r

source of X and the point f (x) =: βX is the target of X. We denote by πs ,

r s r

0 ¤ s ¤ r, the projection jx f ’ jx f of r-jets into s-jets. By Jx (M, N ) or

J r (M, N )y we mean the set of all r-jets of M into N with source x ∈ M or

target y ∈ N , respectively, and we write Jx (M, N )y = Jx (M, N ) © J r (M, N )y .

r r

The map j r f : M ’ J r (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 ¯

r¯ ¯(x), then j r (g —¦ f ) = j r (¯ —¦ f ).

¯

r r r

satisfy jx f = jx f and jy g = jy g , f (x) = y = f

¯ xg

x

r¯ ¯

r

Proof. Take a curve γ on M with γ(0) = x. Then jx f = jx f implies f —¦γ ∼r f —¦γ,

¯¯ r r

lemma 12.1 gives g —¦ f —¦ γ ∼r g —¦ f —¦ γ and jy g = jy g yields g —¦ f —¦ γ ∼r g —¦ f —¦ γ.

¯ ¯ ¯

¯¯

Hence g —¦ f —¦ γ ∼r g —¦ f —¦ γ.

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

r r r r

X ∈ Jx (M, N )y and Y ∈ Jy (N, Q)z are of the form X = jx f and Y = jy g, we

r

can de¬ne the composition Y —¦ X ∈ Jx (M, Q)z by

r

Y —¦ X = jx (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

r

are the r-jets of the identity maps of manifolds. An element X ∈ Jx (M, N )y

is invertible, if there exists X ’1 ∈ Jy (N, M )x such that X ’1 —¦ X = jx (idM )

r r

and X —¦ X ’1 = jy (idN ). By the implicit function theorem, X ∈ J r (M, N ) is

r

r

invertible if and only if the underlying 1-jet π1 X is invertible. The existence of

such a jet implies dim M = dim N . We denote by invJ r (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 J r (f, g) : J r (M, N ) ’ J r (M , N ) de¬ned by

J r (f, g)(X) = (jy g) —¦ X —¦ (jx f )’1

r r

where x = ±X and y = βX are the source and target of X ∈ J r (M, N ). Since

the jet composition is associative, J r is a functor de¬ned on the product category

Mfm —Mf . (We shall see in 12.6 that the values of J r 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 |±| = ±1 + · · · + ±m , ±! = ±1 ! · · · ±m ! (with 0! = 1), x± =

(x1 )±1 . . . (xm )±m for x = (x1 , . . . , xm ) ∈ Rm . We denote by

‚ |±| f

D± f =

(‚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 y p on N in a neighborhood of f (x), two maps f ,

r r

g : M ’ N satisfy jx f = jx g if and only if all the partial derivatives up to order

r of the components f p and g p of their coordinate expressions coincide at x.

Proof. We ¬rst deduce that two curves γ(t), δ(t) : R ’ N satisfy γ ∼r δ if and

only if

dk (y p —¦ γ)(0) dk (y p —¦ δ)(0)

(1) =

dtk dtk

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

12. Jets 119

k = 0, 1, . . . , r, for all coordinate functions y p . On one hand, if γ ∼r δ, then

y p —¦ γ ’ y p —¦ δ vanishes to order r, i.e. (1) is true. On the other hand, let (1)

hold. Given a function • on N with coordinate expression •(y 1 , . . . , y n ), we ¬nd

by the chain rule that all derivatives up to order r of • —¦ δ depend only on the

partial derivatives up to order r of • at γ(0) and on (1). Hence • —¦ γ ’ • —¦ δ

vanishes to order r at 0.

If the partial derivatives up to the order r of f p and g p coincide at x, then

r r

the chain rule implies f —¦ γ ∼r g —¦ γ by (1). This means jx f = jx g. Conversely,

assume jx f = jx g. Consider the curves xi = ai t with arbitrary ai . Then the

r r

coordinate condition for f —¦ γ ∼r g —¦ γ reads

(D± f p (x))a± = (D± g p (x))a±

(2)

|±|=k |±|=k

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

to order r of f p and g p coincide at x.

Now we can easily prove that the auxiliary relation γ ∼r δ can be expressed

in terms of r-jets.

r r

Corollary. Two curves γ, δ : R ’ M satisfy γ ∼r δ if and only if j0 γ = j0 δ.

Proof. Since xi —¦ γ and xi —¦ δ are the coordinate expressions of γ and δ, (1) is

r r

equivalent to j0 γ = j0 δ.

12.6. Write Lr r m n r

m,n = J0 (R , R )0 . By proposition 12.5, the elements of Lm,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

ap x±

±

1¤|±|¤r

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 ).