12.4 transforms JrY into JrZ. Indeed, X 2 JrY , _X = y is characterized

by (jr

yp) _ X = jrx

idM, x = p(y), and q _ f = f0 _ p implies

􀀀

f(y)q

􀀀

(jr

yf) _

X _ (jr

f0(x)f􀀀1

0 )

= (jrx

f0) _ (jr

yp) _ X _ jr

f0(x)f􀀀1

0 = jr

f0(x)idN. The restricted

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

12. Jets 125

map will be denoted by Jrf : JrY ! JrZ and called the r-th jet prolongation

of f. Let FMm denote the category of _bered manifolds with m-dimensional

bases and their morphisms with the additional property that the base maps are

local di_eomorphisms. Then the construction of the r-th jet prolongations can

be interpreted as a functor Jr : FMm ! FM. (If there will be a danger of

confusion with the bifunctor Jr of spaces of r-jets between pairs of manifolds,

we shall write Jr

_b for the _bered manifolds case.)

By proposition 12.11, _r

r􀀀1 : Jr(M; Y ) ! Jr􀀀1(M; Y ) is an a_ne bundle,

the associated vector bundle of which is the pullback of TY SrT_M over

Jr􀀀1(M; Y ). Taking into account the local trivializations of Y , we _nd that

r􀀀1 : JrY ! Jr􀀀1Y is an a_ne subbundle of Jr(M; Y ) and its modelling vector

bundle is the pullback of V Y SrT_M over Jr􀀀1Y , where V Y denotes the

vertical tangent bundle of Y . For r = 1 it is useful to give a direct description

of the a_ne bundle structure on J1Y ! Y because of its great importance in

the theory of connections. The space J1(M; Y ) coincides with the vector bundle

TY T_M = L(TM; TY ). A 1-jet X: TxM ! TyY , x = p(y), belongs to J1Y

if and only if Tp _X = idTxM. The kernel of such a projection induced by Tp is

VyY T_

xM, so that the pre-image of idTxM in TyY T_

xM is an a_ne subspace

with modelling vector space VyY T_

xM.

If we specialize corollary 12.11 to the case of a _bered manifold Y , we deduce

that for every X 2 JrY the kernel of the restriction of T_r

r􀀀1 : TJrY ! TJr􀀀1Y

to TXJrY is V_XY SrT_

_XM.

In conclusion we describe the relation between the contact (n; r)-elements

on a manifold M and the elements of the r-th jet prolongation of a suitable

local _bration on M. In a su_ciently small neighborhood U of an arbitrary

x 2 M there exists a _bration p: U ! N over an n-dimensional manifold N.

By the de_nition of contact elements, every X 2 Krn

M transversal to p (i.e.

the underlying contact 1-element of X is transversal to p) is identi_ed with an

element of Jr(U ! N) and vice versa. In particular, if we take U _= Rn_Rm􀀀n,

then the latter identi_cation induces some simple local coordinates on Krn

12.17. If E ! M is a vector bundle, then JrE ! M is also a vector bundle,

provided we de_ne jrx

s1(u) + jrx

s2(u) = jrx

(s1(u) + s2(u)), where u belongs to a

neighborhood of x 2 M, and kjrx

s(u) = jrx

ks(u), k 2 R.

Let Z ! M be an a_ne bundle with the modelling vector bundle E ! M.

Then JrZ ! M is an a_ne bundle with the modelling vector bundle JrE ! M.

Given jrx

s 2 JrZ and jrx

_ 2 JrE, we set jrx

s(u)+jrx

_(u) = jrx

(s(u)+_(u)), where

the sum s(u) + _(u) is de_ned by the canonical map Z _M E ! Z.

12.18. In_nite jets. Consider an in_nite sequence

(1) A1;A2; : : : ;Ar; : : :

of jets Ai 2 Ji(M;N) satisfying Ai = _i+1

i (Ai+1) for all i = 1; : : : . Such a

sequence is called a jet of order 1 or an in_nite jet of M into N. Hence the set

J1(M;N) of all in_nite jets of M into N is the projective limit of the sequence

J1(M;N) _2

􀀀􀀀1 J2(M;N) _3

􀀀􀀀2 : : :

r􀀀1 􀀀􀀀􀀀 Jr(M;N) _r+1

􀀀r􀀀􀀀 : : :

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

126 Chapter IV. Jets and natural bundles

We denote by _1

r : J1(M;N) ! Jr(M;N) the projection transforming the

sequence (1) into its r-th term. In this book we usually treat J1(M;N) as a

set only, i.e. we consider no topological or smooth structure on J1(M;N). (For

the latter subject the reader can consult e.g. [Michor, 80].)

Given a smooth map f : M ! N, the sequence

xf j2

xf _ _ _ jrx

f : : :

x 2 M, which is denoted by j1

x f or j1f(x), is called the in_nite jet of f at

x. The classical Borel theorem, see 19.4, implies directly that every element of

J1(M;N) is the in_nite jet of a smooth map of M into N, see also 19.4.

The spaces T1

k M of all k-dimensional velocities of in_nite order and the in_-

nite di_erential group G1

m in dimension m are de_ned in the same way. Having

a _bered manifold Y ! M, the in_nite jets of its sections form the in_nite jet

prolongation J1Y of Y .

12.19. Jets of _bered manifold morphisms. If we consider the jets of morphisms

of _bered manifolds, we can formulate additional conditions concerning

the restrictions to the _bers or the induced base maps. In the _rst place, if we

have two maps f, g of a _bered manifold Y into another manifold, we say they

determine the same (r; s)-jet at y 2 Y , s _ r, if

(1) jr

yf = jr

yg and js

y(fjYx) = js

y(gjYx);

where Yx is the _ber passing through y. The corresponding equivalence class will

be denoted by jr;s

y f. Clearly (r; s)-jets of FM-morphisms form a category, and

the bundle projection determines a functor from this category into the category

of r-jets. We denote by Jr;s(Y; _ Y ) the space of all (r; s)-jets of the _bered

manifold morphisms of Y into another _bered manifold _ Y .

Moreover, let q _ r be another integer. We say that two FM-morphisms

f; g : Y ! _ Y determine the same (r; s; q)-jet at y, if it holds (1) and

(2) jqx

Bf = jqx

Bg;

where Bf and Bg are the induced base maps and x is the projection of y to the

base BY of Y . We denote by jr;s;q

y f such an equivalence class and by Jr;s;q(Y; _ Y )

the space of all (r; s; q)-jets of the _bered manifold morphisms between Y and

_ Y . The bundle projection determines a functor from the category of (r; s; q)-jets

of FM-morphisms into the category of q-jets. Obviously, it holds

(3) Jr;s;q(Y; _ Y ) = Jr;s(Y; _ Y ) _

Jr(BY;B _ Y ) Jq(BY;B _ Y )

where we consider the above mentioned projection Jr;s(Y; _ Y ) ! Jr(BY;B _ Y )

and the jet projection _q

r : Jq(BY;B _ Y ) ! Jr(BY;B _ Y ).

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

12. Jets 127

12.20. An abstract characterization of the jet spaces. We remark that

[Kol_a_r, to appear c] has recently deduced that the r-th order jets can be characterized

as homomorphic images of germs of smooth maps in the following way.

According to 12.3, the rule jr de_ned by

jr(germxf) = jrx

transforms germs of smooth maps into r-jets and preserves the compositions.

By 12.6, Jr(M;N) is a _bered manifold over M _N for every pair of manifolds

M, N. So if we denote by G(M;N) the set of all germs of smooth maps of M

into N, jr can be interpreted as a map

jr = jrM

;N : G(M;N) ! Jr(M;N):

More generally, consider a rule F transforming every pair M, N of manifolds

into a _bered manifold F(M;N) over M _ N and a system ' of maps

'M;N : G(M;N) ! F(M;N) commuting with the projections G(M;N) ! M _

N and F(M;N) ! M _N for all M, N. Let us formulate the following requirements

I{IV.

I. Every 'M;N : G(M;N) ! F(M;N) is surjective.

II. For every pairs of composable germs B1, B2 and _B1, _B2, '(B1) = '(_B1)

and '(B2) = '(_B2) imply '(B2 _ B1) = '(_B2 _ _B1).

By I and II we have a well de_ned composition (denoted by the same symbol

as the composition of germs and maps)

X2 _ X1 = '(B2 _ B1)

for every X1 = '(B1) 2 Fx(M;N)y and X2 = '(B2) 2 Fy(N; P)z. Every local

di_eomorphism f : M ! _M and every smooth map g : N ! _N induces a map

F(f; g) : F(M;N) ! F( _M ; _N ) de_ned by

F(f; g)(X) = '(germyg) _ X _ '((germxf)􀀀1); X 2 Fx(M;N)y:

III. Each map F(f; g) is smooth.

Consider the product N1

p1 􀀀 N1 _ N2

p2 􀀀! N2 of two manifolds. Then

we have the induced maps F(idM; p1) : F(M;N1 _ N2) ! F(M;N1) and

F(idM; p2) : F(M;N1 _ N2) ! F(M;N2). Both F(M;N1) and F(M;N2) are

_bered manifolds over M.

IV. F(M;N1_N2) coincides with the _bered product F(M;N1)_MF(M;N2)

and F(idM; p1), F(idM; p2) are the induced projections.

Then it holds: For every pair (F; ') satisfying I{IV there exists an integer

r _ 0 such that (F; ') = (Jr; jr). (The proof is heavily based on the theory of

Weil functors presented in chapter VIII below.)

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

128 Chapter IV. Jets and natural bundles

13. Jet groups

In spite of the fact that the jet groups lie at the core of considerations concerning

geometric objects and operations, they have not been studied very extensively.

The paper [Terng, 78] is one of the exceptions and many results presented

in this section appeared there for the _rst time.

13.1. Let us recall the jet groups Gk

m = invJk

0 (Rm;Rm)0 with the multiplication

de_ned by the composition of jets, cf. 12.6. The jet projections _l+1

l de_ne the

sequence

(1) Gk

! Gk􀀀1

! _ _ _ ! G1

! 1

and the normal subgroups Bl = ker _k

l (or Bk

l if more suitable) form the _ltration

(2) Gk

m = B0 _ B1 _ _ _ _ _ Bk􀀀1 _ Bk = 1.

Since we identify Jk

0 (Rm;Rm) with the space of polynomial maps Rm ! Rm of

degree less then or equal to k, we can write Gk

m = ff = f1 +f2 +_ _ _+fk ; fi 2

sym(Rm;Rm), 1 _ i _ k, and f1 2 GL(m) = G1

g, where Li

sym(Rm;Rn) is the

space of all homogeneous polynomial maps Rm ! Rn of degree i. Hence Gk

m is

identi_ed with an open subset of an Euclidean space consisting of two connected

components. The connected component of the unit, i.e. the space of all invertible

jets of orientation preserving di_eomorphisms, will be denoted by Gk

+. It

follows that the Lie algebra gk

m is identi_ed with the whole space Jk

0 (Rm;Rm)0,

or equivalently with the space of k-jets of vector _elds on Rm at the origin that

vanish at the origin. Since each jk

0X, X 2 X(Rm), has a canonical polynomial

representative, the elements of gk

m can also be viewed as polynomial vector _elds

X =

_x_ @

@xi

. Here the sum goes over i and all multi indices _ with 1 _

j_j _ k.

For technical reasons, we shall not use any summation convention in the rest of

this section and we shall use only subscripts for the indices of the space variables

x 2 Rn, i.e. if (x1; : : : ; xn) 2 Rn, then x21

always means x1:x1, etc.

13.2. The tangent maps to the jet projections turn out to be jet projections

as well. Hence the sequence 13.1.(1) gives rise to the sequence of Lie algebra

homomorphisms

k􀀀1 􀀀􀀀􀀀! gk􀀀1

_k􀀀1

k􀀀2 􀀀􀀀􀀀! _ _ _

􀀀􀀀!1 g1

! 0

and we get the _ltration by ideals bl = ker _k

l (or bkl

if more suitable)

m = b0 _ b1 _ _ _ _ _ bk􀀀1 _ bk = 0.

Let us de_ne gp _ gk

m, 0 _ p _ k􀀀1, as the space of all homogeneous polynomial

vector _elds of degree p+1, i.e. gp = Lp+1

sym(Rm;Rm). By de_nition, gp is identi_ed

with the quotient bp=bp+1 and at the level of vector spaces we have

(1) gk

m = g0 _ g1 _ _ _ _ _ gk􀀀1.

For any two subsets L1, L2 in a Lie algebra g we write [L1;L2] for the linear

subspace generated by the brackets [l1; l2] of elements l1 2 L1, l2 2 L2. A

decomposition g = g0_g1_: : : of a Lie algebra is called a grading if [gi; gj ] _ gi+j

for all 0 _ i; j < 1. In our decomposition of gk

m we take gi = 0 for all i _ k.

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

13. Jet groups 129

Proposition. The Lie algebra gk

m of the Lie group Gk

m is the vector space

fjk

0X ; X 2 X(Rm); X(0) = 0g with the bracket

(2) [jk

0X; jk

0 Y ] = 􀀀jk

0 [X; Y ]

and with the exponential mapping

(3) exp(jk

0X) = jk

0 FlX1

; jk

0X 2 gk

The decomposition (1) is a grading and for all indices 0 _ i; j < k we have

(4) [gi; gj ] = gi+j if m > 1, or if m = 1 and i 6= j.

Proof. For every vector _eld X 2 X(Rm), the map t 7! jk

0 FlXt

is a one-parameter

subgroup in Gk

m and the corresponding element in gk

m is

0 jk

0 FlXt

= jk

0 FlXt

= jk

0X.

Hence exp(t:jk

0X) = jk

0 FlXt

, see 4.18. Now, let us consider vector _elds X, Y

on Rm vanishing at the origin and let us write briey a := jk

0X, b := jk

0 Y .

According to 3.16 and 4.18.(3) we have

􀀀2jk

0 [X; Y ] = 2jk

0 [Y;X] = jk

@t2

___

FlX􀀀

_ FlY

􀀀t

_ FlXt

_ FlYt

= @2

@t2

___

0 FlX􀀀

_jk

0 FlY

􀀀t

_jk

0 FlXt

_jk

0 FlYt

= @2

@t2

___

􀀀

exp(􀀀ta) _ exp(􀀀tb) _ exp(ta) _ exp(tb)

= @2

@t2

___

FlLb

_ FlLa

_ FlLb

􀀀t

_ FlLa

􀀀t

(e) = 2[jk

0X; jk

0 Y ].

So we have P proved formulas (2) and (3). For all polynomial vector _elds a =

_x_ @

@xi

, b =

_x_ @

@xi

2 gk

m the coordinate formula for the Lie bracket of

vector _elds, see 3.4, and formula (2) imply

(5)

[a; b] =

x @

@xi

where

1_j_m

_+_􀀀1j=

􀀀

_jbj

_ai

􀀀 _jaj

_bi

Here 1j means the multi index _ with _i = _ij

and there is no implicit summation

in the brackets. This formula shows that (1) is a grading. Let us evaluate

x_ @

@xi

; x_ @

@xi

= (_i 􀀀 _i)x_+_􀀀1i @

@xi

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

130 Chapter IV. Jets and natural bundles

and consider two degrees p, q, 0 _ p + q _ k 􀀀 1. If p 6= q then for every with

jj = p + q + 1 and for every index 1 _ i _ m, we are able to _nd some _ and

_ with j_j = p + 1, j_j = q + 1 and _ + _ = + 1i, _i 6= _i. Since the vector

_elds x @

@xi

, 1 _ i _ m, jj = p + q + 1, form a linear base of the homogeneous

component gp+q, we get equality (4). If p = q, then the above consideration fails

only in the case i = jj. But if m > 1, then we can take the bracket

[xjxp

@xi

; xq+1

@xj

] = xp+q+1

@xi

􀀀 (q + 1)xp+q

i xj

@xj

j 6= i.

