. 6
( 20)



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
m,n =
’ 1 . The coordinates on Lr will sometimes be denoted more explic-
n m,n
itly by ap , ap , . . . , ap1 ...ir , symmetric in all subscripts. The projection πs : Lr
i ij i
’ Ls consists in suppressing all terms of degree > s.
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
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 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
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

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

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-
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.
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).
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

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

with a pair of maps Rk ’ M1 and Rk ’ M2 , functor Tk preserves products.
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
(T1 M )x coincide.
The space Tk M = J r (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 T1 = T r— . Since Rk is a
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
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
the coordinate expression of a jet X ∈ Jx (M, N )y and one veri¬es easily ¦ =
hom(X). Thus we have proved
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

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

0 ’ S T M ’ T M ’ ’ T r’1— M ’ 0
r r—
’ ’ ’’ ’

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

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
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 ) ’

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

(πr’1 )’1 (j0 x) = Tx M — S r Rk— .
(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,

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(¦))
= 0 and π0 (X(¦)) = 0. This implies X(¦)A(Ψ) = 0 and A(¦)A(Ψ) = 0
for any other Ψ ∈ Ty N . Hence X(¦Ψ) + A(¦Ψ) = X(¦)X(Ψ) = (X(¦) +
A(¦))(X(Ψ) + A(Ψ)), so that X + A is also an algebra homomorphism Ty N ’
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 )

’ 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
every j0 •, j0 ψ ∈ Px M there is a unique element j0 (•’1 —¦ ψ) ∈ Gr satisfying
r r r r
(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
isomorphisms Rm ’ Tx M and G1 = GL(m), so that P 1 M coincides with the
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-

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

{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
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
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

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.
’ ’ ’ ’

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

wT wT w S TM w0
(r’1) (r) r
0 M M

u u u
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.

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 Kn M the set of all contact (n, r)-elements on M . We have
a canonical projection ˜point of contact™ Kn M ’ M .
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
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
a regular (n, r)-velocity on M . There is a unique structure of a smooth ¬bered
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

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
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
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 .
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

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

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,
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,

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

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

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 .
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 ,
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 , . . .
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
π2 π3 π r+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
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 ),

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

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 ).

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-
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

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 ).

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

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

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

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.
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
gk ’ ’ gk’1 ’ ’ · · · ’ 1 g1 ’ 0
’’ ’’ ’m

m m
and we get the ¬ltration by ideals bl = ker πl (or bk if more suitable)
gk = b0 ⊃ b1 ⊃ · · · ⊃ bk’1 ⊃ bk = 0.
Let us de¬ne gp ‚ gk , 0 ¤ p ¤ k ’1, as the space of all homogeneous polynomial
vector ¬elds of degree p+1, i.e. gp = Lp+1 (Rm , Rm ). By de¬nition, gp is identi¬ed
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.

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
[j0 X, j0 Y ] = ’j0 [X, Y ]

and with the exponential mapping

exp(j0 X) = j0 FlX ,
k k
j0 X ∈ gk .
(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
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
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

FlX —¦ FlY —¦ FlX —¦ FlY
k k k
’2j0 [X, Y ] = 2j0 [Y, X] = j0 ’t ’t t t
j0 FlX —¦j0 FlY —¦j0 FlX —¦j0 FlY
k k k k

= ’t ’t t t

exp(’ta) —¦ exp(’tb) —¦ exp(ta) —¦ exp(tb)
= ‚t2
FlLb —¦ FlLa —¦ FlLb —¦ FlLa (e) = 2[j0 X, j0 Y ].
k k
= ’t ’t
t t

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
»j bj ai ’ µj aj bi .
ci =
γ µ» »µ
µ+»’1j =γ

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)
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

(1) e B1 Gm 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 is the semidirect product GL(m) 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
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)-
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 ∈
gk determines the (k ’ 1)-jet j0 (divX). Hence we can de¬ne div(j0 X) = k
j0 (divX) for all j0 X ∈ gk . If X is the canonical polynomial representative
k r
of j0 X of degree k, then divX is a polynomial of degree k ’ 1. Let C1 ‚ gr be
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
= (X(divY ) ’ Y (divX))ω.
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
» »

