will be called the polynomial representative of an r-jet. Hence Lr m,n is a nu-

merical space of the variables a± . Standard combinatorics yields dim Lr

p

m,n =

m+r

’ 1 . The coordinates on Lr will sometimes be denoted more explic-

n m,n

m

itly by ap , ap , . . . , ap1 ...ir , symmetric in all subscripts. The projection πs : Lr

r

m,n

i ij i

’ Ls consists in suppressing all terms of degree > s.

m,n

The jet composition Lr — Lr ’ Lr is evaluated by taking the composi-

m,n n,q m,q

tion 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 —Lr ’ Lr is a polynomial map of the numerical spaces

m,n n,q m,q

in question. The sets Lr can be viewed as the sets of morphisms of a category

m,n

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

r

group Gm called the r-th di¬erential group or the r-th jet group in dimension m.

For r = 1 the group G1 is identi¬ed with GL(m, R). That is why some authors

m

use GLr (m, R) for Gr . m

In the case M = Rm , we can identify every X ∈ J r (Rm , Rn ) with a triple

(±X, (jβX t’1 ) —¦ X —¦ (j0 t±X ), βX) ∈ Rm — Lr — Rn , where tx means the

r r

m,n

βX

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 J r (Rm , Rn ) as well as the structure of

a ¬bered manifold π0 : J r (Rm , Rn ) ’ Rm — Rn . Since the jet composition in

r

Lr is polynomial, the induced map J r (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 ψ : V ’ Rn

determine an identi¬cation (π0 )’1 (U —V ) ∼ J r (Rm , Rn ). Since the chart chang-

r

=

ings are smooth maps, this de¬nes the structure of a smooth ¬bered manifold on

π0 : J r (M, N ) ’ M — N . Now we see that J r is a functor Mfm — Mf ’ FM.

r

r

Obviously, all jet projections πs 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 Cx (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 Cx (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 ∈ M(M, x)r+1 for every • ∈ Cy (N, R).

r

12.8. Velocities and covelocities. The elements of the manifold Tk M :=

J0 (Rk , M ) are said to be the k-dimensional velocities of order r on M , in short

r

(k, r)-velocities. The inclusion Tk M ‚ J r (Rm , M ) de¬nes the structure of a

r

r

smooth ¬ber bundle on Tk M ’ M . Every smooth map f : M ’ N is extended

r r r r r r

into an FM-morphism Tk f : Tk M ’ Tk N de¬ned by Tk f (j0 g) = j0 (f —¦ g).

Hence Tk is a functor Mf ’ FM. Since every map Rk ’ M1 — M2 coincides

r

with a pair of maps Rk ’ M1 and Rk ’ M2 , functor Tk preserves products.

r

1

For k = r = 1 we obtain another de¬nition of the tangent functor T = T1 .

We remark that we can now express the contents of de¬nition 12.2 by saying

r r r r

that jx f = jx g holds if and only if the restrictions of both T1 f and T1 g to

r

(T1 M )x coincide.

The space Tk M = J r (M, Rk )0 is called the space of all (k, r)-covelocities on

r—

M . In the most important case k = 1 we write in short T1 = T r— . Since Rk is a

r—

r— r r r

vector space, Tk M ’ M is a vector bundle with jx •(u) + jx ψ(u) = jx (•(u) +

r r

ψ(u)), u ∈ M , and kjx •(u) = jx k•(u), k ∈ R. Every local di¬eomorphism

r— r— r—

f : M ’ N is extended to a vector bundle morphism Tk f : Tk M ’ Tk N ,

jx • ’ jf (x) (• —¦ f ’1 ), where f ’1 is constructed locally. In this sense Tk is a

r r r—

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

bundles as a functor T1 = T — on Mfm . We remark that the behavior of Tk on

1— r—

arbitrary smooth maps will be re¬‚ected in the concept of star bundle functors

we shall introduce in 41.2.

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

r—

a multiplication in every vector space Tx M by

r r r

(jx •(u))(jx ψ(u)) = jx (•(u)ψ(u)),

r— r r

which turns Tx M into an algebra. Every jx f ∈ Jx (M, N )y de¬nes an algebra

r r— r— r r

homomorphism hom(jx f ) : Ty N ’ Tx M by jy • ’ jx (• —¦ 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 y p on N

centered at x and y. The algebra Ty N is generated by j0 y p . If we prescribe

r— r

quite arbitrarily the images ¦(j0 y p ) in Tx M , this is extended into a unique

r r—

algebra homomorphism ¦ : Ty N ’ Tx M . The n-tuple ¦(j0 y p ) represents

r— r— r

r

the coordinate expression of a jet X ∈ Jx (M, N )y and one veri¬es easily ¦ =

hom(X). Thus we have proved

r

Proposition. There is a canonical bijection between Jx (M, N )y and the set of

r— r—

all algebra homomorphisms Hom(Ty N, Tx M ).

—

For r = 1 the product of any two elements in Tx M is zero. Hence the algebra

— —

homomorphisms coincide with the linear maps Ty N ’ Tx M . This gives an

identi¬cation J 1 (M, N ) = T N — T — M (which can be deduced by several other

ways as well).

12.10. Kernel descriptions. The projection πr’1 : T r— M ’ T r’1— M is a

r

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

sequence of vector bundles over M

r

πr’1

—

0 ’ S T M ’ T M ’ ’ T r’1— M ’ 0

r r—

’ ’ ’’ ’

(1)

where S r indicates the r-th symmetric tensor power. To prove it, we ¬rst con-

r

struct a map p : — T — M ’ T r— 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

r 1 1 r

that jx (f1 . . . fr ) depends on jx f1 , . . . , jx fr only and lies in ker(πr’1 ). We have

r 1 1

···

jx (f1 . . . fr ) = jx f1 jx fr , where means the symmetric tensor prod-

uct, so that p is uniquely extended into a linear isomorphism of S r T — M into

r

ker(πr’1 ).

