14. Natural bundles and operators 139

˜

de¬ned by F f (p, ) = F fp , p ∈ P , is smooth, i.e. smoothly parameterized systems

of local di¬eomorphisms are transformed into smoothly parameterized systems

of ¬bered local isomorphisms.

In sections 10 and 12 we met several bundle functors on Mfm .

14.2. Now let F be a natural bundle. We shall denote by tx : Rm ’ Rm the

translation y ’ y + x and for any manifold M and point x ∈ M we shall write

Fx M for the pre image p’1 (x). In particular, F0 Rm will be called the standard

M

¬ber of the bundle functor F . Every bundle functor F : Mfm ’ FM determines

an action „ of the abelian group Rm on F Rm via „x = F tx .

Proposition. Let F : Mfm ’ FM be a bundle functor on Mfm and let S :=

F0 Rm be the standard ¬ber of F . Then there is a canonical isomorphism Rm —

S ∼ F Rm , (x, z) ’ F tx (z), and for every m-dimensional manifold M the value

=

F M is a locally trivial ¬ber bundle with standard ¬ber S.

Proof. The map ψ : F Rm ’ Rm — S de¬ned by z ’ (x, F t’x (z)), x = p(z), is

the inverse to the map de¬ned in the proposition and both maps are smooth ac-

cording to the regularity condition 14.1.(iii). The rest of the proposition follows

from the locality condition 14.1.(ii). Indeed, a ¬bered atlas of F M is formed by

the values of F on the charts of any atlas of M .

14.3. De¬nition. A natural bundle F : Mfm ’ FM is said to be of ¬nite

order r, 0 ¤ r < ∞, if for all local di¬eomorphisms f , g : M ’ N and every

r r

point x ∈ M , the equality jx f = jx g implies F f |Fx M = F g|Fx M .

14.4. Associated maps. Let us consider a natural bundle F : Mfm ’ FM

of order r. For all m-dimensional manifolds M , N we de¬ne the mapping

FM,N : invJ r (M, N ) —M F M ’ F N , (jx f, y) ’ F f (y). The mappings FM,N

r

are called the associated maps of the bundle functor F .

Proposition. The associated maps are smooth.

Proof. For m = 0 the assertion is trivial. Let us assume m > 0. Since smooth-

ness is a local property, we may restrict ourselves to M = N = Rm . Indeed,

chosen local charts on M and N we get local trivializations on F M and F N and

the induced local chart on invJ r (M, N ). Hence we have

∼ ∼

FU,V

= =

invJ r (Rm , Rm ) —Rm F Rm ’ invJ r (U, V ) —U F U ’ ’ F V ’ F Rm

’ ’’ ’

and we can apply the locality condition.

Now, let us recall that every jet in J r (Rm , Rn ) has a canonical polynomial

representative and that this space coincides with the cartesian product of Rm and

the Euclidean space of coe¬cients of these polynomials, as a smooth manifold. If

we consider the map ev : invJ r (Rm , Rm ) — Rm ’ Rm , evx (j0 f ) = f (x), then the

r

˜

associated map FRm ,Rm coincides with the map F (ev) appearing in the regularity

condition.

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140 Chapter IV. Jets and natural bundles

14.5. Induced action. According to proposition 14.4 the restriction =

FRm ,Rm |Gr — S is a smooth left action of the jet group Gr on the standard

m m

¬ber S.

Let us de¬ne qM = FRm ,M |invJ0 (Rm , M ) — S : P r M — S ’ F M . For every

r

u = j0 g ∈ invJ0 (Rm , M ), s ∈ S and j0 f ∈ Gr we have

r r r

m

qM (j0 g —¦ j0 f, (j0 f ’1 , s)) = qM (j0 g, s)

r r r r

(1)

r

and the restriction (qM )u := qM (j0 g, ) is a di¬eomorphism. Hence q determines

the structure of the associated ¬ber bundle P r M [S; ] on F M , cf. 10.7.

