= j0 (F g —¦ F tg’1 (x) —¦ s(g ’1 (x)))

k

∈ J 0 F Rm

k

˜

= j0 (F t’x —¦ F g —¦ F tg’1 (x) —¦ s(g ’1 (x)))

k k

∈ Tm S

˜

j0 (t’x —¦ g —¦ tg’1 (x) ), s(g ’1 (x)) .

k r

= j0 ˜

r+k

In particular, if a = j0 g ∈ G1 ‚ Gr+k , i.e. g is linear, then

m m

(a, j0 s) = j0 ( (j0 g, s —¦ g ’1 (x))) = j0 ( —¦ s —¦ g ’1 ).

k k k r k

(2) ˜ ˜ ˜

a

As a consequence of the last two propositions we get the basic result for

¬nding natural operators of prescribed types. Consider natural bundles F or F

on Mfm of ¬nite orders r or r , with standard ¬bers S or S and induced actions

or of Gr or Gr , respectively. If q = max{r + k, r } with some ¬xed k ∈ N

m m

then the actions k and trivially extend to actions of Gq on both Tm S and

k

m

S and we have

Theorem. There is a canonical bijective correspondence between the set of

F and the set of all smooth Gq -

all k-th order natural operators A : F m

equivariant maps between the left Gq -spaces Tm S and S .

k

m

14.19. Examples.

1. By the construction in 3.4, the Lie bracket of vector ¬elds is a bilinear

natural operator [ , ] : T • T T of order one, see also corollary 3.11. The

2

corresponding bilinear Gm -equivariant map is

b = (b1 , . . . , bm ) : Tm Rm — Tm Rm ’ Rm

1 1

bj (X i , X k ; Y m , Ypn ) = X i Yij ’ Y i Xi .

j

Later on we shall be able to prove that every bilinear equivariant map b : Tm Rm —

r

Tm Rm ’ Rm is a constant multiple of b composed with the jet projections and,

r

moreover, every natural bilinear operator is of a ¬nite order, so that all bilinear

natural operators on vector ¬elds are the constant multiples of the Lie bracket.

On the other hand, if we drop the bilinearity, then we can iterate the Lie bracket

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146 Chapter IV. Jets and natural bundles

to get operators of higher orders. But nevertheless, one can prove that there are

no other G2 -equivariant maps b : Tm Rm — Tm Rm ’ Rm beside the constant

1 1

m

multiples of b and the projections Tm Rm — Rm ’ Rm . This implies, that the

1

constant multiples of the Lie bracket are essentially the only natural operators

T •T T of order 1.

2. The exterior derivative introduced in 7.8 is a ¬rst order natural oper-

ator d : Λk T — Λk+1 T — . Formula 7.8.(1) expresses the corresponding G2 -m

equivariant map

Tm (Λk Rm— ) ’ Λk+1 Rm—

1

(’1)j+1 •i1 ...ij ...ik+1 ,ij

(•i1 ...ik , •i1 ...ik ,ik+1 ) ’

j

where the hat denotes that the index is omitted. We shall derive in 25.4 that

for k > 0 this is the only G2 -equivariant map up to constant multiples. Con-

m

sequently, the constant multiples of the exterior derivative are the only natural

operators of the type in question.

14.20. In concrete problems we often meet a situation where the representa-

tions of Gr are linear, or at least their restrictions to G1 ‚ Gr turn the

m m m

standard ¬bers into GL(m)-modules. Then the linear equivariant maps between

the standard ¬bers are GL(m)-module homomorphisms and so the structure of

the modules in question is often a very useful information for ¬nding all equi-

variant maps. Given a G1 -module V and linear coordinates y p on V , there are

m

|±| p

the induced coordinates y± = ‚‚x± on Tm V , where xi are the canonical coor-

y

p k

dinates on Rm and 0 ¤ |±| ¤ k. Then the linear subspace in Tm V de¬ned by

k

y± = 0, |±| = i, coincides with V — S i Rm— . Clearly, these identi¬cations do not

p

depend on our choice of the linear coordinates y p . Formula 14.18.(2) shows that

Tm V = V • · · · • V — S k Rm— is a decomposition of Tm V into G1 -submodules

k k

m

and the same formula implies the following result.

