[’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

the regularity assumption, see [Nijenhuis, 72], [Terng, 78], [Palais, Terng, 77],

but we shall prove in chapter V that every bundle functor on Mfm is of ¬nite

order and that the regularity condition 14.1.(iii) follows from the other axioms.

Since the presentation of these results needs rather long and technical analytical

considerations, we prefer to derive ¬rst geometric properties of bundle functors

in the best known situations under stronger assumptions. In fact the material of

this section presents a model for the more general situation treated in the next

chapter.

14.1. De¬nition. A bundle functor on Mfm or a natural bundle over m-

manifolds, is a covariant functor F : Mfm ’ FM satisfying the following con-

ditions

(i) (Prolongation) B —¦ F = IdMfm , where B : FM ’ Mf is the base functor.

Hence the induced projections form a natural transformation p : F ’ IdMfm .

(ii) (Locality) If i : U ’ M is an inclusion of an open submanifold, then

F U = p’1 (U ) and F i is the inclusion of p’1 (U ) into F M .

M M

(iii) (Regularity) If f : P —M ’ N is a smooth map such that for all p ∈ P the

˜

maps fp = f (p, ) : M ’ N are local di¬eomorphisms, then F f : P —F M ’ F N ,