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