Proposition. Let V be a G1 -invariant subspace in —p Rm — —q Rm— and let us

m

consider a representation : Gr ’ Di¬(V ) such that its restriction to G1 ‚ Gr

m m m

is the canonical tensorial action. Then the restriction of the induced action k

of Gr+k on Tm V = V • · · · • V — S k Rm— to G1 ‚ Gr+k is also the canonical

k

m m m

tensorial action.

14.21. Some geometric constructions are performed on the whole category Mf

of smooth manifolds and smooth maps. Similarly to natural bundles, the bundle

functors on the category Mf present a special case of the more general concept

of bundle functors.

De¬nition. A bundle functor on the category Mf is a covariant functor F : Mf

’ FM satisfying the following conditions

(i) B —¦ F = IdMf , so that the ¬ber projections form a natural transformation

p : F ’ IdMf .

(ii) If i : U ’ M is an inclusion of an open submanifold, then F U = p’1 (U )

M

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

M

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

14. Natural bundles and operators 147

˜

(iii) If f : P — M ’ N is a smooth map, then F f : P — F M ’ F N , de¬ned

˜

by F f (p, ) = F fp , p ∈ P , is smooth.

For every non-negative integer m the restriction Fm of a bundle functor F

on Mf to the subcategory Mfm ‚ Mf is a natural bundle. Let us call the

sequence S = {S0 , S1 , . . . , Sm , . . . } of the standard ¬bers of the natural bundles

Fm the system of standard ¬bers of the bundle functor F . Proposition 14.2

implies that for every m there is the canonical isomorphism Rm — Sm ∼ F Rm , =

(x, s) ’ F tx (s), and given an m-dimensional manifold M , pM : F M ’ M is a

locally trivial bundle with standard ¬ber Sm .

Analogously to 14.3 and 14.4, a bundle functor F on Mf is said to be of

order r if for every smooth map f : M ’ N and point x ∈ M the restriction

F f |Fx M depends only on jx f . Then the maps FM,N : J r (M, N )—M F M ’ F N ,

r

r

FM,N (jx f, y) = F f (y) are called the associated maps to the r-th order functor

F . Since in the proof of proposition 14.3 we never used the invertibility of

the jets in question, the same proof applies to the present situation and so the

associated maps to any ¬nite order bundle functor on Mf are smooth. For every

m-dimensional manifold M , there is the canonical structure of the associated

bundle F M ∼ P r M [Sm ], cf. 14.5.

=

Let S = {S0 , S1 , . . . } be the system of standard ¬bers of an r-th order bundle

functor F on Mf . The restrictions m,n of the associated maps FRm ,Rn to

J0 (Rm , Rn )0 — Sm have the following property. For every A ∈ J0 (Rm , Rn )0 ,

r r

B ∈ J0 (Rn , Rp )0 and s ∈ Sm

r

—¦ A, s) =

(1) m,p (B n,p (B, m,n (A, s)).

Hence instead of the action of one group Gr on the standard ¬ber in the case

m

of bundle functors on Mfm , we get an action of the category Lr on S, see

below and 12.6 for the de¬nitions. We recall that the objects of Lr are the

non-negative integers and the set of morphisms between m and n is the set

Lr = J0 (Rm , Rn )0 .

r

m,n

Let S = {S0 , S1 , . . . } be a system of manifolds. A left action of the category

L on S is de¬ned as a system of maps m,n : Lr — Sm ’ Sn satisfying (1).

r

m,n

The action is called smooth if all maps m,n are smooth. The canonical action of

Lr on the system of standard ¬bers of a bundle functor F is called the induced

action. Every induced action of a ¬nite order bundle functor is smooth.

14.22. Consider a system of smooth manifolds S = {S0 , S1 , . . . } and a smooth

action of the category Lr on S. We shall construct a bundle functor L deter-

mined by this action. The restrictions m of the maps m,m to invertible jets

form smooth left actions of the jet groups Gr on manifolds Sm . Hence for every

m

m-dimensional manifold M we can de¬ne LM = P r M [Sm ; m ]. Let us recall the

notation {u, s} for the elements in P r M —Gr Sm , i.e. {u, s} = {u—¦A, m (A’1 , s)}