Proposition. For every bundle functor F : Mfm ’ FM of order r and every

m-dimensional manifold M there is a canonical structure of an associated bundle

P r M [S; ] on F M given by the map qM and the values of the functor F lie in

the category of bundles with structure group Gr and standard ¬ber S.

m

Proof. The ¬rst part was already proved. Consider a local di¬eomorphism

f : M ’ N . For every j0 g ∈ P r M , s ∈ S we have

r

r r

F f —¦ qM (j0 g, s) = F f —¦ F g(s) = qN (j0 (f —¦ g), s).

So we identify F f with {P r f, idS } : P r M —Gr S ’ P r N —Gr S.

m m

14.6. Description of r-th order natural bundles. Every smooth left action

of Gr on a manifold S determines a covariant functor L : PB(Gr ) ’ FMm ,

m m

LP = P [S; ], Lf = {f, idS }. An r-th order bundle functor F with standard

¬ber S induces an action of Gr on S and we can construct a natural bundle

m

G = L —¦ P r : Mfm ’ FM.

r r

We claim that F is naturally equivalent to G. For every u = j0 g ∈ Px M

there is the di¬eomorphism (qM )u : S ’ Fx M which we shall denote F u. Hence

we can de¬ne maps χM : GM ’ F M by

r

χM ({u, s}) = F u(s) = qM (j0 g, s) = F g(s).

According to 14.5.(1), this is a correct de¬nition, and by the construction, the

maps χM are ¬bered isomorphisms. Since Gf = {P r f, idS } for every local

r

di¬eomorphism f : M ’ N , we have F f —¦ χM ({j0 g, s}) = F (f —¦ g)(s) = χN —¦

r

Gf ({j0 g, s}).

From the geometrical point of view, naturally equivalent functors can be

identi¬ed. Hence we have proved

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

natural bundles on m-dimensional manifolds and the set of smooth left actions

of the jet group Gr on smooth manifolds.

m

In the next examples, we demonstrate on well known natural bundles, that

the identi¬cation in the theorem is exactly what the geometers usually do.

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

14. Natural bundles and operators 141

14.7. Examples.

1. The reader should reconsider that in the case of frame bundles P r the

identi¬cation used in 14.6, i.e. the relation of the functor P r to the functor

G constructed from the induced action, is exactly the usual identi¬cation of

principal ¬ber bundles (P, p, M, G) with their associated bundles P [G, »], where

» is the left action of G on itself.

1 1

2. For the tangent bundle T , the map (qM )u with u = j0 g ∈ Px M is just the

linear map T0 g : T0 Rm ’ Tx M determined by j0 g, i.e. the linear coordinates

1

on Tx M induced by local chart g. Hence the tangent bundle corresponds to the

canonical action of G1 = GL(m, R) on Rm .

m

r

3. Further well known natural bundles are the functors Tk of r-th order k-

velocities. More precisely, we consider the restrictions of the functors de¬ned in

12.8 to the category Mfm . Let us recall that Tk M = J0 (Rk , M ) and the action

r r

on morphisms is given by the composition of jets. Hence, in this case, for every

r r

u = j0 g ∈ Px M the map (qM )u transforms the classes of r-equivalent maps

(Rk , 0) ’ (M, x) into their induced coordinate expressions in the local chart g,

i.e. (qM )’1 (j0 f ) = j0 (g ’1 —¦ f ).

r r

u

14.8. Vector bundle functors. In accordance with 6.14, a bundle functor

F : Mfm ’ FM is called a vector bundle functor on Mfm , or natural vector

bundle, if there is a canonical vector bundle structure on each value F M and

the values F f on morphisms are morphisms of vector bundles. Let F be an

r-th order natural vector bundle with standard ¬ber V and with induced action

: Gr — V ’ V . Then is a group homomorphism Gr ’ GL(V ) and so V

m m

carries a structure of Gm -module. On the other hand, every Gr -module V gives

r

m

rise to a natural bundle F , see the construction in 14.6, and an application of F

to charts of any atlas on a manifold M yields a vector bundle atlas on the value

F M ’ M . Therefore proposition 14.6 implies

Proposition. There is a bijective correspondence between r-th order vector

bundle functors on Mfm and Gr -modules.

m

14.9. Examples.

1. In our setting, the p-covariant and q-contravariant tensor ¬elds on a man-

ifold M are just the smooth global sections of F M ’ M , where F is the vector

bundle functor corresponding to the GL(m)-module —p Rm— — —q Rm , cf. 7.2.