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

ti¬cation, which is usually justi¬ed by the coordinate evaluations only. Let y ˆ

denote the constant map of M into y ∈ N .

Proposition. The subspace (πr’1 )’1 (jx y ) ‚ Jx (M, N )y is canonically iden-

r r’1 r

ˆ

—

ti¬ed with Ty N — S r Tx M .

—

1 r r—

Proof. Let B ∈ Ty N and jx fp ∈ Tx M , p = 1, . . . , r. For every jy • ∈ Ty N ,

take the value B• ∈ R of the derivative of • in direction B and construct a

r r

function (B•)f1 (u) . . . fr (u) on M . It is easy to see that jy • ’ jx ((B•)f1 . . . fr )

r— r—

is an algebra homomorphism Ty N ’ Tx M . This de¬nes a map p : Ty N —

r-times

— — r

Tx M — . . . —Tx M ’ Jx (M, N )y . Using coordinates one veri¬es that p generates

linearly the required identi¬cation.

For r = 1 we have a distinguished element jx y in every ¬ber of J 1 (M, N ) ’

1

ˆ

—

1

M — N . This identi¬es J (M, N ) with T N — T M .

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

r’1

r r

πr’1 : (Tk M )x ’ (Tk M )x , x ∈ M , we ¬nd

(πr’1 )’1 (j0 x) = Tx M — S r Rk— .

r’1

r

(2) ˆ

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

122 Chapter IV. Jets and natural bundles

12.11. Proposition. πr’1 : J r (M, N ) ’ J r’1 (M, N ) is an a¬ne bundle,

r

the modelling vector bundle of which is the pullback of T N — S r T — M over

J r’1 (M, N ).

—

Proof. Interpret X ∈ Jx (M, N )y and A ∈ Ty N — S r Tx M ‚ Jx (M, N )y as alge-

r r

r— r— r— r

bra homomorphisms Ty N ’ Tx M . For every ¦ ∈ Ty N we have πr’1 (A(¦))

r

= 0 and π0 (X(¦)) = 0. This implies X(¦)A(Ψ) = 0 and A(¦)A(Ψ) = 0

r—

for any other Ψ ∈ Ty N . Hence X(¦Ψ) + A(¦Ψ) = X(¦)X(Ψ) = (X(¦) +

r—

A(¦))(X(Ψ) + A(Ψ)), so that X + A is also an algebra homomorphism Ty N ’

r—

Tx M . Using coordinates we ¬nd easily that the map (X, A) ’ 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 ob-

tain immediately the following property of the tangent map T πr’1 : T J r (M, N )

r

’ T J r’1 (M, N ).

r r

Corollary. For every X ∈ Jx (M, N )y , the kernel of the restriction of T πr’1 to

—

TX J r (M, N ) is Ty N — S r Tx M .

12.12. The frame bundle of order r. The set P r M 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, P r M = invTm (M ) is an open subset of Tm (M ),

r r

which de¬nes a structure of a smooth ¬ber bundle on P r M ’ M . The group

Gr acts smoothly on P r M on the right by the jet composition. Since for

m

every j0 •, j0 ψ ∈ Px M there is a unique element j0 (•’1 —¦ ψ) ∈ Gr satisfying

r r r r

m

(j0 •)—¦(j0 (•’1 —¦ψ)) = j0 ψ, P r M is a principal ¬ber bundle with structure group

r r r

Gr . For r = 1, the elements of invJ0 (Rm , M )x are identi¬ed with the linear

1

m

isomorphisms Rm ’ Tx M and G1 = GL(m), so that P 1 M coincides with the

m

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

Every velocities space Tk M is a ¬ber bundle associated with P r M with stan-

r

dard ¬ber Lr . The basic idea consists in the fact that for every j0 f ∈ (Tk M )x

r r

k,m

and j0 • ∈ Px M we have j0 (•’1 —¦ f ) ∈ Lr , and conversely, every j0 g ∈ Lr

r r r r

k,m k,m

r r r r

and j0 • ∈ Px M determine j0 (•—¦g) ∈ (Tk M )x . Thus, if we formally de¬ne a left

action Gr — Lr ’ Lr by (j0 h, j0 g) ’ j0 (h —¦ g), then Tk M is canonically

r r r r

m k,m k,m

identi¬ed with the associated ¬ber bundle P r M [Lr ]. k,m

r—

Quite similarly, every covelocities space Tk M is a ¬ber bundle associated

with P r M with standard ¬ber Lr with respect to the left action Gr —Lr ’

m

m,k m,k

Lr , (j0 h, j0 g) ’ j0 (g —¦ h’1 ). Furthermore, P r M — P r N is a principal ¬ber

r r r

m,k

bundle over M — N with structure group Gr — Gr . The space J r (M, N ) is a

m n

¬ber bundle associated with P r M — P r N with standard ¬ber Lr with respect

m,n

to the left action (Gr — Gr ) — Lr ’ Lr , ((j0 •, j0 ψ), j0 f ) ’ j0 (ψ —¦ f —¦ •’1 ).