m

for all u ∈ P r M , A ∈ Gr , s ∈ Sm . For every smooth map f : M ’ N we de¬ne

m

Lf : F M ’ F N by

’1

Lf ({u, s}) = {v, —¦ A —¦ u, s)}

m,n (v

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

148 Chapter IV. Jets and natural bundles

r r r

where m = dimM , n = dimN , u ∈ Px M , A = jx f , and v ∈ Pf (x) N is an arbi-

trary element. We claim that this is a correct de¬nition. Indeed, chosen another

representative for {u, s} and another frame v ∈ Pf (x) , say {u —¦ B, m (B ’1 , s)},

r

and v = v —¦ C, formula 14.21.(1) implies

’1

Lf ({u —¦ B, m (B , s)} =

’1

—¦ v ’1 —¦ A —¦ u —¦ B, ’1

= {v —¦ C, m,n (C m (B , s))} =

’1 ’1

= {v —¦ C, —¦ A —¦ u, s))} =

n (C , m,n (v

’1

= {v, —¦ A —¦ u, s)}.

m,n (v

One veri¬es easily all the axioms of bundle functors, this is left to the reader.

On the other hand, consider an r-th order bundle functor F on Mf and

its induced action . Let L be the corresponding bundle functor, we have

just constructed. Analogously to 14.6, there is a canonical natural equivalence

χ : L ’ F . In fact, we have the restrictions of χ to manifolds of any ¬xed di-

mension which consists of the maps qM determining the canonical structures of

associated bundles on the values F M , see 14.6. It remains only to show that

r r

F f —¦ χM = χN —¦ F f for all smooth maps f : M ’ N . But given j0 g ∈ Px M ,

r r

j0 h ∈ Pf (x) N and s ∈ Sm , we have

F f —¦ χM ({j0 g, s}) = F f —¦ F g(s) = F h —¦ F (h’1 —¦ f —¦ g)(s)

r

r ’1

r r

—¦ f —¦ g), s)) = χN —¦ Lf ({j0 g, s}).

= χN (j0 h, m,n (j0 (h

Since in geometry we usually identify naturally equivalent functors, we have

proved

Theorem. There is a bijective correspondence between the set of r-th order

bundle functors on Mf and the set of smooth left actions of the category Lr on

systems S = {S0 , S1 , . . . } of smooth manifolds.

14.23. Natural transformations. Consider a smooth action or of the

r

category L on a system S = {S0 , S1 , . . . } or S = {S0 , S1 , . . . } of smooth

manifolds, respectively. A sequence • of smooth maps •i : Si ’ Si is called a

smooth Lr -equivariant map between and if for every s ∈ Sm , A ∈ Lr it m,n

holds

•n ( m,n (A, s)) = m,n (A, •m (s)).

Theorem. There is a bijective correspondence between the set of natural trans-

formations of two r-th order bundle functors on Mf and the set of smooth Lr -

equivariant maps between the induced actions of Lr on the systems of standard

¬bers.

Proof. Let χ : F ’ G be a natural transformation, or k be the induced action

on the system of standard ¬bers S = {S0 , S1 , . . . } or Q = {Q0 , Q1 , . . . }, respec-

tively. As we proved in 14.11, all maps χM : F M ’ GM are over identities. Let

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

15. Prolongations of principal ¬ber bundles 149

us de¬ne •n : Sn ’ Qn as the restriction of χRn to Sn . If j0 f ∈ Lr , s ∈ Sm ,

r

m,n

then

r r

= χRn —¦ F f (s) = Gf —¦ χRm (s) = km,n (j0 f, •m (s)),

•n ( m,n (j0 f, s))

so that the maps •m form a smooth Lr -equivariant map between and k. More-

over, the arguments used in 14.11 imply that χ is completely determined by the

maps •m .

Conversely, by virtue of 14.22, we may assume that the functors F and G

coincide with the functors L and K constructed from the induced actions. Con-

sider a smooth Lr -equivariant map • between and k. Then we can de¬ne for

all m-dimensional manifolds M maps χM : F M ’ GM by

χM := {idP r M , •m }.

The reader should verify easily that the maps χM form a natural transforma-

tion.

