<<

. 6
( 20)



>>

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 ).
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
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

<<

. 6
( 20)



>>