r r r r

m n m,n m,n

Every local di¬eomorphism f : M ’ N induces a map P r f : P r M ’ P r N

by P r f (j0 •) = j0 (f —¦ •). Since Gr acts on the right on both P r M and P r N ,

r r

m

P r f is a local principal ¬ber bundle isomorphism. Hence P r is a functor from

Mfm into the category PB(Gr ). m

Given a left action of Gr on a manifold S, we have an induced map

m

{P r f, idS } : P r M [S] ’ P r N [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 ’ P r M [S], f ’ {P r f, idS } is a bundle functor on Mfm as de¬ned in 14.1. A

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

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

results.

r

12.13. For every Lie group G, Tk G is also a Lie group with multiplication

(j0 f (u))(j0 g(u)) = j0 (f (u)g(u)), u ∈ Rk , where f (u)g(u) is the product in

r r r

G. Clearly, if we consider the multiplication map µ : G — G ’ G, then the

r r r r r

multiplication map of Tk G is Tk µ : Tk G — Tk G ’ Tk G. The jet projections

r r s

πs : Tk G ’ Tk G are group homomorphisms. For s = 0, there is a splitting

r r r r

ι : G ’ Tk G of π0 = β : Tk G ’ G de¬ned by ι(g) = j0 g , where g means the

ˆ ˆ

k r

constant map of R into g ∈ G. Hence Tk G is a semidirect product of G and of

r

the kernel of β : Tk G ’ G.

r r

If G acts on the left on a manifold M , then Tk G acts on Tk M by

r r r

(j0 f (u))(j0 g(u)) = j0 f (u)(g(u)) ,

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

the action map : G — M ’ M , then the action map of the induced action is

r r r r

Tk : Tk G — Tk M ’ Tk M . The same is true for right actions.

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

bundle Tk M = (Tk M )— of the (k, r)-covelocities bundle on M . For every map

r r—

r r—

f : M ’ N the jet composition A ’ A —¦ (jx f ), x ∈ M , A ∈ (Tk N )f (x) de¬nes

a linear map »(jx f ) : (Tk N )f (x) ’ (Tk M )x . The dual map (»(jx f ))— =:

r r— r— r

r r r r

(Tk f )x : (Tk M )x ’ (Tk N )f (x) determines a functor Tk on Mf 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 T1 = T (r) (in order to distinguish from the r-th iteration of T ). The

r

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

for r = 1 the formula T M = (T — M )— can be used for introducing the vector

bundle structure on T M .

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

0 ’ T (r’1) M ’ T (r) M ’ S r T M ’ 0.

’ ’ ’ ’

(1)

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

wT wT w S TM w0

(r’1) (r) r

0 M M

u u u

r

T (r’1) f T (r) f

(2) S Tf

wT wT w S TN w0

(r’1) (r) r

0 N N

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 man-

ifold M . For every local chart • : N ’ Rn , the rule x ’ •’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

¯ ¯ r¯

¯

ψ : Rn ’ M of N and ψ : Rn ’ M of N , ψ(0) = x = ψ(0), such that j0 ψ = j0 ψ.

r

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

r

on M . We denote by Kn M the set of all contact (n, r)-elements on M . We have

r

a canonical projection ˜point of contact™ Kn M ’ M .

r

An (n, r)-velocity A ∈ (Tn M )x is called regular, if its underlying 1-jet corre-

sponds to a linear map Rn ’ Tx M of rank n. For every local parametrization

r

ψ of an n-dimensional submanifold, j0 ψ is a regular (n, r)-velocity. Since in

¯

the above de¬nition we can reparametrize ψ and ψ 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 , where A is

n

a regular (n, r)-velocity on M . There is a unique structure of a smooth ¬bered

r

manifold on Kn M ’ M with the property that the factor projection from the

r r r

subbundle regTn M ‚ Tn M of all regular (n, r)-velocities into Kn M is a surjec-

tive submersion. (The simplest way how to check it is to use the identi¬cation

of an open subset in Kn Rm with the r-th jet prolongation of ¬bered manifold

r

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.

r¯

r r

This induces a map Kn f : Kn M ’ Kn M , which is a ¬bered manifold morphism

r 1

over f . Hence Kn is a bundle functor on Mfm . For r = 1 each ¬ber (Kn M )x

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

1

why Kn 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, dim M = m, dim Y = m+n. The set J r Y (also written as J r (Y ’ M )

or J r (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 .

r

Using polynomial representatives we ¬nd easily that an element X ∈ Jx (M, Y )

belongs to J r Y if and only if (jβX p) —¦ X = jx (idM ). Hence J r Y ‚ J r (M, Y ) is a

r r

closed submanifold. For every section s of Y ’ M , j r s is a section of J r Y ’ M .

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

local ¬ber chart • : U ’ Rm+n on Y identi¬es (π0 )’1 (U ) with J r (Rm , Rn ). This

r

de¬nes the induced local coordinates y± on J r Y , 1 ¤ |±| ¤ r, where ± is any

p

multi index of range m.

Let q : Z ’ N be another ¬bered manifold and f : Y ’ Z be an FM-

morphism with the property that the base map f0 : M ’ N is a local dif-

feomorphism. Then the map J r (f, f0 ) : J r (M, Y ) ’ J r (N, Z) constructed in

12.4 transforms J r Y into J r Z. Indeed, X ∈ J r Y , βX = y is characterized

r r r r

by (jy p) —¦ X = jx idM , x = p(y), and q —¦ f = f0 —¦ p implies jf (y) q —¦ (jy f ) —¦

’1 ’1

r r r r r

X —¦ (jf0 (x) f0 ) = (jx f0 ) —¦ (jy p) —¦ X —¦ jf0 (x) f0 = jf0 (x) idN . The restricted

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

12. Jets 125

map will be denoted by J r f : J r Y ’ J r Z 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 J r : FMm ’ FM. (If there will be a danger of

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

r

we shall write J¬b for the ¬bered manifolds case.)