14.24. Remark. Let F be an r-th order bundle functor on Mf . Its in-

duced action can be interpreted as a smooth functor Finf : Lr ’ Mf , where

the smoothness means that all the maps Lr — Finf (m) ’ Finf (n) de¬ned by

m,n

(A, x) ’ Finf A(x) are smooth. Then the concept of smooth Lr -equivariant maps

between the actions coincides with that of a natural transformation. Hence we

can reformulate theorems 14.22 and 14.23 as follows. The full subcategory of

r-th order bundle functors on Mf in the category of functors and natural trans-

formations is naturally equivalent to the full subcategory of smooth functors

Lr ’ Mf . Let us also remark, that the Lr -objects can be viewed as numerical

spaces Rm , 0 ¤ m < ∞, with distinguished origins. Then every Mf -object is

locally isomorphic to exactly one Lr -object and, up to local di¬eomorphisms,

Lr contains all r-jets of smooth maps. Therefore, we can call Lr the r-th order

skeleton of Mf . We shall work out this point of view in our treatment of general

bundle functors in the next chapter. Let us mention that the bundle functors

on Mfm also admit such a description. Indeed, the r-th order skeleton then

consists of the group Gr only.

m

15. Prolongations of principal ¬ber bundles

15.1. In the present section, we shall mostly deal with the category PBm (G)

consisting of principal ¬ber bundles with m-dimensional bases and a ¬xed struc-

ture group G, with PB(G)-morphisms which cover local di¬eomorphisms be-

tween the base manifolds. So a PBm (G)-morphism • : (P, p, M ) ’ (P , p , M )

is a smooth ¬bered map over a local di¬eomorphism •0 : M ’ M satisfying

• —¦ ρg = ρg —¦ • for all g ∈ G, where ρ and ρ are the principal actions on P and

P . In particular, every automorphism • : Rm — G ’ Rm — G is fully determined

by its restriction • : Rm ’ G, •(x) = pr2 —¦ •(x, e), where e ∈ G is the unit, and

¯ ¯

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150 Chapter IV. Jets and natural bundles

by the underlying map •0 : Rm ’ Rm . We shall identify the morphism • with

the couple (•0 , •), i.e. we have

¯

(1) •(x, a) = (•0 (x), •(x).a).

¯

Analogously, every morphism ψ : Rm — G ’ P , i.e. every local trivialization of

˜

P , is determined by ψ0 and ψ := ψ|(Rm — {e}) : Rm ’ P covering ψ0 . Further

˜ ’1

we de¬ne ψ1 = ψ —¦ ψ0 , so that ψ1 is a local section of the principal bundle P ,

and we identify the morphism ψ with the couple (ψ0 , ψ1 ). We have

ψ(x, a) = (ψ1 —¦ ψ0 (x)).a .

(2)

Of course, for an automorphism • on Rm — G we have • = pr2 —¦ •.

¯ ˜

15.2. Principal prolongations of Lie groups. We shall apply the construc-

tion of r-jets to such a situation. Since all PB m (G)-objects are locally isomorphic

to the trivial principal bundle Rm — G and all PBm (G)-morphisms are local iso-

r

morphisms, we ¬rst have to consider the group Wm G of r-jets at (0, e) of all

automorphisms • : Rm — G ’ Rm — G with •0 (0) = 0, where the multiplication

µ is de¬ned by the composition of jets,

µ(j r •(0, e), j r ψ(0, e)) = j r (ψ —¦ •)(0, e).

This is a correct de¬nition according to 15.1.(1) and the inverse elements are

the jets of inverse maps (which always exist locally). The identi¬cation 15.1 of

automorphisms on Rm — G with couples (•0 , •) determines the identi¬cation

¯

Wm G ∼ Gr — Tm G,

r r

j r •(0, e) ’ (j0 •0 , j0 •).

r r

(1) ¯

=m

Let us describe the multiplication µ in this identi¬cation. For every •, ψ ∈

PBm (G)(Rm — G, Rm — G) we have

¯

ψ —¦ •(x, a) = ψ(•0 (x), •(x).a) = (ψ0 —¦ •0 (x), ψ(•0 (x)).•(x).a)

¯ ¯

so that given any (A, B), (A , B ) ∈ Gr — Tm G we get

r

m

µ (A, B), (A , B ) = A —¦ A , (B —¦ A ).B .

(2)

r

Here the dot means the multiplication in the Lie group Tm G, cf. 12.13. Hence

r

there is the structure of a semi direct product of Lie groups on Wm G. The Lie

group Wm G = Gr Tm G is called the (m, r)-principal prolongation of Lie group

r r

m

G.

15.3. Principal prolongations of principal bundles. For every principal

¬ber bundle (P, p, M, G) ∈ ObPB m (G) we de¬ne