Since the second summand belongs to [gp; gq] this completes the proof. _

13.3. Let us recall some general concepts. The commutator of elements a1, a2

of a Lie group G is the element a1a2a􀀀1

1 a􀀀1

2 G. The closed subgroup K(S1; S2)

generated by all commutators of elements s1 2 S1 _ G, s2 2 S2 _ G is called

the commutator of the subsets S1 and S2. In particular, G0 := K(G;G) is called

the derived group of the Lie group G. We get two sequences of closed subgroups

G(0) = G = G(0)

G(n) = (G(n􀀀1))0 n 2 N

G(n) = K(G;G(n􀀀1)) n 2 N:

A Lie group G is called solvable if G(n) = feg and nilpotent if G(n) = feg for

some n 2 N. Since always G(n)

_ G(n), every nilpotent Lie group is solvable.

The Lie bracket determines in each Lie algebra g the following two sequences

of Lie subalgebras

g = g(0) = g(0)

g(n) = [g(n􀀀1); g(n􀀀1)] n 2 N

g(n) = [g; g(n􀀀1)] n 2 N:

The sequence g(n) is called the descending central sequence of g. A Lie algebra g

is called solvable if g(n) = 0 and nilpotent if g(n) = 0 for some n 2 N, respectively.

Every nilpotent Lie algebra is solvable. If b is an ideal in g(n) such that the factor

g(n)=b is commutative, then b _ g(n+1). Consequently Lie algebra g is solvable

if and only if there is a sequence of subalgebras g = b0 _ b1 _ _ _ _ _ bl = 0

where bk+1 _ bk is an ideal, 0 _ k < l, and all factors bk=bk+1 are commutative.

Proposition. [Naymark, 76, p. 516] A connected Lie group is solvable, or nilpotent

if and only if its Lie algebra is solvable, or nilpotent, respectively.

13.4. Let i : GL(m) ! Gk

m be the map transforming every matrix A 2 GL(m)

into the r-jet at zero of the linear isomorphism x 7! A(x), x 2 Rm. This is a

splitting of the short exact sequence of Lie groups

(1) e wB1 wGk

m w

u i

m w e

so that we have the situation of 5.16.

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

13. Jet groups 131

Proposition. The Lie group Gk

m is the semidirect product GL(m) o B1 with

the action of GL(m) on B1 given by (1). The normal subgroup B1 is connected,

simply connected and nilpotent. The exponential map exp: b1 ! B1 is a global

di_eomorphism.

Proof. Since the normal subgroup B1 is di_eomorphic to a Euclidean space,

see 13.1, it is connected and simply connected. Hence B1 is also nilpotent, for

its Lie algebra b1 is nilpotent, see 13.2.(4) and 13.3. By a general theorem, see

[Naymark, 76, p. 516], the exponential map of a connected and simply connected

solvable Lie group is a global di_eomorphism. Since our group is even nilpotent

this also follows from the Baker-Campbell-Hausdor_ formula, see 4.29. _

13.5. We shall need some very basic concepts from representation theory. A

representation _ of a Lie group G on a _nite dimensional vector space V is a

Lie group homomorphism _ : G ! GL(V ). Analogously, a representation of

a Lie algebra g on V is a Lie algebra homomorphism g ! gl(V ). For every

representation _ : G ! GL(V ) of a Lie group, the tangent map at the identity

T_ : g ! gl(V ) is a representation of its Lie algebra, cf. 4.24.

Given two representations _1 on V1 and _2 on V2 of a Lie group G, or a Lie

algebra g, a linear map f : V1 ! V2 is called a G-module or g-module homomorphism,

if f(_1(a)(x)) = _2(a)(f(x)) for all a 2 G or a 2 g and all x 2 V ,

respectively. We say that the representations _1 and _2 are equivalent, if there

is a G-module isomorphism or g-module isomorphism f : V1 ! V2, respectively.

A linear subspace W _ V in the representation space V is called invariant if

_(a)(W) _ W for all a 2 G (or a 2 g) and _ is called irreducible if there is no

proper invariant subspace W _ V . A representation _ is said to be completely

reducible if V decomposes into a direct sum of irreducible invariant subspaces.

A decomposition of a completely reducible representation is unique up to the

ordering and equivalences. A classical result reads that the standard action of

GL(V ) on every invariant linear subspace of pV qV _ is completely reducible

for each p and q, see e.g. [Boerner, 67].

A representation _ of a connected Lie group G is irreducible, or completely

reducible if and only if the induced representation T_ of its Lie algebra g is

irreducible, or completely reducible, respectively, see [Naymark, 76, p. 346].

A representation _ : GL(m) ! GL(V ) is said to have homogeneous degree r if

_(t:idRm) = tridV for all t 2 R n f0g. Obviously, two irreducible representations

with di_erent homogeneous degrees cannot be equivalent.

13.6. The GL(m)-module structure on b1 _ gk

m. Since B1 _ Gk

m is a

normal subgroup, the corresponding subalgebra b1 = g1 __ _ __gk􀀀1 is an ideal.

The (lower case) adjoint action ad of g0 = gl(m) on b1 and the adjoint action

Ad of GL(m) = G1

m on b1 determine structures of a g0-module and a GL(m)-

module on b1. As we proved in 13.2, all homogeneous components gr _ b1 are

g0-submodules.

Let us consider the canonical volume form ! = dx1 ^ _ _ _ ^ dxm on Rm and

recall that for every vector _eld X on Rm its divergence is a function divX on

Rm de_ned by LX! = (divX)!.

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

132 Chapter IV. Jets and natural bundles

In coordinates we have div(

_i@=@xi) =

@_i=@xi and so every k-jet jk

0X 2

m determines the (k 􀀀 1)-jet jk􀀀1

0 (divX). Hence we can de_ne div(jk

0X) =

jk􀀀1

0 (divX) for all jk

0X 2 gk

m. If X is the canonical polynomial representative

of jk

0X of degree k, then divX is a polynomial of degree k 􀀀 1. Let Cr

_ gr be

the subspace of all elements jk

0X 2 gr with divergence zero. By de_nition,

(1)

div[X; Y ]! = L

[X;Y ]! = LXLY ! 􀀀 LY LX!

= (X(divY ) 􀀀 Y (divX))!:

Since every linear vector _eld X 2 g0 has constant divergence, Cr

_ gr is a

gl(m)-submodule. In coordinates,

_x_ @

@xi

2 Cr

1 if and only if

i;_

_iai

_x_􀀀1i = 0;

i.e.

i(_i + 1)ai

_+1i = 0 for each _ with j_j = r.

Further, let us notice that the Lie bracket of the _eld Y0 =

j xj

@xj

with

any linear _eld X 2 g0 is zero. Hence, also the subspace Cr

2 of all vector _elds

Y 2 gr of the form Y = fY0 with an arbitrary polynomial f =

f_x_ of degree

r is g0-invariant. Indeed, it holds [X; fY0] = 􀀀(Xf)Y0.

Since div(fY0) =

j(_j +1)f_x_, we see that gr = Cr

_Cr

2 . In coordinates,

we have linear generators of Cr

(2) X_ = x_(

@xk

); j_j = r;

and if m > 1 then there are linear generators of Cr

(3)

X_;k = x_

(_k + 1)x1

@x1

􀀀 (_1 + 1)xk

@xk

;

j_j = r;

k = 2; : : : ;m

Y_;k = x_ @

@xk

; k = 1; : : : ; m; j_j = r + 1; _k = 0:

We shall write C1 = C1

_ C2

_ _ _ _ _ Ck􀀀1

1 and C2 = C1

_ C2

_ _ _ _ _ Ck􀀀1

2 .

According to (1), C1 _ b1 is a Lie subalgebra. Since for smooth functions f, g on

Rm we have [fX; gX] = (g(Xf)+f(Xg))X, C2 _ b1 is a Lie subalgebra as well.

So we have got a decomposition b1 = C1 _ C2. According to the general theory

this is also a decomposition into G1

+-submodules, but as all the spaces Cr

j are

invariant with respect to the adjoint action of any exchange of two coordinates,

the latter spaces are even GL(m)-submodules.

Proposition. If m > 1, then the GL(m)-submodules Cr

1 , Cr

2 in gr, 1 _ r _

k 􀀀 1, are irreducible and inequivalent. For m = 1, Cr

1 = 0, 1 _ r _ k 􀀀 1, and

all Cr

2 are irreducible inequivalent GL(1)-modules.

Proof. Assume _rst m > 1. A reader familiar with linear representation theory

could verify that the modules Cr

2 are equivalent to the irreducible modules

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

13. Jet groups 133

det􀀀rCm

(r;r;:::;r;0), where the symbol Cm

(r;:::;r;0) corresponds to the Young's diagram

(r; : : : ; r; 0), while Cr

1 are equivalent to det􀀀(r+1)Cm

(r+2;r+1;:::;r+1;0), see e.g.

[Dieudonn_e, Carrell, 71]. We shall present an elementary proof of the proposition.

Let us _rst discuss the modules Cr

2 . Consider one of the linear generators X_

de_ned in (2) and a linear vector _eld xi

@xj

2 gl(m). We have

(4) [􀀀xi

@xj

; x_(

@xk

)] = _jxix_􀀀1j

(xk

@xk

If j = i, we get a scalar multiplication, but in all other cases the index _j

decreases while _i increases by one and if _j = 0, then the bracket is zero.

Hence an iterated action of suitable linear vector _elds on an arbitrary linear

combination of the base elements X_ yields one of the base elements. Further,

formula (4) implies that the submodule generated by any X_ is the whole Cr

2 .

This proves the irreducibility of the GL(m)-modules Cr

2 .

In a similar way we shall prove the irreducibility of Cr

1 . Let us evaluate the

action of Zi;j = xi

@xj

on the linear generators X_;k, Y_;k.

[􀀀Zi;j ;X_;k] = (_k + 1)(_j + _j

1)x_+11+1i􀀀1j @

@x1

􀀀

􀀀 (_1 + 1)(_j + _j

k)x_+1k+1i􀀀1j @

@xk

􀀀

􀀀 _i1

(_k + 1)x_+11 @

@xj

+ _ik

(_1 + 1)x_+1k @

@xj

[􀀀Zi;j ; Y_;k] = _jx_􀀀1j+1i @

@xk

􀀀 _ik

x_ @

@xj

In particular, we get

[􀀀Zi;1; Y_;1] = 0

[􀀀Zi;1;X_;k] =

(_1 + 1)X_+1i􀀀11;k if _1 6= 0, i 6= 1

(_k + 1 + _ik

)Y_+1i;1 if _1 = 0, i 6= 1

[􀀀Zi;j ; Y_;k] =

8><

_jY_􀀀1j+1i;k if i 6= k

X_􀀀1j ;j if i = k, _j 6= 0

􀀀Y_;j if i = k, _j = 0.

Hence starting with an arbitrary linear combination of the base elements, an

iterated action of suitable vector _elds leads to one of the base elements Y_;k.

Then any other base element can be reached by further actions. Therefore also

the modules Cr

2 are irreducible.

If m = 1, then all Cr

1 = 0 by the de_nition and for all 0 _ r _ k 􀀀 1 we have

2 = gr = R with the action of g0 given by [ax @

@x ; bxr+1 @

@x ] = 􀀀rabxr+1 @

@x .

The submodules Cr

1 and Cr

2 cannot be equivalent for dimension reasons. The

adjoint action Ad of GL(m) on gk

m is given by Ad(a)(jk

0X) = jk

0 (a _ X _ a􀀀1).

So each irreducible component of gr has homogeneous degree 􀀀r. Therefore the

modules Cr

i with di_erent r are inequivalent. _

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

134 Chapter IV. Jets and natural bundles

13.7. Corollary. The normal subgroup B1 _ Gk

m is generated by two closed

Lie subgroups D1, D2 invariant under the canonical action of G1

m. The group

D1 is formed by the jets of volume preserving di_eomorphisms and D2 consists

of the jets of di_eomorphisms keeping all the one-dimensional linear subspaces

in Rm. The corresponding Lie subalgebras are the subalgebras with grading

C1 = C1

_ _ _ _ _ Ck􀀀1

1 and C2 = C1

_ _ _ _ _ Ck􀀀1

2 where all the homogeneous

components are irreducible GL(m)-modules with respect to the adjoint action

and b1 = C1 _ C2.

Let us point out that an element jk

0 f 2 Gk

m belongs to D1 or D2 if and

only if its polynomial representative is of the form f = idRm + f2 + _ _ _ + fk

with fi 2 C1 \ Li

sym(Rm;Rm) = Ci􀀀1

1 or fi 2 C2 \ Li

sym(Rm;Rm) = Ci􀀀1

2 ,

respectively.

13.8. Proposition. If m _ 2 and l > 1, or m = 1 and l > 2, then there is no

splitting in the exact sequence e ! Bl ! Gk

! Gl

! e. In dimension m = 1,

there is the exceptional projective splitting G21

! Gk1

de_ned by

(1) ax + bx2 ! a

x + b

x2 + _ _ _ + bk􀀀1

ak􀀀1 xk

Proof. Let us assume there is a splitting j in the exact sequence of Lie algebra

homomorphisms 0 ! bl ! gk

! gl

! 0, l > 1. So j : g0 _ _ _ _ _ gl􀀀1 !

g0 _ _ _ _ _ gk􀀀1 and the restrictions jp

t;q of the components jq : gl

! gq to

the g0-submodules Cp

t in the homogeneous component gp are morphisms of g0-

modules. Hence jp

t;q = 0 whenever p 6= q. Since j is a splitting the maps jp

t;p are

the identities.

Assume now m > 1. Since [gl􀀀1; g1] equals gl in gk

m but at the same time this

bracket equals zero in gl

m, we have got a contradiction.

