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 ,

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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.

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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

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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-

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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

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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) =

˜ ˜

= 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

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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

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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

W r P := {j r ψ(0, e); ψ ∈ PB m (G)(Rm — G, P )}.

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15. Prolongations of principal ¬ber bundles 151

In particular, W r (Rm — G) is identi¬ed with Rm — Wm G by the rule

r

Rm — W m G

r

(x, j r •(0, e)) ’ j r („x —¦ •)(0, e) ∈ W r (Rm — G)

where „x = tx —idG , and so there is a well de¬ned structure of a smooth manifold

on W r (Rm — G). Furthermore, if we de¬ne the action of W r on PB m (G)-

morphisms by the composition of jets, i.e.

W r χ(j r ψ(0, e)) := j r (χ —¦ ψ)(0, e),

W r becomes a functor. Now, taking any principle atlas on a principal bundle

P , the application of the functor W r to the local trivializations yields a ¬bered

atlas on W r . Finally, there is the right action of Wm G on W r P de¬ned for

r

every j r •(0, e) ∈ Wm G and j r ψ(0, e) ∈ W r P by (j r ψ(0, e))(j r •(0, e)) = j r (ψ —¦

r

•)(0, e). Since all the jets in question are invertible, this action is free and

transitive on the individual ¬bers and therefore we have got principal bundle

(W r P, p —¦ β, M, Wm G) called the r-th principal prolongation of the principal

r

bundle (P, p, M, G). By the de¬nition, for a morphism • the mapping W r •

r

always commutes with the right principal action of Wm G and we have de¬ned the

functor W r : PBm (G) ’ PB m (Wm G) of r-th principal prolongation of principal

r

bundles.

15.4. Every PBm (G)-morphism ψ : Rm — G ’ P is identi¬ed with a couple

(ψ0 , ψ1 ), see 15.1.(2). This yields the identi¬cation

W r P = P r M —M J r P

(1)

and also the smooth structures on both sides coincide. Let us express the corre-

sponding action of Gr Tm G on P r M —M J r P . If (u, v) = (j0 ψ0 , j r ψ1 (ψ0 (0))) ∈

r r

m

P r M —M J r P and (A, B) = (j0 •0 , j0 •) ∈ Gr

r r r

¯ Tm G, then 15.2.(2) implies

m

ψ —¦ •(x, a) = ψ(•0 (x), •(x).a) = ψ1 (ψ0 —¦ •0 (x)).•(x).a

¯ ¯

= (ρ —¦ (ψ1 , • —¦ •’1 —¦ ψ0 ) —¦ (ψ0 —¦ •0 )(x)).a

’1

¯ 0

where ρ is the principal right action on P . Hence we have

(u, v)(A, B) = (u —¦ A, v.(B —¦ A’1 —¦ u’1 ))

(2)

where ™.™ is the multiplication

m : J r P —M J r (M, G) ’ J r P, r r r

(jx σ, jx s) ’ jx (ρ —¦ (σ, s)).

The decomposition (1) is natural in the following sense. For every PBm (G)-

morphism ψ : (P, p, M, G) ’ (P , p , M, G), the PB m (Wm G)-morphism W r ψ

r

has the form (P r ψ0 , J r ψ). Indeed, given • : Rm — G ’ P , we have (ψ —¦ •)0 =

’1

ψ0 •0 , (ψ —¦ •)1 = ψ —¦ • —¦ (ψ0 —¦ •0 )’1 = ψ —¦ •1 —¦ ψ0 . Therefore, in the category

˜

of functors and natural transformations, the following diagram is a pullback

wJ

Wr r

u u

wB

Pr —¦ B

Here B : PBm (G) ’ Mfm is the base functor, the upper and left-hand natural

transformations are given by the above decomposition and the right-hand and

bottom arrows are the usual projections.