By proposition 12.11, πr’1 : J r (M, Y ) ’ J r’1 (M, Y ) is an a¬ne bundle,

r

the associated vector bundle of which is the pullback of T Y — S r T — M over

J r’1 (M, Y ). Taking into account the local trivializations of Y , we ¬nd that

πr’1 : J r Y ’ J r’1 Y is an a¬ne subbundle of J r (M, Y ) and its modelling vector

r

bundle is the pullback of V Y — S r T — M over J r’1 Y , 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 J 1 Y ’ Y because of its great importance in

the theory of connections. The space J 1 (M, Y ) coincides with the vector bundle

T Y — T — M = L(T M, T Y ). A 1-jet X : Tx M ’ Ty Y , x = p(y), belongs to J 1 Y

if and only if T p —¦ X = idTx M . The kernel of such a projection induced by T p is

— —

Vy Y — Tx M , so that the pre-image of idTx M in Ty Y — Tx M is an a¬ne subspace

—

with modelling vector space Vy Y — Tx M .

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

that for every X ∈ J r Y the kernel of the restriction of T πr’1 : T J r Y ’ T J r’1 Y

r

—

to TX J r Y is VβX Y — S r T±X M .

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 ∈ M there exists a ¬bration p : U ’ N over an n-dimensional manifold N .

r

By the de¬nition of contact elements, every X ∈ Kn M transversal to p (i.e.

the underlying contact 1-element of X is transversal to p) is identi¬ed with an

element of J r (U ’ N ) and vice versa. In particular, if we take U ∼ Rn — Rm’n ,

=

r

then the latter identi¬cation induces some simple local coordinates on Kn M .

12.17. If E ’ M is a vector bundle, then J r E ’ M is also a vector bundle,

r r r

provided we de¬ne jx s1 (u) + jx s2 (u) = jx (s1 (u) + s2 (u)), where u belongs to a

r r

neighborhood of x ∈ M , and kjx s(u) = jx ks(u), k ∈ R.

Let Z ’ M be an a¬ne bundle with the modelling vector bundle E ’ M .

Then J r Z ’ M is an a¬ne bundle with the modelling vector bundle J r E ’ M .

Given jx s ∈ J r Z and jx σ ∈ J r E, we set jx s(u)+jx σ(u) = jx (s(u)+σ(u)), where

r r r r r

the sum s(u) + σ(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 , . . .

i+1

of jets Ai ∈ J i (M, N ) satisfying Ai = πi (Ai+1 ) for all i = 1, . . . . Such a

sequence is called a jet of order ∞ or an in¬nite jet of M into N . Hence the set

J ∞ (M, N ) of all in¬nite jets of M into N is the projective limit of the sequence

r

π2 π3 π r+1

πr’1

J 1 (M, N ) ←1 J 2 (M, N ) ←2 . . . ← ’ J r (M, N ) ←r ’ . . .

’

’ ’

’ ’’ ’’

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

126 Chapter IV. Jets and natural bundles

We denote by πr : J ∞ (M, N ) ’ J r (M, N ) the projection transforming the

∞

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

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

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

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

1 2 r

jx f ← jx f ← · · · ← jx f ← . . .

x ∈ M , which is denoted by jx f or j ∞ f (x), is called the in¬nite jet of f at

∞

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

J ∞ (M, N ) is the in¬nite jet of a smooth map of M into N , see also 19.4.

∞

The spaces Tk M of all k-dimensional velocities of in¬nite order and the in¬-

nite di¬erential group G∞ in dimension m are de¬ned in the same way. Having

m

a ¬bered manifold Y ’ M , the in¬nite jets of its sections form the in¬nite jet

prolongation J ∞ Y of Y .

12.19. Jets of ¬bered manifold morphisms. If we consider the jets of mor-

phisms 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 ∈ Y , s ≥ r, if

r r s s

jy f = jy g and jy (f |Yx ) = jy (g|Yx ),

(1)

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

r,s

be denoted by jy 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 J r,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

q q

(2) jx Bf = jx 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 jy f such an equivalence class and by J r,s,q (Y, Y )

r,s,q

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

¯ ¯ ¯

J r,s,q (Y, Y ) = J r,s (Y, Y ) —J r (BY,B Y ) J q (BY, B Y )

(3) ¯

¯ ¯

where we consider the above mentioned projection J r,s (Y, Y ) ’ J r (BY, B Y )

¯ ¯

and the jet projection πr : J q (BY, B Y ) ’ J r (BY, B Y ).

q

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´ˇ, to appear c] has recently deduced that the r-th order jets can be charac-

ar

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

According to 12.3, the rule j r de¬ned by

j r (germx f ) = jx f

r

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

By 12.6, J r (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 , j r can be interpreted as a map

j r = jM,N : G(M, N ) ’ J r (M, N ).

r

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

folds 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 require-

ments 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 ) ∈ Fx (M, N )y and X2 = •(B2 ) ∈ 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) = •(germy g) —¦ X —¦ •((germx f )’1 ), X ∈ Fx (M, N )y .

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

p1 p2

Consider the product N1 ← N1 — N2 ’ 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 )—M F (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, •) = (J r , j r ). (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 concern-

ing geometric objects and operations, they have not been studied very exten-

sively. 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 = invJ0 (Rm , Rm )0 with the multiplication

k

m

l+1

de¬ned by the composition of jets, cf. 12.6. The jet projections πl de¬ne the

sequence

Gk ’ Gk’1 ’ · · · ’ G1 ’ 1

(1) m m m

k k

and the normal subgroups Bl = ker πl (or Bl if more suitable) form the ¬ltration

Gk = B0 ⊃ B1 ⊃ · · · ⊃ Bk’1 ⊃ Bk = 1.