2. In 6.7 we discussed constructions with vector bundles corresponding to a

smooth covariant functor F on the category of ¬nite dimensional vector spaces

and these constructions can be applied to the values of any natural vector bundle

to get new natural vector bundles, cf. 6.14. There we applied F to the cocycle of

transition functions. Let us look what happens on the level of the corresponding

Gr -modules. If we apply F to a Gr -module V with action : Gr ’ GL(V ),

m m m

˜: Gr ’ GL(FV ), ˜(g) = F( (g)), i.e.

we get a vector space FV with action m

a new Gr -module FV . Let us assume that G and FG are the natural vector

m

bundles corresponding to V and FV . The canonical vector bundle structure on

(FG)M = P r M —Gr FV coincides with that on F(GM ) by 10.7.(4). Similarly,

m

we can handle contravariant functors and bifunctors on the category of vector

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

142 Chapter IV. Jets and natural bundles

spaces, cf. 6.7. In particular, the values of natural vector bundles corresponding

to direct sums of the modules are just ¬bered products over the base manifolds

of the individual bundles. Let us also note that C ∞ •i Fi M = •i (C ∞ (Fi M )).

3. There are also well known examples of higher order natural vector bun-

dles. First of all, we recall the functor of r-th order k-dimensional covelocities

J r ( , Rk )0 = Tk introduced in 12.8. If r, k = 1, we get the dual bundles to

r—

1

the tangent bundles J0 (R, M ) = T M . So the vector bundle structure on the

cotangent bundle is natural and the tangent spaces are the duals, from our point

of view. But we can apply the construction of a dual module to any Gr -module

m

and this leads to dual natural vector bundles according to 14.6. In this way we

get the r-th order tangent bundles T (r) := (T r— )— or, more general the bundle

functors Tk = (Tk )— , see 12.14.

r r—

14.10. A¬ne bundle functors. A bundle functor F : Mfm ’ FM is called

an a¬ne bundle functor on Mfm , or natural a¬ne bundle, if each value F M ’

M is an a¬ne bundle and the values on morphisms are a¬ne maps. Hence the

standard ¬ber V of an r-th order natural a¬ne bundle is an a¬ne space and the

induced action is a representation of Gr in the group of a¬ne transformations

m

r

of V . So for each g ∈ Gm there is a unique linear map (g) : V ’ V satisfying

(g)(y) = (g)(x) + (g)(y ’ x) for all x, y ∈ V . It follows that is a linear rep-

resentation of Gr on the vector space V and there is the corresponding natural

m

vector bundle F . By the construction, for every m-dimensional manifold M the

value F M is just the modelling vector bundle to F M and for every morphism

f : M ’ N , F f is the modelling linear map to F f . Hence two arbitrary sections

of F M ˜di¬er™ by a section of F M . The best known example of a second order

natural a¬ne bundle is the bundle of elements of linear connections QP 1 which

’’’

we shall study in section 17. The modelling natural vector bundle QP 1 is the

tensor bundle T — T — — T — corresponding to GL(m)-module Rm — Rm— — Rm— .

Next we shall describe all natural transformations between natural bundles

in the terms of Gr -equivariant maps.

m

14.11. Lemma. For every natural transformation χ : F ’ G between two

natural bundles on Mfm all mappings χM : F M ’ GM cover the identities

idM .