If m = 1 and l > 2 the same argument applies, but the inclusion j : g0_g1 !

g0_g1__ _ __gk􀀀1 is a Lie algebra homomorphism, for in gk1

the bracket [g1; g1]

equals zero. Let us _nd the splitting on the Lie group level. The germs of

transformations f_;_(x) = x

_x+_ , _ 6= 0, are determined by their second jets,

so we can view them as elements in G21

. Since the composition of two such

transformations is a transformation of the same type, they give rise to Lie group

homomorphisms G21

! Gr

1 for all r 2 N. One computes easily the derivatives

f(n)

_;_(0) = (􀀀1)n􀀀1n!_n􀀀1_􀀀n. Hence the 2-jet ax+bx2 corresponds to f_;_ with

_ = 􀀀ba􀀀2, _ = a􀀀1. Consequently, the homomorphism G21

! Gr

1 has the form

(1) and its tangent at the unit is the inclusion j. _

We remark that a geometric de_nition of the exceptional splitting (1) is based

on the fact that the construction of the second order jets determines a bijection

between G21

and the germs at zero of the origine preserving projective transformations

of R.

13.9. Proposition. The Lie group Gk1

is solvable. Its Lie algebra gk1

can be

characterized as a Lie algebra generated by three elements

X0 = x d

2 g0; X1 = x2 d

2 g1; X2 = x3 d

2 g2

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

13. Jet groups 135

with relations

(1) [X0;X1] = 􀀀X1

(2) [X0;X2] = 􀀀2X2

(3) (ad(X1))iX2 = 0 for i _ k 􀀀 2.

Proof. The _ltration gk1

= b0 _ _ _ _ _ bk􀀀1 _ 0 from 13.2 is a descending chain

of ideals with dim(bi=bi+1) = 1. Hence gk1

is solvable.

Let us write Xi = xi+1 d

2 gi. Since [X1;Xi] = (1 􀀀 i)Xi+1, we have

Xi =

(􀀀1)i􀀀2

(i 􀀀 2)!

(4) (ad(X1))i􀀀2X2 for k 􀀀 1 _ i _ 3

(5) [Xi;Xj ] = (i 􀀀 j)Xi+j :

Now, let g be a Lie algebra generated by _X0, _X1, _X2 which satisfy relations

(1){(3) and let us de_ne _Xi, i > 2 by (4). Consider the linear map _: gk1

! g,

Xi ! _Xi, 0 _ i _ k 􀀀1. Then [_X1; _Xi] = (1􀀀i)_Xi+1 and using Jacobi identity,

the induction on i yields [_X0; _Xi] = 􀀀i_Xi. A further application of Jacobi

identity and induction on i lead to [_Xi; _Xj ] = (i 􀀀 j)_Xi+j . Hence the map _ is

an isomorphism. _

13.10. The group Gk

m with m _ 2 has a more complicated structure. In particular

m cannot be solvable, for [gk

m; gk

m] contains the whole homogeneous

component g0, so that this cannot be nilpotent. But we have

Proposition. The Lie algebra gk

m, m _ 2, k _ 2, is generated by g0 and any

element a 2 g1 with a =2 C1

[ C1

2 . In particular, we can take a = x21

@x1

Proof. Let g be the Lie algebra generated by g0 and a. Since g1 = C1

_ C1

2 is

a decomposition into irreducible g0-modules, g1 _ g. But then 13.2.(4) implies

g = gk

m. _

13.11. Normal subgroup structure. Let us _rst describe several normal

subgroups of Gk

m. For every r 2 N, 1 _ r _ k 􀀀 1, we de_ne Br;1 _ Br,

Br;1 = fjr

0f; f = idRm + fr+1 + _ _ _ + fk; fr+1 2 Cr

1 ; fi 2 Li

sym(Rm;Rm)g.

The corresponding Lie subalgebra in gk

m is the ideal Cr

_ gr+1 _ _ _ _ _ gk􀀀1

so that Br;1 is a normal subgroup. Analogously, we set Br;2 = fjr

0f; f =

idRm +fr+1 +_ _ _+fk; fr+1 2 Cr

2 ; fi 2 Li

sym(Rm;Rm)g with the corresponding

Lie subalgebra Cr

_gr+1__ _ __gk􀀀1. We can characterize the normal subgroups

Br;j as the subgroups in Br with the projections _k

r+1(Br;j) belonging to the

subgroups Dj _ Gr+1

m , j = 1; 2, cf. 13.7.

Proposition. Every connected normal subgroup H of Gk

m, m _ 2, is one of the

following:

(1) feg, the identity subgroup,

(2) Br, 1 _ r < k, the kernel of the projection _k

r : Gk

! Gr

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

136 Chapter IV. Jets and natural bundles

(3) Br;1, 1 _ r < k, the subgroup in Br of jets of di_eomorphisms keeping

the standard volume form up to the order r + 1 at the origin,

(4) Br;2, 1 _ r < k, the subgroup in Br of jets of di_eomorphisms keeping

the linear one-dimensional subspaces in Rm up to the order r + 1 at the origin,

(5) N o B1, where N is a normal subgroup of GL(m) = G1

Proof. Since we deal with connected subgroups H _ Gk

m, we can prove the

proposition on the Lie algebra level.

Let us _rst assume that H _ B1. Then it su_ces to prove that the ideal in

m generated by Cr

j , j = 1; 2, is the whole Cr

_br+1. But the whole algebra gk

is generated by g0 and X1 = x21

@x1

, and [g1; gi] = gi+1 for all 2 _ i < k. That

is why we have only to prove that gr+1 is contained in the subalgebra generated

by g0; X1 and Cr

j for both j = 1 and j = 2. Since Cr+1

j are irreducible g0-

submodules, it su_ces to _nd an element Y 2 Cr

j such that [X1; Y ] =2 Cr+1

1 and

at the same time [X1; Y ] =2 Cr+1

2 .

Let us take _rst j = 2, i.e. Y = fY0 for certain polynomial f. Since

[fY0;X1] = (X1f)Y0 +f[Y0;X1] = (X1f)Y0 􀀀fX1, the choice f(x) = 􀀀xr

2 gives

[Y;X1] = xr

2x21

@x1

which does not belong to Cr+1

[ Cr+1

2 , for its divergence

equals to 2x1xr

6= 0, cf. 13.5.

Further, consider Y = xr+1

@x1

2 Cr

1 and let us evaluate [xr+1

@x1

; x21

@x1

] =

􀀀2x1xr+1

@x1

. Since the divergence of the latter _eld does not vanish, [Y;X2] =2

Cr+1

[ Cr+1

2 as required. Hence we have proved that all connected normal

subgroups H _ Gk

m contained in B1 are of the form (1){(4).

Consider now an arbitrary ideal h in gk

m and let us denote n = h\g0 _ g0. By

virtue of 13.2.(4), if h contains a vector which generates g1 as a g0-module, then

b1 _ h. We shall prove that for every X 2 g0 any of the equalities [X;C1

1 ] = 0

and [X;C1

2 ] = 0 implies X = 0. Therefore either h _ b1 or n = 0 which concludes

the proof of the proposition.

Let X =

i;j bijxj

@xi

2 g0 and Y = xk

j xj

@xj

2 C1

2 . Then [X; Y ] =

􀀀(

j bkjxj)Y0. Hence [X;C1

2 ] = 0 implies X = 0. Similarly, for Y = x2l

@xk

1 and X 2 g0, the equalities [X; Y ] = 0 for all k 6= l yield X = 0. The simple

computation is left to the reader. _

13.12. Gk

m-modules. In the next sections we shall see that the actions of

the jet groups on manifolds correspond to bundles of geometric objects. In

particular, the vector bundle functors on m-dimensional manifolds correspond

to linear representations of Gk

m, i.e. to Gk

m-modules. Since there is a well known

representation theory of GL(m) which is a subgroup in Gk

m, we should try to

describe possible extensions of a given representation of GL(m) on a vector

space V to a representation of Gk

m. A step towards such description was done

in [Terng, 78], we shall present only an observation showing that the study

of geometric operations on irreducible vector bundles restricts in fact to the

case of irreducible GL(m)-modules (with trivial action of the normal subgroup

B1). According to 5.4, there is a bijective correspondence between Lie group

homomorphisms from B1 to GL(V ) and Lie algebra homomorphisms from b1 to

gl(V ), for B1 is connected and simply connected. Further, there is the semidirect

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

13. Jet groups 137

product structure gk

m = gl(m)ob1 with the adjoint action of gl(m) on b1 which

is tangent to the adjoint action of GL(m) and every representation of GL(m) on

V induces a GL(m)-module structure on gl(V ) via the adjoint action of GL(V )

on gl(V ). This implies immediately

Proposition. For every representation _: GL(m) ! GL(V ) there is a bijection

between the representations __: Gk

! GL(V ) with __jGL(m) = _ and the set

of mappings T : b1 ! gl(V ) which are both Lie algebra homomorphisms and

homomorphisms of GL(m)-modules.

13.13. A G-module is called primary if it is equivalent to a direct sum of copies

of a single irreducible G-module.

Proposition. If V is a Gk

m-module such that the induced GL(m)-module is

primary, then the action of the normal subgroup B1 _ Gk

m is trivial.

Proof. Assume that the GL(m)-module V equals sW, where W is an irreducible

GL(m)-module. Then each irreducible component of the GL(m)-module

gl(V ) = V V _ has homogeneous degree zero. But all the irreducible components

of b1 have negative homogeneous degrees. So there are no non-zero homomorphisms

between the GL(m)-modules b1 and gl(V ) and 13.12 implies the

proposition. _

13.14. Proposition. Let _: Gk

! GL(V ) be a linear representation such

tPhat the corresponding GL(m)-module is completely reducible and let V = r

i=1 niVi, where Vi are inequivalent irreducible GL(m)-modules ordered by

their homogeneous degrees, i.e. the homogeneous degree of Vi is less than or equal

to the homogeneous degree of Vj whenever i _ j. Then W = (

Pl􀀀1

i=1 niVi) _ nVl

is a Gk

m-submodule of V for all 1 _ l _ r and n _ nl.

Proof. By de_nition, (

Pl􀀀1

i=1 niVi) _ nVl is a GL(m)-submodule. Since every irreducible

component of the GL(m)-module b1 has negative homogeneous degree

and for all 1 _ i _ l the homogeneous degree of L(Vi; Vl) is non-negative, we get

Te_(X)((

Xl􀀀1

i=1

niVi) _ nVl) _

Xl􀀀1

i=1

niVi

for all n _ nl and for every X 2 b1. Now the proposition follows from 13.12 and

13.5. _

13.15. Corollary. Every irreducible Gk

m-module which is completely reducible

as a GL(m)-module is an irreducible GL(m)-module with a trivial action of the

normal nilpotent subgroup B1 _ Gk

Proof. Let V be an irreducible Gk

m-module. Then V is irreducible when viewed

as a GL(m)-module, cf. proposition 13.14. But then B1 acts trivially on V by

virtue of proposition 13.13. _

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

138 Chapter IV. Jets and natural bundles

13.16. Remark. In the sequel we shall often work with various subgroups in

the group of all di_eomorphisms Rm ! Rm which determine Lie subgroups in

the jet groups Gk

m. Proposition 13.2 describes the bracket and the exponential

map in the corresponding Lie algebras and also their gradings g = g0 _ _ _ _ _

gk􀀀1. Let us mention at least volume preserving di_eomorphisms, symplectic

di_eomorphisms, isometries and _bered isomorphisms on the _brations Rm+n !

Rm. We shall essentially need the latter case in the next chapter, see 18.8. The

r-th jet group of the category FMm;n is Gr

m;n

_ Gr

m+n and the corresponding

Lie subalgebra gk

m;n

_ gk

m+n consists of all polynomial vector _elds

i;_ ai

_x_ @

@xi

with ai

_ = 0 whenever i _ m and _j 6= 0 for some j > m. The arguments from

the end of the proof of proposition 13.2 imply that even 13.2.(4) remains valid

in the following formulation.

The decomposition gk

m;n = g0 _ _ _ _ _ gk􀀀1 is a grading and for every indices

0 _ i; j < k it holds

(1) [gi; gj ] = gi+j if m > 1, n > 1, or if i 6= j.

14. Natural bundles and operators

In the preface and in the introduction to this chapter, we mentioned that

geometric objects are in fact functors de_ned on a category of manifolds with

values in category FM of _bered manifolds. Therefore we shall use the name

bundle functors, in general. But the best known among them are de_ned on

category Mfm of m-dimensional manifolds and local di_eomorphisms and in

this case many authors keep the traditional name natural bundles. Throughout

this section, we shall use the original de_nition of natural bundles including

the regularity assumption, see [Nijenhuis, 72], [Terng, 78], [Palais, Terng, 77],

but we shall prove in chapter V that every bundle functor on Mfm is of _nite

order and that the regularity condition 14.1.(iii) follows from the other axioms.

Since the presentation of these results needs rather long and technical analytical

considerations, we prefer to derive _rst geometric properties of bundle functors

in the best known situations under stronger assumptions. In fact the material of

this section presents a model for the more general situation treated in the next

chapter.

14.1. De_nition. A bundle functor on Mfm or a natural bundle over mmanifolds,

is a covariant functor F : Mfm ! FM satisfying the following conditions

(i) (Prolongation) B_F = IdMfm, where B: FM!Mf is the base functor.

Hence the induced projections form a natural transformation p: F ! IdMfm.

(ii) (Locality) If i : U ! M is an inclusion of an open submanifold, then

FU = p􀀀1

M (U) and Fi is the inclusion of p􀀀1

M (U) into FM.

(iii) (Regularity) If f : P_M ! N is a smooth map such that for all p 2 P the

maps fp = f(p; ) : M ! N are local di_eomorphisms, then ~ Ff : P_FM ! FN,

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

14. Natural bundles and operators 139

de_ned by ~ Ff(p; ) = Ffp, p 2 P, is smooth, i.e. smoothly parameterized systems

of local di_eomorphisms are transformed into smoothly parameterized systems

of _bered local isomorphisms.

In sections 10 and 12 we met several bundle functors on Mfm.

14.2. Now let F be a natural bundle. We shall denote by tx : Rm ! Rm the

translation y 7! y + x and for any manifold M and point x 2 M we shall write

FxM for the pre image p􀀀1

M (x). In particular, F0Rm will be called the standard

_ber of the bundle functor F. Every bundle functor F : Mfm ! FMdetermines

an action _ of the abelian group Rm on FRm via _x = Ftx.

Proposition. Let F : Mfm ! FM be a bundle functor on Mfm and let S :=

F0Rm be the standard _ber of F. Then there is a canonical isomorphism Rm _

S _= FRm, (x; z) 7! Ftx(z), and for every m-dimensional manifold M the value

FM is a locally trivial _ber bundle with standard _ber S.

Proof. The map : FRm ! Rm _ S de_ned by z 7! (x; Ft􀀀x(z)), x = p(z), is

the inverse to the map de_ned in the proposition and both maps are smooth according

to the regularity condition 14.1.(iii). The rest of the proposition follows

from the locality condition 14.1.(ii). Indeed, a _bered atlas of FM is formed by

the values of F on the charts of any atlas of M. _

14.3. De_nition. A natural bundle F : Mfm ! FM is said to be of _nite

order r, 0 _ r < 1, if for all local di_eomorphisms f, g : M ! N and every

point x 2 M, the equality jrx

f = jrx

g implies FfjFxM = FgjFxM.