(2) m

Since we identify J0 (Rm , Rm ) with the space of polynomial maps Rm ’ Rm of

k

degree less then or equal to k, we can write Gk = {f = f1 + f2 + · · · + fk ; fi ∈

m

Li (Rm , Rm ), 1 ¤ i ¤ k, and f1 ∈ GL(m) = G1 }, where Li (Rm , Rn ) is the

sym m sym

space of all homogeneous polynomial maps R ’ R of degree i. Hence Gk is

m n

m

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 m

follows that the Lie algebra gk is identi¬ed with the whole space J0 (Rm , Rm )0 ,

k

m

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

vanish at the origin. Since each j0 X, X ∈ X(Rm ), has a canonical polynomial

k

representative, the elements of gk can also be viewed as polynomial vector ¬elds

m

‚

ai xµ ‚xi . Here the sum goes over i and all multi indices µ with 1 ¤

X= µ

|µ| ¤ 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 ∈ Rn , i.e. if (x1 , . . . , xn ) ∈ Rn , then x2 always means x1 .x1 , etc.

1

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

k

π2

πk’2

πk’1

gk ’ ’ gk’1 ’ ’ · · · ’ 1 g1 ’ 0

’’ ’’ ’m

’

m m

and we get the ¬ltration by ideals bl = ker πl (or bk if more suitable)

k

l

gk = b0 ⊃ b1 ⊃ · · · ⊃ bk’1 ⊃ bk = 0.

m

Let us de¬ne gp ‚ gk , 0 ¤ p ¤ k ’1, as the space of all homogeneous polynomial

m

vector ¬elds of degree p+1, i.e. gp = Lp+1 (Rm , Rm ). By de¬nition, gp is identi¬ed

sym

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

gk = g0 • g1 • · · · • gk’1 .

(1) m

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 ∈ L1 , l2 ∈ L2 . A

decomposition g = g0 •g1 •. . . of a Lie algebra is called a grading if [gi , gj ] ‚ gi+j

for all 0 ¤ i, j < ∞. In our decomposition of gk we take gi = 0 for all i ≥ k.

m

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

13. Jet groups 129

Proposition. The Lie algebra gk of the Lie group Gk is the vector space

m m

{j0 X ; X ∈ X(Rm ), X(0) = 0} with the bracket

k

k k k

[j0 X, j0 Y ] = ’j0 [X, Y ]

(2)

and with the exponential mapping

exp(j0 X) = j0 FlX ,

k k

j0 X ∈ gk .

k

(3) 1 m

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 = j.

Proof. For every vector ¬eld X ∈ X(Rm ), the map t ’ j0 FlX is a one-parameter

k

t

subgroup in Gk and the corresponding element in gk is

m m

FlX = j0 FlX = j0 X.

k k k

‚ ‚

‚t 0 j0 t t

‚t 0

Hence exp(t.j0 X) = j0 FlX , see 4.18. Now, let us consider vector ¬elds X, Y

k k

t

on Rm vanishing at the origin and let us write brie¬‚y a := j0 X, b := j0 Y .

k k

According to 3.16 and 4.18.(3) we have

‚2

FlX —¦ FlY —¦ FlX —¦ FlY

k k k

’2j0 [X, Y ] = 2j0 [Y, X] = j0 ’t ’t t t

‚t2

0

2

j0 FlX —¦j0 FlY —¦j0 FlX —¦j0 FlY

k k k k

‚

= ’t ’t t t

‚t2

0

2

‚

exp(’ta) —¦ exp(’tb) —¦ exp(ta) —¦ exp(tb)

= ‚t2

0

‚2

FlLb —¦ FlLa —¦ FlLb —¦ FlLa (e) = 2[j0 X, j0 Y ].

k k

= ’t ’t

t t

‚t2

0

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

‚ ‚

ai x» ‚xi , b = bi xµ ‚xi ∈ gk the coordinate formula for the Lie bracket of

µ m

»

vector ¬elds, see 3.4, and formula (2) imply

‚

ci xγ

[a, b] = where

γ

‚xi

i,γ

(5)

»j bj ai ’ µj aj bi .

ci =

γ µ» »µ

1¤j¤m

µ+»’1j =γ

i

Here 1j means the multi index ± with ±i = δj and there is no implicit summation

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

‚ ‚ ‚

x± , xβ = (±i ’ βi )x±+β’1i

‚xi ‚xi ‚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 = q then for every γ with

|γ| = p + q + 1 and for every index 1 ¤ i ¤ m, we are able to ¬nd some ± and

β with |±| = p + 1, |β| = q + 1 and ± + β = γ + 1i , βi = ±i . Since the vector

‚

¬elds xγ ‚xi , 1 ¤ i ¤ m, |γ| = 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 γi = |γ|. But if m > 1, then we can take the bracket

[xj xp ‚xi , xq+1 ‚xj ] = xp+q+1 ‚xi ’ (q + 1)xp+q xj ‚xj

‚ ‚ ‚ ‚

j = i.

i i i 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 a1 a2 a’1 a’1 ∈ G. The closed subgroup K(S1 , S2 )

1 2

generated by all commutators of elements s1 ∈ S1 ‚ G, s2 ∈ S2 ‚ G is called

the commutator of the subsets S1 and S2 . In particular, G := 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) ) n∈N

n ∈ N.