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152 Chapter IV. Jets and natural bundles

15.5. For every associated bundle E = P [S; ] to a principal bundle (P, p, M, G)

there is a canonical left action r : Wm G — Tm S ’ Tm S of Wm G = Gr

r r r r r

Tm G

m

r r r r

on Tm S. We simply compose the prolonged action Tm of Tm G on Tm S, see

12.13, with the canonical left action of Gr on both Tm G and Tm S, i.e. we set

r r

m

(j r •(0, e), j0 s) = j0 ( —¦ (• —¦ •’1 , s —¦ •’1 ))

r r r

(1) ¯ 0 0

for every j r •(0, e) = (j0 •0 , j0 •) ∈ Gr

r r r

¯ Tm G.

m

Proposition. For every associated bundle E = P [S; ], there is a canonical

structure of the associated bundle W r P [Tm S; r ] on the r-th jet prolongation

r

J r E.

Proof. Similarly to 14.6, every action : G — S ’ S determines the functor L

on PBm (G), P ’ P [S, ] and • ’ {•, idS }, with values in the category of

the associated bundles with standard ¬ber S and structure group G. We shall

essentially use the identi¬cation

Tm S ∼ J0 (Rm — S) ∼ J0 ((Rm — G)[S; ])

r

=r =r

r r r

j0 s ’ j0 (idRm , s) ’ j0 {ˆ, s}

(2) e

where e : Rm ’ Rm — G, e(x) = (x, e). Then the action r

ˆ ˆ becomes the form

(j r •(0, e), j0 {ˆ, s}) = j0 {ˆ, —¦ (• —¦ •’1 , s —¦ •’1 )}

r r r

(3) e e ¯ 0 0

= J r (L•)(j0 {ˆ, s}).

r

e

Now we can de¬ne a map q : W r P — Tm S ’ J r E determining the required

r

structure on J r E. Given u = j r ψ(0, e) ∈ W r P and B = j0 s ∈ Tm S, we set

r r

q(u, B) = J r (Lψ)(j0 {ˆ, s}).

r

e

Since the map ψ is a local trivialization of the principal bundle P , the restriction

r r

qu = q(u, ) : Tm S ’ Jψ0 (0) E is a di¬eomorphism. Moreover, for every A =

j r •(0, e) ∈ Wm G, formula (3) implies

r

(A’1 , B)) = J r (L(ψ —¦ •)) J r (L•’1 )(j0 {ˆ, s}) = q(u, B)

r r

q(u.A, e

and the proposition is proved.

For later purposes, let us express the corresponding map „ : W r P —M J r E ’

r

Tm S. It holds

r r

„ (u, jx s) = j0 („E —¦ (ψ —¦ e, s —¦ ψ0 ))

ˆ

where „E : P —M E ’ S is the canonical map of E and u = j r ψ(0, e) ∈ Wx P .

r

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

15. Prolongations of principal ¬ber bundles 153

15.6. First order principal prolongation. We shall point out some special

properties of the groups Wm G and the bundles W 1 P . Let us start with the group

1

Tm G. Every map s : Rm ’ G can be identi¬ed with the couple (s(0), »s(0)’1 —¦s),

r

and for a second map s : Rm ’ G we have (we recall that »a and ρa are the left

and right translations by a in G, µ is the multiplication on G)

µ —¦ (s , s)(x) = s (0)s (0)’1 s (x)s(0)s(0)’1 s(x)

(1)

= s (0)s(0) conjs(0)’1 (s (0)’1 s (x)) s(0)’1 s(x) .

It follows that Tm G is the semi direct product G J0 (Rm , G)e . This can be

r r

described easily in more details in the case r = 1. Namely, the ¬rst order jets

are identi¬ed with linear maps between the tangent spaces, so that (1) implies

Tm G = G (g — Rm— ) with the multiplication

1

(a , Z ).(a, Z) = (a a, Ad(a’1 )(Z ) + Z),

(2)

where a, a ∈ G, Z, Z ∈ Hom(Rm , g). Taking into account the decomposition

15.2.(1) and formula 15.2.(2), we get

1

(g — Rm— )

Wm G = (GL(m) — G)

with multiplication

(A , a , Z ).(A, a, Z) = (A —¦ A, a a, Ad(a’1 )(Z ) —¦ A + Z).