14.4. Associated maps. Let us consider a natural bundle F : Mfm ! FM

of order r. For all m-dimensional manifolds M, N we de_ne the mapping

FM;N : invJr(M;N) _M FM ! FN, (jrx

f; y) 7! Ff(y). The mappings FM;N

are called the associated maps of the bundle functor F.

Proposition. The associated maps are smooth.

Proof. For m = 0 the assertion is trivial. Let us assume m > 0. Since smoothness

is a local property, we may restrict ourselves to M = N = Rm. Indeed,

chosen local charts on M and N we get local trivializations on FM and FN and

the induced local chart on invJr(M;N). Hence we have

invJr(Rm;Rm) _

Rm FRm

􀀀! invJr(U; V ) _U FU

FU;V 􀀀􀀀􀀀! FV

􀀀! FRm

and we can apply the locality condition.

Now, let us recall that every jet in Jr(Rm;Rn) has a canonical polynomial

representative and that this space coincides with the cartesian product of Rm and

the Euclidean space of coe_cients of these polynomials, as a smooth manifold. If

we consider the map ev: invJr(Rm;Rm)_Rm ! Rm, evx(jr

0f) = f(x), then the

associated map FRm;Rm coincides with the map ~ F(ev) appearing in the regularity

condition. _

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

140 Chapter IV. Jets and natural bundles

14.5. Induced action. According to proposition 14.4 the restriction ` =

FRm;RmjGr

_ S is a smooth left action of the jet group Gr

m on the standard

_ber S.

Let us de_ne qM = FRm;M jinvJr

0 (Rm;M) _ S : PrM _ S ! FM. For every

u = jr

0g 2 invJr

0 (Rm;M), s 2 S and jr

0f 2 Gr

m we have

(1) qM(jr

0g _ jr

0f; `(jr

0f􀀀1; s)) = qM(jr

0g; s)

and the restriction (qM)u := qM(jr

0g; ) is a di_eomorphism. Hence q determines

the structure of the associated _ber bundle PrM[S; `] on FM, cf. 10.7.

Proposition. For every bundle functor F : Mfm ! FM of order r and every

m-dimensional manifold M there is a canonical structure of an associated bundle

PrM[S; `] on FM given by the map qM and the values of the functor F lie in

the category of bundles with structure group Gr

m and standard _ber S.

Proof. The _rst part was already proved. Consider a local di_eomorphism

f : M ! N. For every jr

0g 2 PrM, s 2 S we have

Ff _ qM(jr

0g; s) = Ff _ Fg(s) = qN(jr

0 (f _ g); s).

So we identify Ff with fPrf; idSg: PrM _Gr

m S ! PrN _Gr

m S. _

14.6. Description of r-th order natural bundles. Every smooth left action

` of Gr

m on a manifold S determines a covariant functor L: PB(Gr

m) ! FMm,

LP = P[S; `], Lf = ff; idSg. An r-th order bundle functor F with standard

_ber S induces an action ` of Gr

m on S and we can construct a natural bundle

G = L _ Pr : Mfm ! FM.

We claim that F is naturally equivalent to G. For every u = jr

0g 2 Pr

there is the di_eomorphism (qM)u : S ! FxM which we shall denote Fu. Hence

we can de_ne maps _M : GM ! FM by

_M(fu; sg) = Fu(s) = qM(jr

0g; s) = Fg(s).

According to 14.5.(1), this is a correct de_nition, and by the construction, the

maps _M are _bered isomorphisms. Since Gf = fPrf; idSg for every local

di_eomorphism f : M ! N, we have Ff _ _M(fjr

0g; sg) = F(f _ g)(s) = _N _

Gf(fjr

0g; sg).

From the geometrical point of view, naturally equivalent functors can be

identi_ed. Hence we have proved

Theorem. There is a bijective correspondence between the set of all r-th order

natural bundles on m-dimensional manifolds and the set of smooth left actions

of the jet group Gr

m on smooth manifolds.

In the next examples, we demonstrate on well known natural bundles, that

the identi_cation in the theorem is exactly what the geometers usually do.

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

14. Natural bundles and operators 141

14.7. Examples.

1. The reader should reconsider that in the case of frame bundles Pr the

identi_cation used in 14.6, i.e. the relation of the functor Pr to the functor

G constructed from the induced action, is exactly the usual identi_cation of

principal _ber bundles (P; p;M;G) with their associated bundles P[G; _], where

_ is the left action of G on itself.

2. For the tangent bundle T, the map (qM)u with u = j1

0g 2 P1

xM is just the

linear map T0g : T0Rm ! TxM determined by j1

0g, i.e. the linear coordinates

on TxM induced by local chart g. Hence the tangent bundle corresponds to the

canonical action of G1

m = GL(m;R) on Rm.

3. Further well known natural bundles are the functors Tr

k of r-th order kvelocities.

More precisely, we consider the restrictions of the functors de_ned in

12.8 to the category Mfm. Let us recall that Tr

kM = Jr

0 (Rk;M) and the action

on morphisms is given by the composition of jets. Hence, in this case, for every

u = jr

0g 2 Pr

xM the map (qM)u transforms the classes of r-equivalent maps

(Rk; 0) ! (M; x) into their induced coordinate expressions in the local chart g,

i.e. (qM)􀀀1

u (jr

0f) = jr

0 (g􀀀1 _ f).

14.8. Vector bundle functors. In accordance with 6.14, a bundle functor

F : Mfm ! FM is called a vector bundle functor on Mfm, or natural vector

bundle, if there is a canonical vector bundle structure on each value FM and

the values Ff on morphisms are morphisms of vector bundles. Let F be an

r-th order natural vector bundle with standard _ber V and with induced action

` : Gr

_ V ! V . Then ` is a group homomorphism Gr

! GL(V ) and so V

carries a structure of Gr

m-module. On the other hand, every Gr

m-module V gives

rise to a natural bundle F, see the construction in 14.6, and an application of F

to charts of any atlas on a manifold M yields a vector bundle atlas on the value

FM ! M. Therefore proposition 14.6 implies

Proposition. There is a bijective correspondence between r-th order vector

bundle functors on Mfm and Gr

m-modules.

14.9. Examples.

1. In our setting, the p-covariant and q-contravariant tensor _elds on a manifold

M are just the smooth global sections of FM ! M, where F is the vector

bundle functor corresponding to the GL(m)-module pRm_ qRm, cf. 7.2.

2. In 6.7 we discussed constructions with vector bundles corresponding to a

smooth covariant functor F on the category of _nite dimensional vector spaces

and these constructions can be applied to the values of any natural vector bundle

to get new natural vector bundles, cf. 6.14. There we applied F to the cocycle of

transition functions. Let us look what happens on the level of the corresponding

m-modules. If we apply F to a Gr

m-module V with action ` : Gr

! GL(V ),

we get a vector space FV with action ~` : Gr

! GL(FV ), ~`(g) = F(`(g)), i.e.

a new Gr

m-module FV . Let us assume that G and FG are the natural vector

bundles corresponding to V and FV . The canonical vector bundle structure on

(FG)M = PrM _Gr

FV coincides with that on F(GM) by 10.7.(4). Similarly,

we can handle contravariant functors and bifunctors on the category of vector

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

142 Chapter IV. Jets and natural bundles

spaces, cf. 6.7. In particular, the values of natural vector bundles corresponding

to direct sums of the modules are just _bered products over the base manifolds

of the individual bundles. Let us also note that C1 􀀀

_iFiM

= _i (C1(FiM)).

3. There are also well known examples of higher order natural vector bundles.

First of all, we recall the functor of r-th order k-dimensional covelocities

Jr( ;Rk)0 = Tr_

k introduced in 12.8. If r; k = 1, we get the dual bundles to

the tangent bundles J1

0 (R;M) = TM. So the vector bundle structure on the

cotangent bundle is natural and the tangent spaces are the duals, from our point

of view. But we can apply the construction of a dual module to any Gr

m-module

and this leads to dual natural vector bundles according to 14.6. In this way we

get the r-th order tangent bundles T(r) := (Tr_)_ or, more general the bundle

functors Tr_

k = (Tr_

k )_, see 12.14.

14.10. A_ne bundle functors. A bundle functor F : Mfm ! FM is called

an a_ne bundle functor on Mfm, or natural a_ne bundle, if each value FM !

M is an a_ne bundle and the values on morphisms are a_ne maps. Hence the

standard _ber V of an r-th order natural a_ne bundle is an a_ne space and the

induced action ` is a representation of Gr

m in the group of a_ne transformations

of V . So for each g 2 Gr

m there is a unique linear map ~`

(g) : ~V ! ~V satisfying

`(g)(y) = `(g)(x) +~`

(g)(y 􀀀 x) for all x, y 2 V . It follows that ~`

is a linear representation

of Gr

m on the vector space ~V and there is the corresponding natural

vector bundle ~F. By the construction, for every m-dimensional manifold M the

value ~FM is just the modelling vector bundle to FM and for every morphism

f : M ! N, ~Ff is the modelling linear map to Ff. Hence two arbitrary sections

of FM `di_er' by a section of ~FM. The best known example of a second order

natural a_ne bundle is the bundle of elements of linear connections QP1 which

we shall study in section 17. The modelling natural vector bundle

􀀀􀀀!

QP1 is the

tensor bundle T T_ T_ corresponding to GL(m)-module Rm Rm_ Rm_.

Next we shall describe all natural transformations between natural bundles

in the terms of Gr

m-equivariant maps.

14.11. Lemma. For every natural transformation _: F ! G between two

natural bundles on Mfm all mappings _M : FM ! GM cover the identities

idM.

Proof. Let _: F ! G be a natural transformation and let us write p: FM ! M

and q : GM ! M for the canonical projections onto an m-dimensional manifold

M. If y 2 FM is a point with z := q(_M(y)) 6= p(y), then there is a local

di_eomorphism f : M ! M such that germp(y)f = germp(y)idM and f(z) = _z,

_z 6= z. But now the localization condition implies q__M_Ff(y) 6= q_Gf__M(y),

for q _ Gf = f _ q. This is a contradiction. _

14.12. Theorem. There is a bijective correspondence between the set of all

natural transformations between two r-th order natural bundles on Mfm and

the set of smooth Gr

m-equivariant maps between their standard _bers.

Proof. Let F and G be natural bundles with standard _bers S and Q and let

_: F ! G be a natural transformation. According to 14.11, we have the restric-

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

14. Natural bundles and operators 143

tion _RmjS : S ! Q and we claim that this is Gr

m-equivariant with respect to the

induced actions. Indeed, for any jr

0f 2 Gr

m we get (_RmjS)_Ff = Gf _(_RmjS),

but Ff : S ! S and Gf : Q ! Q are just the induced actions of jr

0f on S and

Q. Now we have to show that the whole transformation _ is determined by the

map _RmjS. First, using translations tx : Rm ! Rm we see this for the map

_Rm. Then, if we choose any atlas (U_; u_) on a manifold M, the maps Fu_

form a _ber bundle atlas on FM and we know _M _ Fu_ = Gu_ _ _Rm. Hence

the locality of bundle functors implies _Mj(pF

M)􀀀1(U_) = Gu_ _ _Rm _ (Fu_)􀀀1.

On the other hand, let _0 : S ! Q be an arbitrary Gr

m-equivariant smooth

map. According to 14.6, the functors F or G are canonically naturally equivalent

to the functors L _ Pr or K _ Pr, where L or K are the functors corresponding

to the induced Gr

m-actions ` or k on the standard _bers S or Q, respectively.

So it su_ces to de_ne a natural transformation _: L _ Pr ! K _ Pr. We

set _M = fidPrM; _0g. It is an easy exercise to verify that _ is a natural

transformation. Moreover, we have _RmjS = _0. _

In general, an operator is a rule transforming sections of a _bered manifold

Y ! M into sections of another _bered manifold _ Y ! _M . We shall deal with

the case M = _M in this section. Let us recall that C1Y means the set of all

smooth sections of a _bered manifold Y ! M.

14.13. De_nition. Let Y

p 􀀀!

M, _ Y

􀀀! M be _bered manifolds. A local

operator A: C1Y ! C1 _ Y is a map such that for every section s: M ! Y

and every point x 2 M the value As(x) depends on the germ of s at x only.

If, moreover, for certain k 2 N or k = 1 the condition jk

xs = jk

xq implies

As(x) = Aq(x), then A is said to be of order k. An operator A: C1Y ! C1 _ Y

is called a regular operator if every smoothly parameterized family of sections of

Y is transformed into a smoothly parameterized family of sections of _ Y .

14.14. Associated maps to an k-th order operator. Consider an operator

A: C1Y ! C1 _ Y of order k. We de_ne a map A: JkY ! _ Y by A(jk

xs) = As(x)

which is called the associated map to the k-th order operator A.

Proposition. The associated map to any _nite order operator A is smooth if

and only if A is regular.

Proof. Let A: C1Y ! C1 _ Y be an operator of order k. If we choose local _bered

coordinates on Y , we also get the induced _bered coordinates on JkY . But

in these local coordinates, the jets of sections are identi_ed with (polynomial)

sections. Thus, a chart on JkY can be viewed as a smoothly parameterized

family of sections in C1Y and so the smoothness of A follows from the regularity.

The converse implication is obvious. _

14.15. Natural operators. A natural operator A: F G between two

natural bundles F and G is a system of regular operators AM : C1(FM) !

C1(GM), M 2 ObMfm, satisfying

(i) for every section s 2 C1(FM ! M) and every di_eomorphism f : M ! N

it holds

AN(Ff _ s _ f􀀀1) = Gf _ AMs _ f􀀀1

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

144 Chapter IV. Jets and natural bundles

(ii) AU(sjU) = (AMs)jU for every s 2 C1(FM) and every open submanifold

U _ M.

In particular, condition (ii) implies that natural operators are formed by local

operators.

A natural operator A: F G is said to be of order k, 0 _ k _ 1, if all

operators AM are of order k. The system of associated maps AM : JkFM ! GM

to the k-th order operators AM is called the system of associated maps to the

natural operator A. The associated maps to _nite order natural operators are

smooth.

We can look at condition (i) even from the viewpoint of the local coordinates

on a manifold M. Given a local chart u: U _ M ! V _ Rm, the di_eomorphisms

f : V ! W _ Rm correspond to the changes of coordinates on U.

Combining this observation with localization property (ii), we conclude that the

natural operators coincide, in fact, with those operators, the local descriptions

of which do not depend on the changes of coordinates.

14.16. Proposition. For every r-th order bundle functor F on Mfm its

composition with the functor of k-th jet prolongations of _bered manifolds

Jk : FM! FM is a natural bundle of order r + k.

Proof. Let f : M ! N be a local di_eomorphism. Then, by de_nition of the

associated maps FM;N , we have

Ff = FM;N _

􀀀

(jrf _ pM) _ idFM

: FM ! FN.

Hence Jk(Ff) depends on (k + r)-jets of f in the underlying points in M only.

It is an easy exercise to verify the axioms of natural bundles. _

14.17. Proposition. There is a bijective correspondence between the set of

k-th order natural operators A: F G between two natural bundles on Mfm

and the set of all natural transformations _: Jk _ F ! G.