G(n) = K(G, G(n’1) )

A Lie group G is called solvable if G(n) = {e} and nilpotent if G(n) = {e} for

some n ∈ 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∈N

n ∈ N.

g(n) = [g, g(n’1) ]

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 ∈ 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 nilpo-

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

13.4. Let i : GL(m) ’ Gk be the map transforming every matrix A ∈ GL(m)

m

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

splitting of the short exact sequence of Lie groups

w

w w w

k

u

π1

k

G1

(1) e B1 Gm e

m

i

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 is the semidirect product GL(m) B1 with

m

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 π of a Lie group G on a ¬nite dimensional vector space V is a

Lie group homomorphism π : G ’ GL(V ). Analogously, a representation of

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

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

T π : g ’ gl(V ) is a representation of its Lie algebra, cf. 4.24.

Given two representations π1 on V1 and π2 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 homo-

morphism, if f (π1 (a)(x)) = π2 (a)(f (x)) for all a ∈ G or a ∈ g and all x ∈ V ,

respectively. We say that the representations π1 and π2 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

π(a)(W ) ‚ W for all a ∈ G (or a ∈ g) and π is called irreducible if there is no

proper invariant subspace W ‚ V . A representation π 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 —p V ——q V — is completely reducible

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

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

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

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

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

π(t.idRm ) = tr idV for all t ∈ R \ {0}. Obviously, two irreducible representations

with di¬erent homogeneous degrees cannot be equivalent.

13.6. The GL(m)-module structure on b1 ‚ gk . Since B1 ‚ Gk is a

m m

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

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 ω = 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 ω = (divX)ω.

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

132 Chapter IV. Jets and natural bundles

In coordinates we have div( ξ i ‚/‚xi ) = ‚ξ i /‚xi and so every k-jet j0 X ∈

k

k’1

gk determines the (k ’ 1)-jet j0 (divX). Hence we can de¬ne div(j0 X) = k

m

k’1

j0 (divX) for all j0 X ∈ gk . If X is the canonical polynomial representative

k

m

k r

of j0 X of degree k, then divX is a polynomial of degree k ’ 1. Let C1 ‚ gr be

k

the subspace of all elements j0 X ∈ gr with divergence zero. By de¬nition,

div[X, Y ]ω = L[X,Y ] ω = LX LY ω ’ LY LX ω

(1)

= (X(divY ) ’ Y (divX))ω.

r

Since every linear vector ¬eld X ∈ g0 has constant divergence, C1 ‚ gr is a

gl(m)-submodule. In coordinates,

‚

ai x» r

»i ai x»’1i = 0,

∈ C1 if and only if

» »

‚xi

i,»

i.e. i (µi + 1)ai i = 0 for each µ with |µ| = r.

µ+1

‚

Further, let us notice that the Lie bracket of the ¬eld Y0 = j xj ‚xj with

r

any linear ¬eld X ∈ g0 is zero. Hence, also the subspace C2 of all vector ¬elds

Y ∈ gr of the form Y = f Y0 with an arbitrary polynomial f = f± x± of degree

r is g0 -invariant. Indeed, it holds [X, f Y0 ] = ’(Xf )Y0 .

Since div(f Y0 ) = j (±j + 1)f± x± , we see that gr = C1 • C2 . In coordinates,

r r

r

we have linear generators of C2

X± = x± ( ‚

|±| = r,

(2) xk ‚xk ),

k

r

and if m > 1 then there are linear generators of C1

|±| = r,

X±,k = x± (±k + 1)x1 ‚x1 ’ (±1 + 1)xk ‚xk ,

‚ ‚

k = 2, . . . , m

(3)

Yµ,k = xµ ‚xk ,

‚

k = 1, . . . , m, |µ| = r + 1, µk = 0.

k’1 k’1

1 2 1 2

We shall write C1 = C1 • C1 • · · · • C1 and C2 = C2 • C2 • · · · • C2 .

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

Rm we have [f X, 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 Cj are

r

m

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

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

r r

Proposition. If m > 1, then the GL(m)-submodules C1 , C2 in gr , 1 ¤ r ¤

r

k ’ 1, are irreducible and inequivalent. For m = 1, C1 = 0, 1 ¤ r ¤ k ’ 1, and

r

all C2 are irreducible inequivalent GL(1)-modules.

Proof. Assume ¬rst m > 1. A reader familiar with linear representation the-

r

ory could verify that the modules C2 are equivalent to the irreducible modules

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

13. Jet groups 133

det’r C(r,r,...,r,0) , where the symbol C(r,...,r,0) corresponds to the Young™s dia-

m m

gram (r, . . . , r, 0), while C1 are equivalent to det’(r+1) C(r+2,r+1,...,r+1,0) , see e.g.

r m

[Dieudonn´, Carrell, 71]. We shall present an elementary proof of the proposi-

e

tion.

r

Let us ¬rst discuss the modules C2 . Consider one of the linear generators X±

‚

de¬ned in (2) and a linear vector ¬eld xi ‚xj ∈ gl(m). We have

[’xi ‚xj , x± ( xk ‚xk )] = ±j xi x±’1j

‚ ‚ ‚

(4) (xk ‚xk ).

k k

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,

r

formula (4) implies that the submodule generated by any X± is the whole C2 .

r

This proves the irreducibility of the GL(m)-modules C2 .

r

In a similar way we shall prove the irreducibility of C1 . Let us evaluate the

‚

action of Zi,j = xi ‚xj on the linear generators X±,k , Yµ,k .