(3)

Now, let us view ¬bers Px M as subsets in Hom(Rm , Tx M ) and elements

1

1

in Jx P as homomorphisms in Hom(Tx M, Ty P ), y ∈ Px . Given any (u, v) ∈

P 1 M —M J 1 P = W 1 P and (A, a, Z) ∈ (G1 — G) (g — Rm— ), 15.4.(2) implies

m

(u, v)(A, a, Z) = (u —¦ A, T ρ(v, T »a —¦ Z —¦ A’1 —¦ u’1 ))

(4)

where ρ is the principal right action on P .

15.7. Principal prolongations of frame bundles. Consider the r-th prin-

cipal prolongation W r (P s M ) of the s-th order frame bundle P s M of a manifold

M . Every local di¬eomorphism • : Rm ’ M induces a principal ¬ber bundle

morphism P s • : P s Rm ’ P s M and we can construct j(0,es ) (P s •) ∈ W r (P s M ),

r

where es denotes the unit of Gs . One sees directly that this element de-

m

r+s r+s

pends on the (r + s)-jet j0 • only. Hence the map j0 • ’ j(0,es ) (P s •)

r

de¬nes an injection iM : P r+s M ’ W r (P s M ). Since the group multiplication

in both Gr+s and Wm Gs is de¬ned by the composition of jets, the restriction

r

m m

i0 : Gr+s ’ Wm Gs of iRm to the ¬bers over 0 ∈ Rm is a group homomor-

r

m m

phism. Thus, the (r + s)-order frames on a manifold M form a natural reduction

iM : P r+s M ’ W r (P s M ) of the r-th principal prolongation of the s-th order

frame bundle of M to the subgroup i0 (Gr+s ) ‚ Wm Gs .r

m m

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154 Chapter IV. Jets and natural bundles

15.8. Coordinate expression of i0 : Gr+s ’ Wm Gs . The canonical coordi-

r

m m

r+s

nates xi on Rm induce coordinates ai , 0 < |±| ¤ r + s, on Gr+s , ai (j0 f ) =

± m ±

|±| i

1‚ f rs r

±! ‚x± (0), and the following coordinates on Wm Gm : Any element j •(0, e) ∈

Wm Gs is given by j0 •0 ∈ Gr and j0 • ∈ Tm Gs , see 15.2. Let us denote the

r r r r

¯

m m m

i

coordinate expression of • by bγ (x), 0 < |γ| ¤ s, so that we have the coordi-

¯

‚ |δ| bi

1

nates bi , 0 < |γ| ¤ s, 0 ¤ |δ| ¤ r on Tm Gs , bi (j0 •) = δ! ‚xδγ (0), and the

r r

¯

m

γ,δ γ,δ

coordinates (ai ; bi ), 0 < |β| ¤ r, 0 < |γ| ¤ s, 0 ¤ |δ| ¤ r, on Wm Gs . By

r

m

β γ,δ

de¬nition, we have

i0 (ai ) = (ai ; ai ).

(1) ± β γ+δ

In the ¬rst order case, i.e. for r = 1, we have to take into account a further

s+1

structure, namely Tm Gs = Gs

1

(gs — Rm— ), cf. 15.6. So given i0 (j0 f ) =

m m m

(j0 f, j0 q), where q : Rm ’ Gs , we are looking for b = q(0) ∈ Gs and Z =

1 1

m m

s m—

T »b’1 —¦ T0 q ∈ gm — R . Let us perform this explicitly for s = 2.