Proof. Let χ : F ’ G be a natural transformation and let us write p : F M ’ M

and q : GM ’ M for the canonical projections onto an m-dimensional manifold

M . If y ∈ F M is a point with z := q(χM (y)) = p(y), then there is a local

di¬eomorphism f : M ’ M such that germp(y) f = germp(y) idM and f (z) = z , ¯

z = z. But now the localization condition implies q—¦χM —¦F f (y) = q—¦Gf —¦χM (y),

¯

for q —¦ Gf = f —¦ q. This is a contradiction.

14.12. Theorem. There is a bijective correspondence between the set of all

natural transformations between two r-th order natural bundles on Mfm and

the set of smooth Gr -equivariant maps between their standard ¬bers.

m

Proof. Let F and G be natural bundles with standard ¬bers S and Q and let

χ : F ’ G be a natural transformation. According to 14.11, we have the restric-

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

14. Natural bundles and operators 143

tion χRm |S : S ’ Q and we claim that this is Gr -equivariant with respect to the

m

induced actions. Indeed, for any j0 f ∈ Gr we get (χRm |S) —¦ F f = Gf —¦ (χRm |S),

r

m

r

but F f : S ’ S and Gf : Q ’ Q are just the induced actions of j0 f on S and

Q. Now we have to show that the whole transformation χ is determined by the

map χRm |S. First, using translations tx : Rm ’ Rm we see this for the map

χRm . Then, if we choose any atlas (U± , u± ) on a manifold M , the maps F u±

form a ¬ber bundle atlas on F M and we know χM —¦ F u± = Gu± —¦ χRm . Hence

the locality of bundle functors implies χM |(pF )’1 (U± ) = Gu± —¦ χRm —¦ (F u± )’1 .

M

On the other hand, let χ0 : S ’ Q be an arbitrary Gr -equivariant smooth

m

map. According to 14.6, the functors F or G are canonically naturally equivalent

to the functors L —¦ P r or K —¦ P r , where L or K are the functors corresponding

to the induced Gr -actions or k on the standard ¬bers S or Q, respectively.

m

So it su¬ces to de¬ne a natural transformation χ : L —¦ P r ’ K —¦ P r . We

set χM = {idP r M , χ0 }. It is an easy exercise to verify that χ is a natural

transformation. Moreover, we have χRm |S = χ0 .

In general, an operator is a rule transforming sections of a ¬bered manifold

¯ ¯

Y ’ M into sections of another ¬bered manifold Y ’ M . We shall deal with

¯ ∞

the case M = M in this section. Let us recall that C Y means the set of all

smooth sections of a ¬bered manifold Y ’ M .

p p

¯

¯’

14.13. De¬nition. Let Y ’ M , Y ’ M be ¬bered manifolds. A local

’

¯

operator A : C ∞ Y ’ C ∞ Y is a map such that for every section s : M ’ Y

and every point x ∈ M the value As(x) depends on the germ of s at x only.

k k

If, moreover, for certain k ∈ N or k = ∞ the condition jx s = jx q implies

¯

As(x) = Aq(x), then A is said to be of order k. An operator A : C ∞ Y ’ C ∞ Y

is called a regular operator if every smoothly parameterized family of sections of

¯

Y is transformed into a smoothly parameterized family of sections of Y .

14.14. Associated maps to an k-th order operator. Consider an operator

¯ ¯

A : C ∞ Y ’ C ∞ Y of order k. We de¬ne a map A : J k Y ’ Y by A(jx s) = As(x)

k

which is called the associated map to the k-th order operator A.

Proposition. The associated map to any ¬nite order operator A is smooth if

and only if A is regular.

¯

Proof. Let A : C ∞ Y ’ C ∞ Y be an operator of order k. If we choose local ¬bered

coordinates on Y , we also get the induced ¬bered coordinates on J k Y . But

in these local coordinates, the jets of sections are identi¬ed with (polynomial)

sections. Thus, a chart on J k Y can be viewed as a smoothly parameterized

family of sections in C ∞ Y and so the smoothness of A follows from the regularity.

The converse implication is obvious.