Proof. Let AM be the associated maps of an k-th order natural operator A: F

G. We claim that these maps form a natural transformation _: JkF ! G. They

are smooth by virtue of 14.14 and we have to verify Gf _ AM = AN _ JkFf for

an arbitrary local di_eomorphism f : M ! N. We have

AN((JkFf)(jk

xs)) = AN(jk(Ff _ s _ f􀀀1)(f(x)))

= AN(Ff _ s _ f􀀀1)(f(x)) = Gf _ AMs(x)

= Gf _ AM(jk

xs).

On the other hand, consider a natural transformation _: JkF ! G. We

de_ne operators AM : FM GM by AMs(x) = _M(jk

xs) for all sections

s 2 C1(FM). Since the maps _M are smooth _bered morphisms and according

to lemma 14.11 they all cover the identities idM, the maps AMs are smooth sections

of GM. The straightforward veri_cation of the axioms of natural operators

is left to the reader. _

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

14. Natural bundles and operators 145

14.18. Let F : Mfm ! FM be an r-th order natural bundle with standard

_ber S and let ` : Gr

_S ! S be the induced action. The identi_cation Rm_S _=

FRm, (x; s) 7! F(tx)(s), induces the identi_cation C1(Rm; S) _= C1(FRm),

(~s: Rm ! S) 7! (s(x) = Ftx(~s(x))) 2 C1(FRm). Hence the standard _ber of

the natural bundle JkF equals Tkm

S. Under these identi_cations, the action of

F on an arbitrary local di_eomorphism is of the form

Fg(x; s) = (g(x); F(t􀀀g(x)

_ g _ tx)(s))

and the induced action `k : Gr+k

_Tkm

S ! Tkm

S determined by the functor JkF

is expressed by the following formula

`k(jr+k

0 g; jk

0 ~s) = `k(jr+k

0 g; jk

(1) 0 (Ftx _ s~(x)))

= jk

0 (Fg _ Ftg􀀀1(x)

_ ~s(g􀀀1(x))) 2 Jk

0 FRm

= jk

0 (Ft􀀀x _ Fg _ Ftg􀀀1(x)

_ ~s(g􀀀1(x))) 2 Tkm

= jk

􀀀

0 (t􀀀x _ g _ tg􀀀1(x)); ~s(g􀀀1(x))

In particular, if a = jr+k

0 g 2 G1

_ Gr+k

m , i.e. g is linear, then

(2) `k(a; jk

0 ~s) = jk

0 (`(jr

0g; ~s _ g􀀀1(x))) = jk

0 (`a _ ~s _ g􀀀1):

As a consequence of the last two propositions we get the basic result for

_nding natural operators of prescribed types. Consider natural bundles F or F0

onMfm of _nite orders r or r0, with standard _bers S or S0 and induced actions

` or `0 of Gr

m or Gr0

m, respectively. If q = maxfr + k; r0g with some _xed k 2 N

then the actions `k and `0 trivially extend to actions of Gq

m on both Tkm

S and

S0 and we have

Theorem. There is a canonical bijective correspondence between the set of

all k-th order natural operators A: F F0 and the set of all smooth Gq

mequivariant

maps between the left Gq

m-spaces Tkm

S and S0.

14.19. Examples.

1. By the construction in 3.4, the Lie bracket of vector _elds is a bilinear

natural operator [ ; ] : T _ T T of order one, see also corollary 3.11. The

corresponding bilinear G2

m-equivariant map is

b = (b1; : : : ; bm) : T1m

Rm _ T1m

Rm ! Rm

bj(Xi;Xk

` ; Y m; Y n

p ) = XiY j

􀀀 Y iXj

i .

Later on we shall be able to prove that every bilinear equivariant map b0 : Trm

Rm_

Trm

Rm ! Rm is a constant multiple of b composed with the jet projections and,

moreover, every natural bilinear operator is of a _nite order, so that all bilinear

natural operators on vector _elds are the constant multiples of the Lie bracket.

On the other hand, if we drop the bilinearity, then we can iterate the Lie bracket

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

146 Chapter IV. Jets and natural bundles

to get operators of higher orders. But nevertheless, one can prove that there are

no other G2

m-equivariant maps b0 : T1m

Rm _ T1m

Rm ! Rm beside the constant

multiples of b and the projections T1m

Rm _ Rm ! Rm. This implies, that the

constant multiples of the Lie bracket are essentially the only natural operators

T _ T T of order 1.

2. The exterior derivative introduced in 7.8 is a _rst order natural operator

d: _kT_ _k+1T_. Formula 7.8.(1) expresses the corresponding G2

equivariant map

T1m

(_kRm_) ! _k+1Rm_

('i1:::ik ; 'i1:::ik;ik+1) 7!

(􀀀1)j+1'i1:::bij :::ik+1;ij

where the hat denotes that the index is omitted. We shall derive in 25.4 that

for k > 0 this is the only G2

m-equivariant map up to constant multiples. Consequently,

the constant multiples of the exterior derivative are the only natural

operators of the type in question.

14.20. In concrete problems we often meet a situation where the representations

of Gr

m are linear, or at least their restrictions to G1

_ Gr

m turn the

standard _bers into GL(m)-modules. Then the linear equivariant maps between

the standard _bers are GL(m)-module homomorphisms and so the structure of

the modules in question is often a very useful information for _nding all equivariant

maps. Given a G1

m-module V and linear coordinates yp on V , there are

the induced coordinates yp_ = @j_jyp

@x_ on Tkm

V , where xi are the canonical coordinates

on Rm and 0 _ j_j _ k. Then the linear subspace in Tkm

V de_ned by

yp_ = 0, j_j 6= i, coincides with V SiRm_. Clearly, these identi_cations do not

depend on our choice of the linear coordinates yp. Formula 14.18.(2) shows that

Tkm

V = V _ _ _ _ _ V SkRm_ is a decomposition of Tkm

V into G1

m-submodules

and the same formula implies the following result.

Proposition. Let V be a G1

m-invariant subspace in pRm qRm_ and let us

consider a representation ` : Gr

! Di_(V ) such that its restriction to G1

_ Gr

is the canonical tensorial action. Then the restriction of the induced action `k

of Gr+k

m on Tkm

V = V _ _ _ _ _ V SkRm_ to G1

_ Gr+k

m is also the canonical

tensorial action.

14.21. Some geometric constructions are performed on the whole categoryMf

of smooth manifolds and smooth maps. Similarly to natural bundles, the bundle

functors on the category Mf present a special case of the more general concept

of bundle functors.

De_nition. A bundle functor on the categoryMf is a covariant functor F : Mf

! FM satisfying the following conditions

(i) B _F = IdMf , so that the _ber projections form a natural transformation

p: F ! IdMf .

(ii) If i : U ! M is an inclusion of an open submanifold, then FU = p􀀀1

M (U)

and Fi is the inclusion of p􀀀1

M (U) into FM.

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

14. Natural bundles and operators 147

(iii) If f : P _M ! N is a smooth map, then ~ Ff : P _ FM ! FN, de_ned

by ~ Ff(p; ) = Ffp, p 2 P, is smooth.

For every non-negative integer m the restriction Fm of a bundle functor F

on Mf to the subcategory Mfm _ Mf is a natural bundle. Let us call the

sequence S = fS0; S1; : : : ; Sm; : : : g of the standard _bers of the natural bundles

Fm the system of standard _bers of the bundle functor F. Proposition 14.2

implies that for every m there is the canonical isomorphism Rm _ Sm

FRm,

(x; s) 7! Ftx(s), and given an m-dimensional manifold M, pM : FM ! M is a

locally trivial bundle with standard _ber Sm.

Analogously to 14.3 and 14.4, a bundle functor F on Mf is said to be of

order r if for every smooth map f : M ! N and point x 2 M the restriction

FfjFxM depends only on jrx

f. Then the maps FM;N : Jr(M;N)_MFM ! FN,

FM;N (jrx

f; y) = Ff(y) are called the associated maps to the r-th order functor

F. Since in the proof of proposition 14.3 we never used the invertibility of

the jets in question, the same proof applies to the present situation and so the

associated maps to any _nite order bundle functor onMf are smooth. For every

m-dimensional manifold M, there is the canonical structure of the associated

bundle FM _= PrM[Sm], cf. 14.5.

Let S = fS0; S1; : : : g be the system of standard _bers of an r-th order bundle

functor F on Mf. The restrictions `m;n of the associated maps FRm;Rn to

0 (Rm;Rn)0 _ Sm have the following property. For every A 2 Jr

0 (Rm;Rn)0,

B 2 Jr

0 (Rn;Rp)0 and s 2 Sm

(1) `m;p(B _ A; s) = `n;p(B; `m;n(A; s)).

Hence instead of the action of one group Gr

m on the standard _ber in the case

of bundle functors on Mfm, we get an action of the category Lr on S, see

below and 12.6 for the de_nitions. We recall that the objects of Lr are the

non-negative integers and the set of morphisms between m and n is the set

m;n = Jr

0 (Rm;Rn)0.

Let S = fS0; S1; : : : g be a system of manifolds. A left action ` of the category

Lr on S is de_ned as a system of maps `m;n : Lr

m;n

_ Sm ! Sn satisfying (1).

The action is called smooth if all maps `m;n are smooth. The canonical action of

Lr on the system of standard _bers of a bundle functor F is called the induced

action. Every induced action of a _nite order bundle functor is smooth.

14.22. Consider a system of smooth manifolds S = fS0; S1; : : : g and a smooth

action ` of the category Lr on S. We shall construct a bundle functor L determined

by this action. The restrictions `m of the maps `m;m to invertible jets

form smooth left actions of the jet groups Gr

m on manifolds Sm. Hence for every

m-dimensional manifold M we can de_ne LM = PrM[Sm; `m]. Let us recall the

notation fu; sg for the elements in PrM_Gr

m Sm, i.e. fu; sg = fu_A; `m(A􀀀1; s)g

for all u 2 PrM, A 2 Gr

m, s 2 Sm. For every smooth map f : M ! N we de_ne

Lf : FM ! FN by

Lf(fu; sg) = fv; `m;n(v􀀀1 _ A _ u; s)g

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

148 Chapter IV. Jets and natural bundles

where m = dimM, n = dimN, u 2 Pr

xM, A = jrx

f, and v 2 Pr

f(x)N is an arbitrary

element. We claim that this is a correct de_nition. Indeed, chosen another

representative for fu; sg and another frame v0 2 Pr

f(x), say fu _ B; `m(B􀀀1; s)g,

and v0 = v _ C, formula 14.21.(1) implies

Lf(fu _ B;`m(B􀀀1; s)g =

= fv _ C; `m;n(C􀀀1 _ v􀀀1 _ A _ u _ B; `m(B􀀀1; s))g =

= fv _ C; `n(C􀀀1; `m;n(v􀀀1 _ A _ u; s))g =

= fv; `m;n(v􀀀1 _ A _ u; s)g.

One veri_es easily all the axioms of bundle functors, this is left to the reader.

On the other hand, consider an r-th order bundle functor F on Mf and

its induced action `. Let L be the corresponding bundle functor, we have

just constructed. Analogously to 14.6, there is a canonical natural equivalence

_: L ! F. In fact, we have the restrictions of _ to manifolds of any _xed dimension

which consists of the maps qM determining the canonical structures of

associated bundles on the values FM, see 14.6. It remains only to show that

Ff _ _M = _N _ Ff for all smooth maps f : M ! N. But given jr

0g 2 Pr

xM,

0h 2 Pr

f(x)N and s 2 Sm, we have

Ff _ _M(fjr

0g; sg) = Ff _ Fg(s) = Fh _ F(h􀀀1 _ f _ g)(s)

= _N(jr

0h; `m;n(jr

0 (h􀀀1 _ f _ g); s)) = _N _ Lf(fjr

0g; sg):

Since in geometry we usually identify naturally equivalent functors, we have

proved

Theorem. There is a bijective correspondence between the set of r-th order

bundle functors on Mf and the set of smooth left actions of the category Lr on

systems S = fS0; S1; : : : g of smooth manifolds.

14.23. Natural transformations. Consider a smooth action ` or `0 of the

category Lr on a system S = fS0; S1; : : : g or S0 = fS0

0; S0

1; : : : g of smooth

manifolds, respectively. A sequence ' of smooth maps 'i : Si ! S0

i is called a

smooth Lr-equivariant map between ` and `0 if for every s 2 Sm, A 2 Lr

m;n it

holds

'n(`m;n(A; s)) = `0

m;n(A; 'm(s)).

Theorem. There is a bijective correspondence between the set of natural transformations

of two r-th order bundle functors on Mf and the set of smooth Lrequivariant

maps between the induced actions of Lr on the systems of standard

_bers.

Proof. Let _: F ! G be a natural transformation, ` or k be the induced action

on the system of standard _bers S = fS0; S1; : : : g or Q = fQ0;Q1; : : : g, respectively.

As we proved in 14.11, all maps _M : FM ! GM are over identities. Let

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

15. Prolongations of principal _ber bundles 149

us de_ne 'n : Sn ! Qn as the restriction of _Rn to Sn. If jr

0f 2 Lr

m;n, s 2 Sm,

then

'n(`m;n(jr

0f; s)) = _Rn _ Ff(s) = Gf _ _Rm(s) = km;n(jr

0f; 'm(s));

so that the maps 'm form a smooth Lr-equivariant map between ` and k. Moreover,

the arguments used in 14.11 imply that _ is completely determined by the

maps 'm.

Conversely, by virtue of 14.22, we may assume that the functors F and G

coincide with the functors L and K constructed from the induced actions. Consider

a smooth Lr-equivariant map ' between ` and k. Then we can de_ne for

all m-dimensional manifolds M maps _M : FM ! GM by

_M := fidPrM; 'mg.

The reader should verify easily that the maps _M form a natural transformation.

14.24. Remark. Let F be an r-th order bundle functor on Mf. Its induced

action can be interpreted as a smooth functor Finf : Lr ! Mf, where

the smoothness means that all the maps Lr

m;n

_ Finf (m) ! Finf (n) de_ned by

(A; x) 7! FinfA(x) are smooth. Then the concept of smooth Lr-equivariant maps

between the actions coincides with that of a natural transformation. Hence we

can reformulate theorems 14.22 and 14.23 as follows. The full subcategory of

r-th order bundle functors onMf in the category of functors and natural transformations

is naturally equivalent to the full subcategory of smooth functors

Lr !Mf. Let us also remark, that the Lr-objects can be viewed as numerical

spaces Rm, 0 _ m < 1, with distinguished origins. Then every Mf-object is

locally isomorphic to exactly one Lr-object and, up to local di_eomorphisms,

Lr contains all r-jets of smooth maps. Therefore, we can call Lr the r-th order

skeleton ofMf. We shall work out this point of view in our treatment of general

bundle functors in the next chapter. Let us mention that the bundle functors

on Mfm also admit such a description. Indeed, the r-th order skeleton then

consists of the group Gr

m only.