In G2 we have (ai , ai )’1 = (˜i , ai ) with ai aj = δk and ai = ’˜i al as ap .

i

aj ˜jk j ˜k ˜jk al ps ˜k ˜j

m j jk

i i i i 2 i i 2

Let X = (ak , ajk , Aj , Ajk ) ∈ T Gm and b = (bk , bjk ) ∈ Gm . It is easy to compute

T »b (X) = (bi ak , bi al + bi ap as , bi Ap , bi Ap + bi Ap as + bi ap As ).

k j l jk ps j k p j p jk ps j k ps j k

Taking into account all our identi¬cations we get a formula for i0 : G3 ’ Wm G2

1

m m

i0 (ai , ai , ai ) = (ai ; ai , ai ; ai ap , ai ap + ai ap as + ai ap as ).

jk ˜p jl ˜p jkl ˜ps jl k ˜ps j kl

j jk jkl j j

If we perform the above consideration up to the ¬rst order terms only, we get

i0 : G2 ’ Wm G1 , i0 (ai , ai ) = (ai ; ai ; ai ap ).

1

j ˜p jl

m m j j

jk

16. Canonical di¬erential forms

16.1. Consider a vector bundle E = P [V, ] associated to a principal bundle

(P, p, M, G) and the space of all E-valued di¬erential forms „¦(M ; E). By theo-

rem 11.14, there is the canonical isomorphism q between „¦(M ; E) and the space

of horizontal G-equivariant V -valued di¬erential forms on P . According to 10.12,

the image ¦ = q (•) ∈ „¦k (P ; V )G is called the frame form of • ∈ „¦k (M ; E).

hor

We have

¦(X1 , . . . , Xk ) = „ (u, ) —¦ •(T pX1 , . . . , T pXk )

(1)

where Xi ∈ Tu P and „ : P —M E ’ V is the canonical map. Conversely, for

¯ ¯

every X1 , . . . , Xk ∈ Tx M , we can choose arbitrary vectors X1 , . . . , Xk ∈ Tu P

¯

with u ∈ Px and T pXi = Xi to get

¯ ¯

•(X1 , . . . , Xk ) = q(u, ) —¦ ¦(X1 , . . . , Xk )

(2)

where q : P —V ’ E is the other canonical map. The elements ¦ ∈ „¦hor (P ; V )G

are sometimes called the tensorial forms of type , while the di¬erential forms

in „¦(P ; V )G are called pseudo tensorial forms of type .

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

16. Canonical di¬erential forms 155

16.2. The canonical form on P 1 M . We de¬ne an Rm -valued one-form θ =

θM on P 1 M for every m-dimensional manifold M as follows. Given u = j0 g ∈

1

P 1 M and X = j0 c ∈ Tu P 1 M we set

1

θM (X) = u’1 —¦ T p(X) = j0 (g ’1 —¦ p —¦ c) ∈ T0 Rm = Rm .

1

In words, the choice of u ∈ P 1 M determines a local chart at x = p(u) up to the

¬rst order and the form θM transforms X ∈ Tu P 1 M into the induced coordinates

of T pX. If we insert • = idT M into 16.1.(1) we get immediately

Proposition. The canonical form θM ∈ „¦1 (P 1 M ; Rm ) is a tensorial form which

is the frame form of the 1-form idT M ∈ „¦1 (M ; T M ).

Consider further a principal connection “ on P 1 M . Then the covariant ex-

terior di¬erential d“ θM is called the torsion form of “. By 11.15, d“ θM is

identi¬ed with a section of T M — Λ2 T — M , which is called the torsion tensor of

“. If d“ θM = 0, connection “ is said to be torsion-free.