14.15. Natural operators. A natural operator A : F G between two

natural bundles F and G is a system of regular operators AM : C ∞ (F M ) ’

C ∞ (GM ), M ∈ ObMfm , satisfying

(i) for every section s ∈ C ∞ (F M ’ M ) and every di¬eomorphism f : M ’ N

it holds

AN (F f —¦ s —¦ f ’1 ) = Gf —¦ AM s —¦ f ’1

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

144 Chapter IV. Jets and natural bundles

(ii) AU (s|U ) = (AM s)|U for every s ∈ C ∞ (F M ) and every open submanifold

U ‚ M.

In particular, condition (ii) implies that natural operators are formed by local

operators.

G is said to be of order k, 0 ¤ k ¤ ∞, if all

A natural operator A : F

operators AM are of order k. The system of associated maps AM : J k F M ’ GM

to the k-th order operators AM is called the system of associated maps to the

natural operator A. The associated maps to ¬nite order natural operators are

smooth.

We can look at condition (i) even from the viewpoint of the local coordinates

on a manifold M . Given a local chart u : U ‚ M ’ V ‚ Rm , the di¬eo-

morphisms f : V ’ W ‚ Rm correspond to the changes of coordinates on U .

Combining this observation with localization property (ii), we conclude that the

natural operators coincide, in fact, with those operators, the local descriptions

of which do not depend on the changes of coordinates.

14.16. Proposition. For every r-th order bundle functor F on Mfm its

composition with the functor of k-th jet prolongations of ¬bered manifolds

J k : FM ’ FM is a natural bundle of order r + k.

Proof. Let f : M ’ N be a local di¬eomorphism. Then, by de¬nition of the

associated maps FM,N , we have

F f = FM,N —¦ (j r f —¦ pM ) — idF M : F M ’ F N .

Hence J k (F f ) depends on (k + r)-jets of f in the underlying points in M only.

It is an easy exercise to verify the axioms of natural bundles.

14.17. Proposition. There is a bijective correspondence between the set of

G between two natural bundles on Mfm

k-th order natural operators A : F

and the set of all natural transformations ± : J k —¦ F ’ G.

Proof. Let AM be the associated maps of an k-th order natural operator A : F

G. We claim that these maps form a natural transformation ± : J k F ’ G. They

are smooth by virtue of 14.14 and we have to verify Gf —¦ AM = AN —¦ J k F f for

an arbitrary local di¬eomorphism f : M ’ N . We have

AN ((J k F f )(jx s)) = AN (j k (F f —¦ s —¦ f ’1 )(f (x)))

k

= AN (F f —¦ s —¦ f ’1 )(f (x)) = Gf —¦ AM s(x)

k

= Gf —¦ AM (jx s).

On the other hand, consider a natural transformation ± : J k F ’ G. We

k

de¬ne operators AM : F M GM by AM s(x) = ±M (jx s) for all sections

s ∈ C ∞ (F M ). Since the maps ±M are smooth ¬bered morphisms and according

to lemma 14.11 they all cover the identities idM , the maps AM s are smooth sec-

tions of GM . The straightforward veri¬cation of the axioms of natural operators

is left to the reader.

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14. Natural bundles and operators 145

14.18. Let F : Mfm ’ FM be an r-th order natural bundle with standard

¬ber S and let : Gr —S ’ S be the induced action. The identi¬cation Rm —S ∼=

m

∼ C ∞ (F Rm ),

∞

m m

F R , (x, s) ’ F (tx )(s), induces the identi¬cation C (R , S) =

(˜ : Rm ’ S) ’ (s(x) = F tx (˜(x))) ∈ C ∞ (F Rm ). Hence the standard ¬ber of

s s

k k

the natural bundle J F equals Tm S. Under these identi¬cations, the action of

F on an arbitrary local di¬eomorphism is of the form

F g(x, s) = (g(x), F (t’g(x) —¦ g —¦ tx )(s))

and the induced action k : Gr+k — Tm S ’ Tm S determined by the functor J k F

k k

m

is expressed by the following formula

r+k r+k

k k k k

(j0 g, j0 (F tx —¦ s(x)))

(1) (j0 g, j0 s) =