15. Prolongations of principal _ber bundles

15.1. In the present section, we shall mostly deal with the category PBm(G)

consisting of principal _ber bundles with m-dimensional bases and a _xed structure

group G, with PB(G)-morphisms which cover local di_eomorphisms between

the base manifolds. So a PBm(G)-morphism ': (P; p;M) ! (P0; p0;M0)

is a smooth _bered map over a local di_eomorphism '0 : M ! M0 satisfying

' _ _g = _0

_ ' for all g 2 G, where _ and _0 are the principal actions on P and

P0. In particular, every automorphism ': Rm_G ! Rm_G is fully determined

by its restriction _': Rm ! G, _'(x) = pr2 _ '(x; e), where e 2 G is the unit, and

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

150 Chapter IV. Jets and natural bundles

by the underlying map '0 : Rm ! Rm. We shall identify the morphism ' with

the couple ('0; _'), i.e. we have

(1) '(x; a) = ('0(x); _'(x):a).

Analogously, every morphism : Rm _ G ! P, i.e. every local trivialization of

P, is determined by 0 and ~ := j(Rm _ feg) : Rm ! P covering 0. Further

we de_ne 1 = ~ _ 􀀀1

0 , so that 1 is a local section of the principal bundle P,

and we identify the morphism with the couple ( 0; 1). We have

(2) (x; a) = ( 1 _ 0(x)):a :

Of course, for an automorphism ' on Rm _ G we have _' = pr2 _ ~'.

15.2. Principal prolongations of Lie groups. We shall apply the construction

of r-jets to such a situation. Since all PBm(G)-objects are locally isomorphic

to the trivial principal bundle Rm _G and all PBm(G)-morphisms are local isomorphisms,

we _rst have to consider the group Wrm

G of r-jets at (0; e) of all

automorphisms ': Rm _G ! Rm _G with '0(0) = 0, where the multiplication

_ is de_ned by the composition of jets,

_(jr'(0; e); jr (0; e)) = jr( _ ')(0; e):

This is a correct de_nition according to 15.1.(1) and the inverse elements are

the jets of inverse maps (which always exist locally). The identi_cation 15.1 of

automorphisms on Rm _ G with couples ('0; _') determines the identi_cation

(1) Wrm

G _= Gr

_ Trm

G; jr'(0; e) 7! (jr

0'0; jr

0 _'):

Let us describe the multiplication _ in this identi_cation. For every ', 2

PBm(G)(Rm _ G;Rm _ G) we have

_ '(x; a) = ('0(x); _'(x):a) = ( 0 _ '0(x); _ ('0(x)): _'(x):a)

so that given any (A;B), (A0;B0) 2 Gr

_ Trm

G we get

(2) _

􀀀

(A;B); (A0;B0)

􀀀

A _ A0; (B _ A0):B0_

Here the dot means the multiplication in the Lie group Trm

G, cf. 12.13. Hence

there is the structure of a semi direct product of Lie groups on Wrm

G. The Lie

group Wrm

G = Gr

moTrm

G is called the (m; r)-principal prolongation of Lie group

15.3. Principal prolongations of principal bundles. For every principal

_ber bundle (P; p;M;G) 2 ObPBm(G) we de_ne

WrP := fjr (0; e); 2 PBm(G)(Rm _ G; P)g.

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15. Prolongations of principal _ber bundles 151

In particular, Wr(Rm _ G) is identi_ed with Rm _Wrm

G by the rule

Rm _Wrm

G 3 (x; jr'(0; e)) 7! jr(_x _ ')(0; e) 2 Wr(Rm _ G)

where _x = tx_idG, and so there is a well de_ned structure of a smooth manifold

on Wr(Rm _ G). Furthermore, if we de_ne the action of Wr on PBm(G)-

morphisms by the composition of jets, i.e.

Wr_(jr (0; e)) := jr(_ _ )(0; e),

Wr becomes a functor. Now, taking any principle atlas on a principal bundle

P, the application of the functor Wr to the local trivializations yields a _bered

atlas on Wr. Finally, there is the right action of Wrm

G on WrP de_ned for

every jr'(0; e) 2 Wrm

G and jr (0; e) 2 WrP by (jr (0; e))(jr'(0; e)) = jr( _

')(0; e). Since all the jets in question are invertible, this action is free and

transitive on the individual _bers and therefore we have got principal bundle

(WrP; p _ _;M;Wrm

G) called the r-th principal prolongation of the principal

bundle (P; p;M;G). By the de_nition, for a morphism ' the mapping Wr'

always commutes with the right principal action ofWrm

G and we have de_ned the

functor Wr : PBm(G) ! PBm(Wrm

G) of r-th principal prolongation of principal

bundles.

15.4. Every PBm(G)-morphism : Rm _ G ! P is identi_ed with a couple

( 0; 1), see 15.1.(2). This yields the identi_cation

(1) WrP = PrM _M JrP

and also the smooth structures on both sides coincide. Let us express the corresponding

action of Gr

moTrm

G on PrM_MJrP. If (u; v) = (jr

0 0; jr 1( 0(0))) 2

PrM _M JrP and (A;B) = (jr

0'0; jr

0 _') 2 Gr

m o Trm

G, then 15.2.(2) implies

_ '(x; a) = ('0(x); _'(x):a) = 1( 0 _ '0(x)): _'(x):a

= (_ _ ( 1; _' _ '􀀀1

_ 􀀀1

0 ) _ ( 0 _ '0)(x)):a

where _ is the principal right action on P. Hence we have

(2) (u; v)(A;B) = (u _ A; v:(B _ A􀀀1 _ u􀀀1))

where '.' is the multiplication

m: JrP _M Jr(M;G) ! JrP; (jrx

_; jrx

s) 7! jrx

(_ _ (_; s)):

The decomposition (1) is natural in the following sense. For every PBm(G)-

morphism : (P; p;M;G) ! (P0; p0;M;0 G), the PBm(Wrm

G)-morphism Wr

has the form (Pr 0; Jr ). Indeed, given ': Rm _ G ! P, we have ( _ ')0 =

0'0, ( _ ')1 = _ ~' _ ( 0 _ '0)􀀀1 = _ '1 _ 􀀀1

0 . Therefore, in the category

of functors and natural transformations, the following diagram is a pullback

Wr w

Pr _ B wB

Here B: PBm(G) !Mfm is the base functor, the upper and left-hand natural

transformations are given by the above decomposition and the right-hand and

bottom arrows are the usual projections.

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

152 Chapter IV. Jets and natural bundles

15.5. For every associated bundle E = P[S; `] to a principal bundle (P; p;M;G)

there is a canonical left action `r : Wrm

G _ Trm

S ! Trm

S of Wrm

G = Gr

m o Trm

on Trm

S. We simply compose the prolonged action Trm

` of Trm

G on Trm

S, see

12.13, with the canonical left action of Gr

m on both Trm

G and Trm

S, i.e. we set

(1) `r(jr'(0; e); jr

0s) = jr

0 (` _ ( _' _ '􀀀1

0 ; s _ '􀀀1

0 ))

for every jr'(0; e) = (jr

0'0; jr

0 _') 2 Gr

m o Trm

Proposition. For every associated bundle E = P[S; `], there is a canonical

structure of the associated bundle WrP[Trm

S; `r] on the r-th jet prolongation

JrE.

Proof. Similarly to 14.6, every action ` : G _ S ! S determines the functor L

on PBm(G), P 7! P[S; `] and ' 7! f'; idSg, with values in the category of

the associated bundles with standard _ber S and structure group G. We shall

essentially use the identi_cation

Trm

S _= Jr

0 (Rm _ S) _= Jr

0 ((Rm _ G)[S; `])

0s 7! jr

0(idRm; s) 7! jr

(2) fe^; sg

where ^e: Rm ! Rm _ G, ^e(x) = (x; e). Then the action `r becomes the form

`r(jr'(0; e); jr

f^e; sg) = jr

f^e; ` _ ( _' _ '􀀀1

0 ; s _ '􀀀1

0 (3) )g

= Jr(L')(jr

f^e; sg):

Now we can de_ne a map q : WrP _ Trm

S ! JrE determining the required

structure on JrE. Given u = jr (0; e) 2 WrP and B = jr

0s 2 Trm

S, we set

q(u;B) = Jr(L )(jr

f^e; sg).

Since the map is a local trivialization of the principal bundle P, the restriction

qu = q(u; ) : Trm

S ! Jr

0(0)E is a di_eomorphism. Moreover, for every A =

jr'(0; e) 2 Wrm

G, formula (3) implies

q(u:A; `r(A􀀀1;B)) = Jr(L( _ '))

􀀀

Jr(L'􀀀1)(jr

f^e; sg)

= q(u;B)

and the proposition is proved. _

For later purposes, let us express the corresponding map _ : WrP _M JrE !

Trm

S. It holds

_ (u; jrx

s) = jr

0 (_E _ ( _ ^e; s _ 0))

where _E : P _M E ! S is the canonical map of E and u = jr (0; e) 2 Wr

xP.

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

15. Prolongations of principal _ber bundles 153

15.6. First order principal prolongation. We shall point out some special

properties of the groups W1m

G and the bundles W1P. Let us start with the group

Trm

G. Every map s: Rm ! G can be identi_ed with the couple (s(0); _s(0)􀀀1 _s),

and for a second map s0 : Rm ! G we have (we recall that _a and _a are the left

and right translations by a in G, _ is the multiplication on G)

_ _ (s0; s)(x) = s0(0)(1) s0(0)􀀀1s0(x)s(0)s(0)􀀀1s(x)

􀀀

s0(0)s(0)

_􀀀

conjs(0)􀀀1 (s0(0)􀀀1s0(x))

_􀀀

s(0)􀀀1s(x)

It follows that Trm

G is the semi direct product G o Jr

0 (Rm;G)e. This can be

described easily in more details in the case r = 1. Namely, the _rst order jets

are identi_ed with linear maps between the tangent spaces, so that (1) implies

T1m

G = G o (g Rm_) with the multiplication

(2) (a0;Z0):(a;Z) = (a0a; Ad(a􀀀1)(Z0) + Z),

where a, a0 2 G, Z, Z0 2 Hom(Rm; g). Taking into account the decomposition

15.2.(1) and formula 15.2.(2), we get

W1m

G = (GL(m) _ G) o (g Rm_)

with multiplication

(3) (A0; a0;Z0):(A; a;Z) = (A0 _ A; a0a; Ad(a􀀀1)(Z0) _ A + Z):

Now, let us view _bers P1

xM as subsets in Hom(Rm; TxM) and elements

in J1

xP as homomorphisms in Hom(TxM; TyP), y 2 Px. Given any (u; v) 2

P1M _M J1P = W1P and (A; a;Z) 2 (G1

_ G) o (g Rm_), 15.4.(2) implies

(4) (u; v)(A; a;Z) = (u _ A; T_(v; T_a _ Z _ A􀀀1 _ u􀀀1))

where _ is the principal right action on P.

15.7. Principal prolongations of frame bundles. Consider the r-th principal

prolongation Wr(PsM) of the s-th order frame bundle PsM of a manifold

M. Every local di_eomorphism ': Rm ! M induces a principal _ber bundle

morphism Ps': PsRm ! PsM and we can construct jr

(0;es)(Ps') 2 Wr(PsM),

where es denotes the unit of Gs

m. One sees directly that this element depends

on the (r + s)-jet jr+s

0 ' only. Hence the map jr+s

0 ' 7! jr

(0;es)(Ps')

de_nes an injection iM : Pr+sM ! Wr(PsM). Since the group multiplication

in both Gr+s

m and Wrm

m is de_ned by the composition of jets, the restriction

i0 : Gr+s

! Wrm

m of iRm to the _bers over 0 2 Rm is a group homomorphism.

Thus, the (r+s)-order frames on a manifold M form a natural reduction

iM : Pr+sM ! Wr(PsM) of the r-th principal prolongation of the s-th order

frame bundle of M to the subgroup i0(Gr+s

m ) _ Wrm

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

154 Chapter IV. Jets and natural bundles

15.8. Coordinate expression of i0 : Gr+s

! Wrm

m. The canonical coordinates

xi on Rm induce coordinates ai

_, 0 < j_j _ r + s, on Gr+s

m , ai

_(jr+s

0 f) =

@j_jfi

@x_ (0), and the following coordinates on Wrm

m: Any element jr'(0; e) 2

Wrm

m is given by jr

0'0 2 Gr

m and jr

0 _' 2 Trm

m, see 15.2. Let us denote the

coordinate expression of _' by bi

(x), 0 < jj _ s, so that we have the coordinates

;_ , 0 < jj _ s, 0 _ j_j _ r on Trm

m, bi

;_(jr

0 _') = 1

@j_jbi

@x_ (0), and the

coordinates (ai

_; bi

;_), 0 < j_j _ r, 0 < jj _ s, 0 _ j_j _ r, on Wrm

m. By

de_nition, we have

(1) i0(ai

_) = (ai

_; ai

+_):

In the _rst order case, i.e. for r = 1, we have to take into account a further

structure, namely T1m

m = Gs

m o (gs

Rm_), cf. 15.6. So given i0(js+1

0 f) =

(j1

0f; j1

0q), where q : Rm ! Gs

m, we are looking for b = q(0) 2 Gs

m and Z =

T_b􀀀1 _ T0q 2 gs

Rm_. Let us perform this explicitly for s = 2.

In G2

m we have (ai

j ; ai

jk)􀀀1 = (~ai

j ; ~ai

jk) with ai

j~aj

k = _ik

and ~ai

jk = 􀀀~ail

ps~as

k~ap

j .

Let X = (ai

k; ai

jk;Ai

j ;Ai

jk) 2 TG2

m and b = (bi

k; bi

jk) 2 G2

m. It is easy to compute

T_b(X) = (bi

kakj

; bil

jk + bi

psap

jas

k; bi

pAp

j ; bi

pAp

jk + bi

psAp

jas

k + bi

psap

jAs

k):

Taking into account all our identi_cations we get a formula for i0 : G3

! W1m

i0(ai

j ; ai

jk; ai

jkl) = (ai

j ; ai

jk; ~ai

pap

jl; ~ai

pap

jkl + ~ai

psap

jlas

k + ~ai

psap

jas

kl):

If we perform the above consideration up to the _rst order terms only, we get

i0 : G2

! W1m

m, i0(ai

j ; ai

jk) = (ai

j ; ai

j ; ~ai

pap

jl).

16. Canonical di_erential forms

16.1. Consider a vector bundle E = P[V; `] associated to a principal bundle

(P; p;M;G) and the space of all E-valued di_erential forms (M;E). By theorem

11.14, there is the canonical isomorphism q] between (M;E) and the space

of horizontal G-equivariant V -valued di_erential forms on P. According to 10.12,

the image _ = q](') 2 k

hor(P; V )G is called the frame form of ' 2 k(M;E).

We have

(1) _(X1; : : : ;Xk) = _ (u; ) _ '(TpX1; : : : ; TpXk)

where Xi 2 TuP and _ : P _M E ! V is the canonical map. Conversely, for

every X1; : : : ;Xk 2 TxM, we can choose arbitrary vectors _X1; : : : ; _Xk 2 TuP

with u 2 Px and Tp_Xi = Xi to get

(2) '(X1; : : : ;Xk) = q(u; ) _ _(_X1; : : : ; _Xk)

where q : P _V ! E is the other canonical map. The elements _ 2 hor(P; V )G

are sometimes called the tensorial forms of type `, while the di_erential forms

in (P; V )G are called pseudo tensorial forms of type `.