j

[’Zi,j , X±,k ] = (±k + 1)(±j + δ1 )x±+11 +1i ’1j ‚x1 ’

‚

j

’ (±1 + 1)(±j + δk )x±+1k +1i ’1j ‚xk ’

‚

’ δ1 (±k + 1)x±+11 ‚xj + δk (±1 + 1)x±+1k ‚xj

i i

‚ ‚

[’Zi,j , Yµ,k ] = µj xµ’1j +1i ‚xk ’ δk xµ ‚xj .

i

‚ ‚

In particular, we get

[’Zi,1 , Yµ,1 ] = 0

(±1 + 1)X±+1i ’11 ,k if ±1 = 0, i = 1

[’Zi,1 , X±,k ] = i

(±k + 1 + δk )Y±+1i ,1 if ±1 = 0, i = 1

±

µj Yµ’1j +1i ,k if i = k

[’Zi,j , Yµ,k ] = Xµ’1j ,j if i = k, µj = 0

’Yµ,j if i = k, µj = 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 Yµ,k .

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

r

the modules C2 are irreducible.

r

If m = 1, then all C1 = 0 by the de¬nition and for all 0 ¤ r ¤ k ’ 1 we have

‚ ‚ ‚

C2 = gr = R with the action of g0 given by [ax ‚x , bxr+1 ‚x ] = ’rabxr+1 ‚x .

r

r r

The submodules C1 and C2 cannot be equivalent for dimension reasons. The

adjoint action Ad of GL(m) on gk is given by Ad(a)(j0 X) = j0 (a —¦ X —¦ a’1 ).

k k

m

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

r

modules Ci 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 is generated by two closed

m

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

m

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

k’1 k’1

1 1

C1 = C1 • · · · • C1 and C2 = C2 • · · · • C2 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 j0 f ∈ Gk belongs to D1 or D2 if and

k

m

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

i’1 i’1

with fi ∈ C1 © Li (Rm , Rm ) = C1 or fi ∈ C2 © Li (Rm , Rm ) = C2 ,

sym sym

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,

m m

there is the exceptional projective splitting G1 ’ Gk de¬ned by

2

1

bk’1

b

ax + bx2 ’ a x + x2 + · · · + k’1 xk .

(1)

a a

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 ’

m m

p

g0 • · · · • gk’1 and the restrictions jt,q of the components jq : gl ’ gq to

m

p

the g0 -submodules Ct in the homogeneous component gp are morphisms of g0 -

p p

modules. Hence jt,q = 0 whenever p = q. Since j is a splitting the maps jt,p are

the identities.

Assume now m > 1. Since [gl’1 , g1 ] equals gl in gk but at the same time this

m

bracket equals zero in gl , we have got a contradiction.

m

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 gk the bracket [g1 , g1 ]

1

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

x

transformations f±,β (x) = ±x+β , β = 0, are determined by their second jets,

so we can view them as elements in G2 . Since the composition of two such

1

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

homomorphisms G2 ’ Gr for all r ∈ N. One computes easily the derivatives

1 1

(n) n’1 ’n

n’1

β . Hence the 2-jet ax+bx2 corresponds to f±,β with

f±,β (0) = (’1) n!±

± = ’ba’2 , β = a’1 . Consequently, the homomorphism G2 ’ Gr has the form

1 1

(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 G2 and the germs at zero of the origine preserving projective transfor-

1

mations of R.

13.9. Proposition. The Lie group Gk is solvable. Its Lie algebra gk can be

1 1

characterized as a Lie algebra generated by three elements

X1 = x2 dx ∈ g1 , X2 = x3 dx ∈ g2

d d d

X0 = x dx ∈ g0 ,

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

13. Jet groups 135

with relations

[X0 , X1 ] = ’X1

(1)

[X0 , X2 ] = ’2X2

(2)

(ad(X1 ))i X2 = 0 for i ≥ k ’ 2.

(3)

Proof. The ¬ltration gk = b0 ⊃ · · · ⊃ bk’1 ⊃ 0 from 13.2 is a descending chain

1

of ideals with dim(bi /bi+1 ) = 1. Hence gk is solvable.

1

d

Let us write Xi = xi+1 dx ∈ gi . Since [X1 , Xi ] = (1 ’ i)Xi+1 , we have

(’1)i’2

(ad(X1 ))i’2 X2 for k ’ 1 ≥ i ≥ 3

(4) Xi =

(i ’ 2)!

[Xi , Xj ] = (i ’ j)Xi+j .

(5)

¯ ¯ ¯

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 ± : gk ’ g,

1

¯ ¯¯ ¯

Xi ’ Xi , 0 ¤ i ¤ k ’ 1. Then [X1 , Xi ] = (1 ’ i)Xi+1 and using Jacobi identity,

¯¯ ¯

the induction on i yields [X0 , Xi ] = ’iXi . 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 with m ≥ 2 has a more complicated structure. In par-

m

ticular Gk cannot be solvable, for [gk , gk ] contains the whole homogeneous

m mm

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

Proposition. The Lie algebra gk , m ≥ 2, k ≥ 2, is generated by g0 and any

m

‚

element a ∈ g1 with a ∈ C1 ∪ C2 . In particular, we can take a = x2 ‚x1 .