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

Further, let us notice that the Lie bracket of the ¬eld Y0 = j xj ‚xj with
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
we have linear generators of C2

X± = x± ( ‚
|±| = r,
(2) xk ‚xk ),

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
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
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 ¤
k ’ 1, are irreducible and inequivalent. For m = 1, C1 = 0, 1 ¤ r ¤ k ’ 1, and
all C2 are irreducible inequivalent GL(1)-modules.
Proof. Assume ¬rst m > 1. A reader familiar with linear representation the-
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-
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,
formula (4) implies that the submodule generated by any X± is the whole C2 .
This proves the irreducibility of the GL(m)-modules C2 .
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 .

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

’ (±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 .
‚ ‚

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
the modules C2 are irreducible.
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
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
So each irreducible component of gr has homogeneous degree ’r. Therefore the
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
Lie subgroups D1 , D2 invariant under the canonical action of G1 . 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
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
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
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

ax + bx2 ’ a x + x2 + · · · + k’1 xk .
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
g0 • · · · • gk’1 and the restrictions jt,q of the components jq : gl ’ gq to
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
bracket equals zero in gl , 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 gk the bracket [g1 , g1 ]
equals zero. Let us ¬nd the splitting on the Lie group level. The germs of
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
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
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-
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
[X0 , X2 ] = ’2X2
(ad(X1 ))i X2 = 0 for i ≥ k ’ 2.

Proof. The ¬ltration gk = b0 ⊃ · · · ⊃ bk’1 ⊃ 0 from 13.2 is a descending chain
of ideals with dim(bi /bi+1 ) = 1. Hence gk is solvable.
Let us write Xi = xi+1 dx ∈ gi . Since [X1 , Xi ] = (1 ’ i)Xi+1 , we have

(ad(X1 ))i’2 X2 for k ’ 1 ≥ i ≥ 3
(4) Xi =
(i ’ 2)!
[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 ± : gk ’ 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 ] = ’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-
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

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 .

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 ,
Br,1 = {j0 f ; f = idRm + fr+1 + · · · + fk , fr+1 ∈ C1 , fi ∈ Li (Rm , Rm )}.
r r
The corresponding Lie subalgebra in gk is the ideal C1 • gr+1 • · · · • gk’1
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
Lie subalgebra C2 •gr+1 •· · ·•gk’1 . We can characterize the normal subgroups
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.

Proposition. Every connected normal subgroup H of Gk , m ≥ 2, is one of the
(1) {e}, the identity subgroup,
(2) Br , 1 ¤ r < k, the kernel of the projection πr : Gk ’ Gr ,
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
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
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
by g0 , X1 and Cj for both j = 1 and j = 2. Since Cj are irreducible g0 -
/ r+1
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
r+1 r+1

[Y, X1 ] = xr x2 ‚x1 which does not belong to C1 ∪ C2 , for its divergence
equals to 2x1 xr = 0, cf. 13.5.
Further, consider Y = xr+1 ‚x1 ∈ C1 and let us evaluate [xr+1 ‚x1 , x2 ‚x1 ] =
‚ ‚ ‚
2 2
’2x1 xr+1 ‚x1 . Since the divergence of the latter ¬eld does not vanish, [Y, X2 ] ∈

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).
Consider now an arbitrary ideal h in gk 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 ∈ g0 any of the equalities [X, C1 ] = 0
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 ∈
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
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
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
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
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
primary, then the action of the normal subgroup B1 ‚ Gk is trivial.

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
13.14. Proposition. Let ρ : Gk ’ GL(V ) be a linear representation such
that the corresponding GL(m)-module is completely reducible and let V =
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
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 .
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
13.15. Corollary. Every irreducible Gk -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 -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 . 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 ‚ Gr m+n and the corresponding

Lie subalgebra gm,n ‚ gm+n consists of all polynomial vector ¬elds i,µ ai xµ ‚xi
k k
with ai = 0 whenever i ¤ m and µj = 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 = 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 = 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


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