16.3. The canonical form on W 1 P . For every principal bundle (P, p, M, G)

we can generalize the above construction to an (Rm • g)-valued one-form on

W 1 P . Consider the target projection β : W 1 P ’ P , an element u = j 1 ψ(0, e) ∈

W 1 P and a tangent vector X = j0 c ∈ Tu (W 1 P ). We de¬ne the form θ = θP by

1

θ(X) = u’1 —¦ T β(X) = j0 (ψ ’1 —¦ β —¦ c) ∈ T(0,e) (Rm — G) = Rm • g.

1

Let us notice that if G = {e} is the trivial structure group, then we get P = M ,

W 1 P = P 1 M and θP = θM .

The principal action ρ on P induces an action of G on the tangent space

T P . We claim that the space of orbits T P/G is the associated vector bundle

E = W 1 P [Rm • g; ] with the left action of Wm G on T(0,e) (Rm — G) = Rm • g,

1

’1

(j 1 •(0, e), j0 c) = j0 (ρ•(0)

1 1 ¯

—¦ • —¦ c).

Indeed, every PBm (G)-morphism commutes with the principal actions, so that

is a left action which is obviously linear and the map q : W 1 P —T(0,e) (Rm —G) ’

E transforming every couple j 1 ψ(0, e) ∈ W 1 P and j0 c ∈ T(0,e) (Rm — G) into the

1

1

orbit in T P/G determined by j0 (ψ —¦ c) describes the associated bundle structure

on E.

Proposition. The canonical form θP on W 1 P is a pseudo tensorial one-form

of type .

1

Proof. We have to prove θP ∈ „¦1 (W 1 P ; Rm • g)Wm G . Let ρ and ρ be the

¯

1 1 1 1

principal actions on P and W P , X = j0 c ∈ Tu W P , u = j ψ(0, e), A =

j 1 •(0, e) ∈ Wm G, a = pr2 —¦ β(A). We have

1

β —¦ ρA = ρa —¦ β

¯

(¯A )— X = j0 (¯A —¦ c) ∈ TuA W 1 P

1

ρ ρ

θP —¦ (¯A )— X = j0 (•’1 —¦ ψ ’1 —¦ β —¦ ρA —¦ c) = j0 (ρa —¦ •’1 —¦ ψ ’1 —¦ β —¦ c).

1 1

ρ ¯

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

156 Chapter IV. Jets and natural bundles

Hence

’1

1

—¦ β —¦ c) = θP —¦ (¯A )— (X).

—¦ θP (X) = A’1 (j0 (ψ ρ

A’1

Unfortunately, θP is not horizontal since the principal bundle projection on

1

W P is p —¦ β.

16.4. Lemma. Let (P, p, M, G) be a principal bundle and q : W 1 P = P 1 M —M

J 1 P ’ P 1 M be the projection onto the ¬rst factor. Then the following diagram

u

commutes

θP

Rm • g T W 1P

u u

pr1 Tq

u θM

Rm T P 1M

Proof. Consider X = j0 c ∈ Tu W 1 P , u = j 1 ψ(0, e). Then T q(X) = j 1 (q —¦ c) and

1

1

q(u) = j0 ψ0 . It holds

’1

pr1 —¦ θP (X) = pr1 (j0 (ψ ’1 —¦ β —¦ c)) = j0 (ψ0 —¦ p —¦ β —¦ c)

1 1

’1

1

= j0 (ψ0 —¦ p —¦ q —¦ c) = θM —¦ T q(X)

¯

where p : P 1 M ’ M is the canonical projection.

¯

16.5. Canonical forms on frame bundles. Let us consider a frame bun-

dle P r M and the ¬rst order principal prolongation W 1 (P r’1 M ). We know

1 r’1

the canonical form θ ∈ „¦1 (W 1 (P r’1 M ); Rm • gr’1 )Wm Gm and the reduction

m

iM : P r M ’ W 1 (P r’1 M ) to the structure group Gr , see 15.7. So we can de¬ne

m

the canonical form θr on P r M to be the pullback i— θ ∈ „¦1 (P r M, Rm • gr’1 ).