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

16. Canonical di_erential forms 155

16.2. The canonical form on P1M. We de_ne an Rm-valued one-form _ =

_M on P1M for every m-dimensional manifold M as follows. Given u = j1

0g 2

P1M and X = j1

0c 2 TuP1M we set

_M(X) = u􀀀1 _ Tp(X) = j1

0 (g􀀀1 _ p _ c) 2 T0Rm = Rm.

In words, the choice of u 2 P1M determines a local chart at x = p(u) up to the

_rst order and the form _M transforms X 2 TuP1M into the induced coordinates

of TpX. If we insert ' = idTM into 16.1.(1) we get immediately

Proposition. The canonical form _M 2 1(P1M;Rm) is a tensorial form which

is the frame form of the 1-form idTM 2 1(M; TM).

Consider further a principal connection 􀀀 on P1M. Then the covariant exterior

di_erential d􀀀_M is called the torsion form of 􀀀. By 11.15, d􀀀_M is

identi_ed with a section of TM _2T_M, which is called the torsion tensor of

􀀀. If d􀀀_M = 0, connection 􀀀 is said to be torsion-free.

16.3. The canonical form on W1P. For every principal bundle (P; p;M;G)

we can generalize the above construction to an (Rm _ g)-valued one-form on

W1P. Consider the target projection _ : W1P ! P, an element u = j1 (0; e) 2

W1P and a tangent vector X = j1

0c 2 Tu(W1P). We de_ne the form _ = _P by

_(X) = u􀀀1 _ T_(X) = j1

0 ( 􀀀1 _ _ _ c) 2 T(0;e)(Rm _ G) = Rm _ g:

Let us notice that if G = feg is the trivial structure group, then we get P = M,

W1P = P1M and _P = _M.

The principal action _ on P induces an action of G on the tangent space

TP. We claim that the space of orbits TP=G is the associated vector bundle

E = W1P[Rm_g; `] with the left action ` of W1m

G on T(0;e)(Rm_G) = Rm_g,

`(j1'(0; e); j1

0c) = j1

0 (__'(0)􀀀1

_ ' _ c).

Indeed, every PBm(G)-morphism commutes with the principal actions, so that `

is a left action which is obviously linear and the map q : W1P _T(0;e)(Rm_G) !

E transforming every couple j1 (0; e) 2 W1P and j1

0c 2 T(0;e)(Rm_G) into the

orbit in TP=G determined by j1

0 ( _ c) describes the associated bundle structure

on E.

Proposition. The canonical form _P on W1P is a pseudo tensorial one-form

of type `.

Proof. We have to prove _P 2 1(W1P;Rm _ g)W1

mG. Let _ and __ be the

principal actions on P and W1P, X = j1

0c 2 TuW1P, u = j1 (0; e), A =

j1'(0; e) 2 W1m

G, a = pr2 _ _(A). We have

_ _ __A = _a _ _

(__A)_X = j1

0 (__A _ c) 2 TuAW1P

_P _ (__A)_X = j1

0 ('􀀀1 _ 􀀀1 _ _ _ __A _ c) = j1

0 (_a _ '􀀀1 _ 􀀀1 _ _ _ c):

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

156 Chapter IV. Jets and natural bundles

Hence

`A􀀀1 _ _P (X) = `A􀀀1 (j1

0 ( 􀀀1 _ _ _ c) = _P _ (__A)_(X). _

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

W1P is p _ _.

16.4. Lemma. Let (P; p;M;G) be a principal bundle and q : W1P = P1M_M

J1P ! P1M be the projection onto the _rst factor. Then the following diagram

commutes

Rm _ g

pr1

u TW1P _P

Rm u TP1M _M

Proof. Consider X = j1

0c 2 TuW1P, u = j1 (0; e). Then Tq(X) = j1(q _ c) and

q(u) = j1

0 0. It holds

pr1 _ _P (X) = pr1(j1

0 ( 􀀀1 _ _ _ c)) = j1

0 ( 􀀀1

_ p _ _ _ c)

= j1

0 ( 􀀀1

_ _p _ q _ c) = _M _ Tq(X)

where _p: P1M ! M is the canonical projection. _

16.5. Canonical forms on frame bundles. Let us consider a frame bundle

PrM and the _rst order principal prolongation W1(Pr􀀀1M). We know

the canonical form _ 2 1(W1(Pr􀀀1M);Rm _ gr􀀀1

m )W1

mGr􀀀1

m and the reduction

iM : PrM ! W1(Pr􀀀1M) to the structure group Gr

m, see 15.7. So we can de_ne

the canonical form _r on PrM to be the pullback i_

M_ 2 1(PrM;Rm _ gr􀀀1

m ).

By virtue of 16.3 there is the linear action _` = ` _ _ where _ is the group homomorphism

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

of type _`. The form _r can also be described directly. Given X 2 TuPrM,

we set _u = _r

r􀀀1u, _X = T_r

r􀀀1(X) 2 T_uPr􀀀1M. Since every u = jr

0f 2 PrM

determines a linear map ~u = T(0;e)Pr􀀀1f : Rm _ gr􀀀1

! Tjr􀀀1

0 fPr􀀀1M we get

_r(X) = ~u􀀀1(_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 _ : P ! S

of a section s: M ! E, _(u) = _E(u; s(p(u))). As we proved in 15.5, JrE =

WrP[Trm

S; `r], m = dimM, and so for every _xed section s: M ! E the frame

form _r of its r-th prolongation jrs is a map _r : WrP ! Trm

S. If we choose

some local coordinates (U; '), ' = (yp), on S, then there are the induced local

coordinates yp_ on (_r

0)􀀀1(U) _ Trm

S, 0 _ j_j _ r, and for every section s: M !

E the compositions yp_

__r de_ne (on the corresponding preimages) the coordinate

functions ap

_ of jrs induced by the local chart (U; '). We deduced in 15.5 that

for every u = jr (0; e) = (jr

0 0; jr 1( 0(0))) 2 WrP

_r(u) = jr

0_E( 1 _ 0; s _ 0):

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

16. Canonical di_erential forms 157

In particular, for the _rst order case we get

ap(u) = yp _ _(u)

i (u) = dyp(j1

0_E( 1 _ 0; s _ 0) _ c)

where c : R ! Rm is the curve t 7! 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 _ on W1P decomposes

into _ = _iei + __e_. Let us further write Y_ for the fundamental vector _elds

on S determined by e_ and let !_ be the dual basis to that induced from e_

on V P. Hence if the coordinate formulas for Y_ are Y_ = _p

_(y) @

@yp , then for

z 2 Ex, u 2 Px, X 2 VuP, y = _E(u; z) we get

_E( ; z)_X = 􀀀Y_(y)!_(X) = 􀀀_p

_(y)!_(X) @

@yp :

The next proposition describes the coordinate functions of j1s on W1P by

means of the canonical form _ 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

i , ap be the coordinate functions of j1s. Then ap = _ap _ _,

where _ : W1P ! P is the target projection, and

dap + _p

_ (aq)__ = ap

i _i:

Proof. The equality ap = _ap _ _ follows directly from the de_nition. We shall

evaluate dap(X) with arbitrary X 2 TuW1P, where u 2 W1P, u = j1 (0; e) =

(j1

0 0; j1 1( 0(0))). The frame u determines the linear isomorphism

~u = T(0;e) : Rm _ g ! T_uP;

_u = _(u). We shall denote _i(X) = _i, __(X) = __, so that _(X) = ~u􀀀1(__X) =

_iei+__e_. Let us write _X = __X = _X1+ _X2 with _X1 = ~u(_iei), _X2 = ~u(__e_)

and let c be the curve t 7! t_iei on Rm. We have

d_ap(_X1) = dyp(j1

0 (_ _ 1 _ 0 _ c))

= dyp(j1

0 (_E( 1 _ 0; s _ 0) _ c)) = ap

i (u)_i

d_ap(_X2) = dyp(_E( ; s(p(_u)))_ _X2) = 􀀀_p

_(aq(_u))__ @

@yp :

Hence

dap(X) = d_ap(__X) = ap

i (u)_i(X) 􀀀 _p

_(aq(u))__(X): _

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

_ 2 1(Y ; V Y ) with _ _ _ = _ and Im_ = V Y . Equivalently, any connection

is determined by the horizontal projection _ = idTY 􀀀 _, or by the horizontal

subspaces _(TyY ) _ TyY in the individual tangent spaces, i.e. by the horizontal

distribution. But every horizontal subspace _(TyY ) is complementary to the

vertical subspace VyY and therefore it is canonically identi_ed with a unique

element j1

ys 2 J1

yY . On the other hand, each j1

ys 2 J1

yY determines a subspace in

TyY complementary to VyY . 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 ! J1Y of the _rst jet prolongation _ : J1Y ! Y .

Now, the horizontal lifting : TM_MY ! TY corresponding to a connection

􀀀 is given by the composition of jets, i.e. for every _x = j1

0c 2 TxM and y 2 Y ,

p(y) = x, we have (_x; y) = 􀀀(y) _ _x. Given a vector _eld _, we get the 􀀀-

lift 􀀀_ 2 X(Y ), 􀀀_(y) = 􀀀(y) _ _(p(y)) which is a projectable vector _eld on

Y ! M. Note that for every connection 􀀀 on p: Y ! M and _ 2 TyY it holds

_(_) = (Tp(_); y) and _ = idTY 􀀀 _.

Since the _rst jet prolongations carry a natural a_ne structure, we can consider

J1 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 manifold

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 J1Y ! 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

J1U_. In this way we obtain the local sections _ : U_ ! (V T_B)(U_) which

correspond to the Christo_el forms introduced in 9.7. More explicitly, if we pull

back the sections _ 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

Frolicher-Nijenhuis bracket, 2R = [_; _]. It holds R[X1;X2] = _([_X1;_X2])

for all vector _elds X1, X2 on Y . In other words, given two vectors A1, A2 2

TyY , we extend them to arbitrary vector _elds X1 and X2 on Y and we have

R(A1;A2) = _([_X1;_X2](y)). Clearly, we can take for X1 and X2 projectable

vector _elds over some vector _elds _1, _2 on M. Then _Xi = _i, i = 1; 2. This

implies that R can be interpreted as a map R(y; _1; _2) = _([_1; _2](y)). Such

a map is identi_ed with a section Y ! V Y _2T_M. Obviously, the latter

formula can be rewritten as

R(y; _1; _2) = [_1; _2](y) 􀀀 ([_1; _2])(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 : J1P _ G ! J1P given by rg(j1

xs) = j1

x(rg _ s) for all g 2 G

and j1

xs 2 J1P. In accordance with 11.1 we de_ne a principal connection 􀀀 on

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

􀀀: P ! J1P of the _rst jet prolongation J1P ! 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 _ 2 X(M) and principal connection 􀀀 the 􀀀-

lift 􀀀_ is a right invariant projectable vector _eld on P. Furthermore, a principal

connection induces an identi_cation J1P _= 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 _ 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 ! J1E by 􀀀Efu; sg = j1

f_; ^sg, where u 2 Px

and s 2 S are arbitrary, 􀀀(u) = j1

x_ and ^s means the constant map M ! S

with value s. It follows immediately that the parallel transport PtE(c; fu; sg) of

an element fu; sg 2 E along a curve c : R ! M is the curve t 7! fPt(c; u; t); sg

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

connection on P.

We recall the canonical principal bundle structure (TP; Tp; TM; TG) on TP

and TE = TP[TS; T`], see 10.18. The horizontal lifting determined by the

induced connection 􀀀E is given for every _ 2 X(M) by

(1) 􀀀E_(fu; sg) = f􀀀_(u); 0sg 2 (TE)_(p(u));

where 0s 2 TsS is the zero tangent vector. Let us now consider an arbitrary

general connection 􀀀E on E. Chosen an auxiliary principal connection

􀀀P on P, we can express the horizontal lifting E in the form 􀀀E_(fu; sg) =

f􀀀P _(u); _(_(p(u)); s)g. The map _ 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 E can be expressed

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

_ 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 2 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 J1P=G. Since the source projection _: J1P ! M is Ginvariant,

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

J1'(j1

x(ra _ s)) = j1'

0(x)(r'1(a) _ ' _ s _ '􀀀1

0 )

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

160 Chapter IV. Jets and natural bundles

for all j1

xs 2 J1P, a 2 G. Hence the map J1': J1P ! J1 _ P factors to a map

Q': QP ! Q _ P and Q becomes a functor with values in _bered sets. More

explicitly, for every j1

xs in an orbit A 2 QP the value Q'(A) is the orbit in J1 _ P

going through J1'(j1

xs). By the construction, we have a bijective correspondence

between the sections of the _bered set QP ! M and the G-equivariant sections

of J1P ! 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 J1(Rm _ G)=G, namely j1

xs with s(x) = (x; e), e 2 G being

the unit. Moreover, J1(Rm _ G) is identi_ed with Rm _ J1

0 (Rm;G), j1

xs 7!

(x; j1

0(pr2 _ s _ tx)). Hence there is the induced smooth structure Q(Rm _ G) _=

Rm _ J1

0 (Rm;G)e and the canonical projection J1(Rm _ G) ! Q(Rm _ 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 j1

xs 2 A 2 Q(Rm_G) with s(x) = (x; e), the orbit Q'(A) is determined

by J1'(j1

xs). This means that

Q'(j1

xs) = j1'

0(x)(ra􀀀1

_ ' _ s _ '􀀀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 J1

0 (Rm;G)e. The smooth sections of QP are in bijection with the principle

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 : Rm _ G ! P.