1 1

/ 1

1 1

Proof. Let g be the Lie algebra generated by g0 and a. Since g1 = C1 • C2 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 . For every r ∈ N, 1 ¤ r ¤ k ’ 1, we de¬ne Br,1 ‚ Br ,

m

Br,1 = {j0 f ; f = idRm + fr+1 + · · · + fk , fr+1 ∈ C1 , fi ∈ Li (Rm , Rm )}.

r r

sym

The corresponding Lie subalgebra in gk is the ideal C1 • gr+1 • · · · • gk’1

r

m

r

so that Br,1 is a normal subgroup. Analogously, we set Br,2 = {j0 f ; f =

idRm + fr+1 + · · · + fk , fr+1 ∈ C2 , fi ∈ Li (Rm , Rm )} with the corresponding

r

sym

r

Lie subalgebra C2 •gr+1 •· · ·•gk’1 . We can characterize the normal subgroups

k

Br,j as the subgroups in Br with the projections πr+1 (Br,j ) belonging to the

subgroups Dj ‚ Gr+1 , j = 1, 2, cf. 13.7.

m

Proposition. Every connected normal subgroup H of Gk , m ≥ 2, is one of the

m

following:

(1) {e}, the identity subgroup,

(2) Br , 1 ¤ r < k, the kernel of the projection πr : Gk ’ Gr ,

k

m m

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

Proof. Since we deal with connected subgroups H ‚ Gk , we can prove the

m

proposition on the Lie algebra level.

Let us ¬rst assume that H ‚ B1 . Then it su¬ces to prove that the ideal in

gm generated by Cj , j = 1, 2, is the whole Cj • br+1 . But the whole algebra gk

k r r

m

2‚

is generated by g0 and X1 = x1 ‚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

r+1

r

by g0 , X1 and Cj for both j = 1 and j = 2. Since Cj are irreducible g0 -

/ r+1

r

submodules, it su¬ces to ¬nd an element Y ∈ Cj such that [X1 , Y ] ∈ C1 and

/ r+1

at the same time [X1 , Y ] ∈ C2 .

Let us take ¬rst j = 2, i.e. Y = f Y0 for certain polynomial f . Since

[f Y0 , X1 ] = (X1 f )Y0 + f [Y0 , X1 ] = (X1 f )Y0 ’ f X1 , the choice f (x) = ’xr gives

2

r+1 r+1

‚

[Y, X1 ] = xr x2 ‚x1 which does not belong to C1 ∪ C2 , for its divergence

21

equals to 2x1 xr = 0, cf. 13.5.

2

Further, consider Y = xr+1 ‚x1 ∈ C1 and let us evaluate [xr+1 ‚x1 , x2 ‚x1 ] =

‚ ‚ ‚

r

1

2 2

’2x1 xr+1 ‚x1 . Since the divergence of the latter ¬eld does not vanish, [Y, X2 ] ∈

‚

/

2

r+1 r+1

C1 ∪ C2 as required. Hence we have proved that all connected normal

subgroups H ‚ Gk contained in B1 are of the form (1)“(4).

m

Consider now an arbitrary ideal h in gk and let us denote n = h © g0 ‚ g0 . By

m

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

1

b1 ‚ h. We shall prove that for every X ∈ g0 any of the equalities [X, C1 ] = 0

1

and [X, C2 ] = 0 implies X = 0. Therefore either h ⊃ b1 or n = 0 which concludes

the proof of the proposition.

‚ ‚ 1

i,j bij xj ‚xi ∈ g0 and Y = xk j xj ‚xj ∈ C2 . Then [X, Y ] =

Let X =

‚

’( j bkj xj )Y0 . Hence [X, C2 ] = 0 implies X = 0. Similarly, for Y = x2 ‚xk ∈

1

l

1

C1 and X ∈ g0 , the equalities [X, Y ] = 0 for all k = l yield X = 0. The simple

computation is left to the reader.

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

m

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 , i.e. to Gk -modules. Since there is a well known

m m

representation theory of GL(m) which is a subgroup in Gk , we should try to

m

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

space V to a representation of Gk . A step towards such description was done

m

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

m

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 ρ : GL(m) ’ GL(V ) there is a bijection

between the representations ρ : Gk ’ GL(V ) with ρ|GL(m) = ρ and the set

¯m ¯

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

m

primary, then the action of the normal subgroup B1 ‚ Gk is trivial.

m

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

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

gl(V ) = V — V — has homogeneous degree zero. But all the irreducible compo-

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

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

proposition.

13.14. Proposition. Let ρ : Gk ’ GL(V ) be a linear representation such

m

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

r

i=1 ni Vi , 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

l’1

to the homogeneous degree of Vj whenever i ¤ j. Then W = ( i=1 ni Vi ) • nVl

is a Gk -submodule of V for all 1 ¤ l ¤ r and n ¤ nl .

m

l’1

Proof. By de¬nition, ( i=1 ni Vi ) • nVl is a GL(m)-submodule. Since every ir-

reducible 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

l’1 l’1

ni Vi ) • nVl ) ‚

Te ρ(X)(( ni V i

i=1 i=1

for all n ¤ nl and for every X ∈ b1 . Now the proposition follows from 13.12 and

13.5.

13.15. Corollary. Every irreducible Gk -module which is completely reducible

m

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

normal nilpotent subgroup B1 ‚ Gk .

m

Proof. Let V be an irreducible Gk -module. Then V is irreducible when viewed

m

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 . Proposition 13.2 describes the bracket and the exponential

m

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 ‚ Gr m+n and the corresponding

m,n

‚

Lie subalgebra gm,n ‚ gm+n consists of all polynomial vector ¬elds i,µ ai xµ ‚xi

k k

µ

with ai = 0 whenever i ¤ m and µj = 0 for some j > m. The arguments from

µ

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

in the following formulation.

The decomposition gk = g0 • · · · • gk’1 is a grading and for every indices

m,n

0 ¤ i, j < k it holds

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