m

M

By virtue of 16.3 there is the linear action ¯ = —¦ κ where κ is the group ho-

momorphism corresponding to iM , see 15.7, and θr is a pseudo tensorial form

of type ¯. The form θr can also be described directly. Given X ∈ Tu P r M ,

¯

we set u = πr’1 u, X = T πr’1 (X) ∈ Tu P r’1 M . Since every u = j0 f ∈ P r M

r r r

¯ ¯

determines a linear map u = T(0,e) P r’1 f : Rm • gr’1 ’ Tj r’1 f P r’1 M we get

˜ m 0

’1 ¯

r

θ (X) = u (X).

˜

16.6. Coordinate functions of sections of associated bundles. Let us

¬x an associated bundle E = P [S; ] to a principal bundle (P, p, M, G). The

canonical map „E : P —M E ’ S determines the so called frame form σ : P ’ S

of a section s : M ’ E, σ(u) = „E (u, s(p(u))). As we proved in 15.5, J r E =

W r P [Tm S; r ], m = dimM , and so for every ¬xed section s : M ’ E the frame

r

form σ r of its r-th prolongation j r s is a map σ r : W r P ’ Tm S. If we choose

r

some local coordinates (U, •), • = (y p ), on S, then there are the induced local

coordinates y± on (π0 )’1 (U ) ‚ Tm S, 0 ¤ |±| ¤ r, and for every section s : M ’

p r r

E the compositions y± —¦σ r de¬ne (on the corresponding preimages) the coordinate

p

functions ap of j r s induced by the local chart (U, •). We deduced in 15.5 that

±

for every u = j r ψ(0, e) = (j0 ψ0 , j r ψ1 (ψ0 (0))) ∈ W r P

r

σ r (u) = j0 „E (ψ1 —¦ ψ0 , s —¦ ψ0 ).

r

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

16. Canonical di¬erential forms 157

In particular, for the ¬rst order case we get

ap (u) = y p —¦ σ(u)

ap (u) = dy p (j0 „E (ψ1 —¦ ψ0 , s —¦ ψ0 ) —¦ c)

1

i

where c : R ’ Rm is the curve t ’ txi .

We shall describe the ¬rst order prolongation in more details. Let us denote ei ,

i = 1, . . . , m, the canonical basis in Rm and let e± , ± = m + 1, . . . , m + dimG, be

a linear basis of the Lie algebra g. So the canonical form θ on W 1 P decomposes

into θ = θi ei + θ± e± . Let us further write Y± for the fundamental vector ¬elds

on S determined by e± and let ω ± be the dual basis to that induced from e±

‚

p

on V P . Hence if the coordinate formulas for Y± are Y± = ·± (y) ‚yp , then for

z ∈ Ex , u ∈ Px , X ∈ Vu P , y = „E (u, z) we get

„E ( , z)— X = ’Y± (y)ω ± (X) = ’·± (y)ω ± (X) ‚yp .

p ‚

The next proposition describes the coordinate functions of j 1 s on W 1 P by

means of the canonical form θ and the coordinate functions ap of s on P .

Proposition. Let ap be the coordinate functions of a geometric object ¬eld

¯

s : M ’ E and let ap , ap be the coordinate functions of j 1 s. Then ap = ap —¦ β,

¯

i

1

where β : W P ’ P is the target projection, and

dap + ·± (aq )θ± = ap θi .

p

i

Proof. The equality ap = ap —¦ β follows directly from the de¬nition. We shall

¯

evaluate da (X) with arbitrary X ∈ Tu W 1 P , where u ∈ W 1 P , u = j 1 ψ(0, e) =

p

(j0 ψ0 , j 1 ψ1 (ψ0 (0))). The frame u determines the linear isomorphism

1

u = T(0,e) ψ : Rm • g ’ Tu P,

˜ ¯

u = β(u). We shall denote θi (X) = ξ i , θ± (X) = ξ ± , so that θ(X) = u’1 (β— X) =