By the de_nition, the restriction of Q to the _ber S := (Q(Rm _ G))0 is a

di_eomorphism onto the _ber QP 0(0). Since the functor Q is of order one, this

di_eomorphism is determined by j1 (0; e) 2 W1P, cf. 15.3. For the same reason,

every element j1'(0; e) 2 W1m

G determines a di_eomorphism Q'jS : S ! S. By

the de_nition of the Lie group structure on W1m

G, this de_nes a left action ` of

W1m

G on S. We de_ne a mapping q : W1P _ S ! QP by

q(j1 (0; e);A) = Q (A):

Since q(j1( _')(0; e);Q'􀀀1(A)) = Q _Q'_Q'􀀀1(A), the map q identi_es QP

with W1P[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 onMfm 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 = J1

0 (Rm _ G)=G _= (Rm _ T1m

G)0=G _= J1

0 (Rm;G)e

g Rm_;

see 17.4, and W1m

G = G1

m o T1m

G. Moreover, we have introduced the identi_cation

T1m

G = Go(gRm_) with the multiplication (a;Z)(_a; _ Z) = (a_a; Ad(_a􀀀1)Z+

_ Z), see 15.6. Let us now express the action ` of W1m

G = (G1

_G)o(gRm_) on

S = (gRm_). Given (A; a;Z) _= j1'(0; e) 2 W1m

G, and Y _= j1

0s 2 J1

0 (Rm_G),

s(0) = (0; e), we have A = j1

0'0, a = pr2 _ '(0; e), Z = T_a􀀀1 _ T0 _' and

Y = T0~s, where ~s = pr2 _ s. By de_nition, Q'(j1

0s) = j1

0q and if we require

~q(0) := pr2 _q(0) = e we have q = _a􀀀1

_'_s_'􀀀1

0 , where _ denotes the principal

right action of G. Then we evaluate

~q = _a􀀀1

_ _ _ ( _ '; ~s) _ '􀀀1

0 = conj(a) _ _ _ (_a􀀀1 _ _ '; ~s) _ '􀀀1

0 :

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

(1) (A; a;Z)(Y ) = Ad(a)(Y + Z) _ A􀀀1:

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

connections QP is the associated _ber bundle W1P[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 C1(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) 7! _A(u) 2 VuP

identi_es V P with P _ g and (ua; Ad(a􀀀1)(A)) 7! TRa _ _A(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

m given by

(1) (a;A)(Y ) = Ad(a)(Y ) _ A􀀀1:

At the same time, we can view g Rm_ as the standard _ber of QP with the

action ` of W1m

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 2 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 W1P with the modelling

vector bundle LP T_M = W1P[g Rm_] corresponding to the action (1) of

the Lie subgroup G1

_ G _ W1m

G via the canonical homomorphism W1m

G !

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

section in C1(V P _2T_M ! P), we can view the curvature as an operator

R: C1(QP ! M) ! C1(LP _2T_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 consider

a frame bundle PrM and the bundle of principal connections QPrM. The

composition Q _ Pr is a bundle functor on Mfm of order r + 1, so that there

is the canonical structure QPrM _= Pr+1M[gr

Rm_], but there also is the

identi_cation QPrM _= W1Pr[gr

Rm_; `] described in 17.6. It is an easy exercise

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

the latter one by the natural reduction iM : Pr+1M ! W1PrM, see proposition

15.7.

The most important case is r = 1, since the functor QP1 associates to each

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

expressions of the actions of W1m

m and G2

m on (g1

Rm_) = Hom(Rm; gl(m)).

Given (A;B;Z) 2 W1m

m, A = (ai

j) 2 G1

m, B = (bi

j) 2 G1

m, Z = (zij

k) 2

(g1

Rm_), 􀀀 = (􀀀i

jk) 2 (g1

Rm_), we have Ad(B)(Z) = (bi

mzm

), so that

17.5.(1) implies

(A;B;Z)(􀀀i

jk) = (bi

m(􀀀m

nl + zm

nl)~al

The coordinate expression of the homomorphism i0 : G2

! W1m

m deduced in

15.8 yields the formula

(ai

j ; ai

jk)(􀀀i

jk) = (ai

m􀀀m

nl~al

k~anj

+ ai

nl~al

k~anj

We remark that the 􀀀i

jk introduced in this way di_er from the classical Christo_el

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

17.15.

Let us mention briey the second order case. We have to deal with (A;B;Z) 2

W1m

m, A = (ai

j) 2 G1

m, B = (bi

j ; bi

jk) 2 G2

m, Z = (zij

k; zij

kl) 2 (g2

Rm_). We

compute

Ad(B)(Z) _ A􀀀1 = (bi

pzp

sm~amk

; bi

pzp

sm~aml

+ bi

pzp

mnq~aq

+ bi

pszp

mn~anl

k + bi

pszs

mn~anl

)

and we have to compose this action with the homomorphism i0 : G3

! W1m

For every a = (ai

j ; ai

jk; ai

jkl) 2 G3

m, the formula derived in 15.8 implies

a:(􀀀i

jk; 􀀀i

jkl) =

􀀀

m􀀀m

nl~al

k~anj

+ ai

nl~al

k~anj;

p􀀀p

mnq~aq

l ~ank

~amj

+ ai

p􀀀p

sm~aml

~asj

k + ai

ps􀀀p

mn~anl

~amj

~as

+ ai

ps􀀀s

nm~aml

~ap

j ~ank

+ ai

mnq~aq

l ~amk

~anj

+ ai

sm~as

kj~aml

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 ! J1P 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 _ : M !

P and s: M ! E with a common domain U _ M and a point x 2 U, there is

the map '_;s: U 3 y 7! g_(x)_g_(y)

􀀀1

_s(y) 2 Ex, i.e. '_;s = q(_(x); )__ _(_; s).

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

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

and _(x) = (0; e), then we get just the projection onto the standard _ber. Since

the principal connection 􀀀 associates to every u 2 Px a 1-jet 􀀀(u) = j1

x_ of a

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

jet of '_;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 2 Px, for �� is right invariant.

So we de_ne the absolute (or covariant) di_erential rs(x) of s at x with respect

to the principal connection 􀀀 by

rs(x) = j1

x'_;s 2 J1

x(M;Ex)s(x)

Hom(TxM; Vs(x)E):

If E is an associated vector bundle, then there is the canonical identi_cation

Vs(x)E = Ex. Then we have rs(x) 2 Hom(TxM;Ex) and we shall see that this

coincides with the values of the covariant derivative r as de_ned in section 11.

We can de_ne a structure of an associated bundle on the union of the manifolds

x(M;Ex), x 2 M, where the mappings rs take their values. Let

us consider the principal _ber bundle P1M _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

the obvious principal actions). We de_ne

_ : (P1M _M P) _M ([x2MJ1

x(M;Ex)) ! T1m

_ ((j1

0f; u); j1

x') = j1

0 (~u􀀀1 _ ' _ f):

Let us further de_ne a left action _` of G1

_G on T1m

S by (remember E = P[S; `])

_`((j1

0 h; a2); j1

0q) = j1

0 (`a2

_ q) _ j1

0h􀀀1:

One veri_es easily that _ determines the structure of the associated bundle

E1 = (P1M _M P)[T1m

S; _`] and that for every section s: M ! E its absolute

di_erential rs with respect to a _xed principal connection 􀀀 on P is a smooth

section of E1. Hence r can be viewed as an operator

r: C1(E) ! C1((P1M _M P)[T1m

S; _`]):

17.9. Absolute di_erentiation along vector _elds. Let E, P, 􀀀 be as in

17.8. Given a tangent vector Xx 2 TxM, we de_ne the absolute di_erentiation

in the direction Xx of a section s: M ! E to be the value rs(x)(Xx) 2 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 2 X(M) we get a map rXs: M !

V E with the following properties

(1) _E _ rXs = s

(2) rfX+gY s = frXs + grY s

for all vector _elds X, Y 2 X(M) and smooth functions f, g on M, _E : V E _

TE ! E being the canonical projection.

So every X 2 X(M) determines an operator rX : C1(E) ! C1(V E) and

the whole procedure of the absolute di_erentiation can be viewed as an operator

r: C1(TM _M E) ! C1(V E).

By the de_nition of the connection form _E of the induced connection 􀀀E, it

holds

(3) rXs = _E _ Ts _ X

r(4) Xs = Ts _ X 􀀀 (􀀀EX) _ s:

17.10. The frame forms. For every vector _eld X 2 X(M) and every map

_s: P ! S we de_ne

rX_s: P ! TS; rX_s = T _s _ 􀀀EX

r_s: P1M _M P ! T1m

S; r_s(v; u) = T _s _ T_ _ v;

where 􀀀(u) = j1

x_, x = p(u). We call r_s the absolute di_erential of _s while rX_s

is called the absolute di_erential along X.

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

r_s is the frame form of rs and for every X 2 X(M), rX_s is the frame form of

rXs.

Proof. The map rXs is a section of V E = P[TS] and _s(u) = _E(u; s _ p(u)),

u 2 P. Further, for every u 2 Px with 􀀀(u) = j1

x_, we have rs(x) = j1

x(~u__s__) 2

Hom(TxM; Vs(x)E). Hence for every X 2 X(M) we get rXs = T ~u _ T(_s _ _) _X

and since the di_eomorphism TS ! (V E)x determined by u 2 P is just T ~u, the

frame form of rXs is rX_s.

In order to prove the other equality, let us evaluate

rs(x) = f(v; u); (j1

x(~u􀀀1 _ ')) _ vg:

Since ' = ~u _ _s _ _, where 􀀀(u) = j1

x_, the frame form of rs is r_s. _

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

identi_cation S _= TyS for each point y 2 S. Consider a section s: M ! E

and its frame form _s: P ! S. Then rs(x) 2 J1

x(M;Ex) can be viewed as a

value of a form Ds 2 1(M;E). The corresponding S-valued tensorial 1-form

D_s: TP ! S is de_ned by D_s = d_s _ _ = (__d)(_s), where _ is the horizontal

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_erentials

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_a_r, 73 b].

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 ! J1P the image of the map 􀀀 de_nes a reduction

R(􀀀): P1M _M P 􀀀 􀀀! P1M _M 􀀀(P) ,! P1M _M J1P = W1P

of the principal bundle W1P to the structure group

_ G ,! G1

m o T1m

G = G1

m o (G o (g Rm_)):

Let us write ~_ for the restriction of the canonical form _ on W1P to P1M_M

􀀀(P), let ! be the connection form of 􀀀 and _M will denote the canonical form

_M 2 1(P1M;Rm).

Lemma. The following diagram is commutative

T(P1M _M 􀀀(P)) w

pr1 u __

TP1M

g u Rm _ g pr2 w

pr1 Rm

Proof. For every u 2 W1P, u = j1 (0; e), _(u) = _u, we have the isomorphism

~u: Rm _ g ! T_uP and for every X 2 TuW1P, _(X) = ~u􀀀1(__X). If X 2

T(P1M _M 􀀀(P)), we denote _(X) = Y1 + Y2 2 Rm _ g. Then ~u(Y1) = T( 1 _

0)Y1 = _(__X) and ~u(Y2) = __X 􀀀 ~u(Y1) = _(__X), where _ and _ are the

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

the second factor in Rm_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 _s1 : W1P ! T1m

S be the frame form of j1s. Then for all u 2 P1M _M P _=

P1M _M 􀀀(P) _ W1P it holds _s1(u) = r_s(u).

Proof. If u = j1 (0; e), _u = _(u), then 􀀀(_u) = j1 1( 0(0)). Since we know

_s1(u) = j1

0 (_E( 1 _ 0; s _ 0)), we get r_s(u) = j1

0 (_s _ 1 _ 0) = _s1(u). _

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

(U; '), ' = (yp), on S. Let ei, i = 1; : : : ;m be the canonical basis in Rm and e_,

_ = m+ 1; : : : ;m+ dimG be a base of Lie algebra g. Let us denote _M = _i

M ei

the canonical form on P1M, ! = !_e_, j1 and j2 be the canonical projections

on P1M _M P. Further, let us write _!_ = j_

2!_, __i

M = j_

1 _i

M and let _p

_(y) @

@yp

be the fundamental vector _elds corresponding to e_. For a section s: M ! E

let ap, ap

i be the coordinate functions of rs on P1M _M P while _ap be those of

s. Then it holds

dap + _p

_(aq)_!_ = ap

__i

M :

Proof. In 16.6 we described the coordinate functions bp, bp

i of j1s de_ned on

W1P, bp = ___ap, dbp + _p

_ (bq)__ = bp

i _i. According to 17.13, the functions ap,

i are restrictions of bp, bp

i to P1M _M P. But then the proposition follows

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 _ on a manifold M

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

standard action _yi = ai

jyj of GL(m) on Rm, the fundamental vector _elds _ij

Rm corresponding to the canonical basis of the Lie algebra of GL(m) are of the

form _k

i yj @

@yk . Every local coordinates (xi) on an open subset U _ M de_ne

a section _: U ! P1M formed by the coordinate frames ( @

@x1 ; : : : ; @

@xm ) and it

holds ___i

M = dxi. On the other hand, from the explicite equation 25.2.(2) of 􀀀

we deduce easily that the restriction of the connection form ! = (!ij

) of 􀀀 to _

is (􀀀􀀀i

jk(x)dxk). Thus, if we consider the coordinate expression _i(x) @

@xi of _ in

our coordinate system and we write rj_i for the additional coordinates of r_,

we obtain from 17.14

rj_i = @_i

@xj

􀀀 􀀀i

kj_k:

Comparing with the classical formula in [Kobayashi, Nomizu, 63, p. 144], we

conclude that our quantities 􀀀i

jk di_er from the classical Christo_el symbols by

sign and by the order of subscripts.

Remarks

The development of the theory of natural bundles and operators is described

in the preface and in the introduction to this chapter. But let us come back

to the jet groups. As mentioned in [Reinhart, 83], it is remarkable how very

little of existing Lie group theory applies to them. The results deduced in our

exposition are mainly due to [Terng, 78] where the reader can _nd some more

information on the classi_cation of Gr

m-modules. For the _rst order jet groups,

it is very useful to study in detail the properties of irreducible representations,

cf. section 34. But in view of 13.15 it is not interesting to extend this approach

to the higher orders. The bundle functors on the whole category Mf were _rst

studied by [Jany_ska, 83]. We shall continue the study of such functors in chapter

IX.

The basic ideas from section 15 were introduced in a slightly modi_ed situation

by [Ehresmann, 55]. Every principal _ber bundle p: P ! M with structure

group G determines the associated groupoid PP􀀀1 which can be de_ned as the

factor space P _P= _ with respect to the equivalence relation (u; v) _ (ug; vg),

u, v 2 P, g 2 G. Writing uv􀀀1 for such an equivalence class, we have two

projections a, b : PP􀀀1 ! M, a(uv􀀀1) = p(v), b(uv􀀀1) = p(u). If E is a _ber

bundle associated with P with standard _ber S, then every _ = uv􀀀1 2 PP􀀀1

determines a di_eomorphism qu _ (qv)􀀀1 : Ea_ ! Eb_, where qv : S ! Ea_ and

qu : S ! Eb_ are the `frame maps' introduced in 10.7. This de_nes an action of

groupoid PP􀀀1 on _ber bundle E. The space PP􀀀1 is a prototype of a smooth

groupoid over M. In [Ehresmann, 55] the r-th prolongation _r of an arbitrary

smooth groupoid _ over M is de_ned and every action of _ on a _ber bundle

E ! M is extended into an action of _r on the r-th jet prolongation JrE

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

Remarks 167

of E ! M. This construction was modi_ed to the principal _ber bundles by

[Libermann, 71], [Virsik, 69] and [Kol_a_r, 71b].

The canonical Rm-valued form on the _rst order frame bundle P1M is one

of the basic concepts of modern di_erential geometry. Its generalization to r-th

order frame bundles was introduced by [Kobayashi, 61]. The canonical form

on W1P (as well as on WrP) was de_ned in [Kol_a_r, 71b] in connection with

some local considerations by [Laptev, 69] and [Gheorghiev, 68]. Those canonical

forms play an important role in a generalization of the Cartan method of moving

frames, see [Kol_a_r, 71c, 73a, 73b, 77].

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

168

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