¯ ˜

¯ ¯ ¯ ¯ ¯

ξ ei + ξ e± . Let us write X = β— X = X1 + X2 with X1 = u(ξ ei ), X2 = u(ξ ± e± )

i ± i

˜ ˜

i m

and let c be the curve t ’ tξ ei on R . We have

a¯

d¯p (X1 ) = dy p (j0 (σ —¦ ψ1 —¦ ψ0 —¦ c))

1

= dy p (j0 („E (ψ1 —¦ ψ0 , s —¦ ψ0 ) —¦ c)) = ap (u)ξ i

1

i

p¯ ¯ 2 ) = ’· p (aq (¯))ξ ± ‚ p .

p

d¯ (X2 ) = dy („E ( , s(p(¯)))— X

a u u ± ‚y

Hence

dap (X) = d¯p (β— X) = ap (u)θi (X) ’ ·± (aq (u))θ± (X).

p

a i

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

158 Chapter IV. Jets and natural bundles

17. Connections and the absolute di¬erentiation

17.1. Jet approach to general connections. The (general) connections on

any ¬ber bundle (Y, p, M, S) were introduced in 9.3 as the vector valued 1-forms

¦ ∈ „¦1 (Y ; V Y ) with ¦ —¦ ¦ = ¦ and Im¦ = V Y . Equivalently, any connection

is determined by the horizontal projection χ = idT Y ’ ¦, or by the horizontal

subspaces χ(Ty Y ) ‚ Ty Y in the individual tangent spaces, i.e. by the horizontal

distribution. But every horizontal subspace χ(Ty Y ) is complementary to the

vertical subspace Vy Y and therefore it is canonically identi¬ed with a unique

1 1 1 1

element jy s ∈ Jy Y . On the other hand, each jy s ∈ Jy Y determines a subspace in

Ty Y complementary to Vy Y . This leads us to the following equivalent de¬nition.

De¬nition. A (general) connection “ on a ¬ber bundle (Y, p, M ) is a section

“ : Y ’ J 1 Y of the ¬rst jet prolongation β : J 1 Y ’ Y .

Now, the horizontal lifting γ : T M —M Y ’ T Y corresponding to a connection

1

“ is given by the composition of jets, i.e. for every ξx = j0 c ∈ Tx M and y ∈ Y ,

p(y) = x, we have γ(ξx , y) = “(y) —¦ ξx . Given a vector ¬eld ξ, we get the “-

lift “ξ ∈ X(Y ), “ξ(y) = “(y) —¦ ξ(p(y)) which is a projectable vector ¬eld on

Y ’ M . Note that for every connection “ on p : Y ’ M and ξ ∈ Ty Y it holds

χ(ξ) = γ(T p(ξ), y) and ¦ = idT Y ’ χ.

Since the ¬rst jet prolongations carry a natural a¬ne structure, we can con-

sider J 1 as an a¬ne bundle functor on the category FMm,n of ¬bered manifolds

with m-dimensional bases and n-dimensional ¬bers and their local ¬bered mani-

fold isomorphisms. The corresponding vector bundle functor is V — T — B, where

B : FMm,n ’ Mfm is the base functor, see 12.11. The choice of a (general)

connection “ on p : Y ’ M yields an identi¬cation of J 1 Y ’ Y with V Y —T — M .

Chosen any ¬bered atlas •± : (Rm+n ’ Rm ) ’ (Y ’ M ) with •± (Rm+n ) = U± ,

we can use the canonical ¬‚at connection on Rm+n to get such identi¬cations on

J 1 U± . In this way we obtain the local sections γ± : U± ’ (V — T — B)(U± ) which

correspond to the Christo¬el forms introduced in 9.7. More explicitly, if we pull

back the sections γ± to Rm+n ’ Rm and use the product structure, then we

obtain exactly the Christo¬el forms.

In 9.4 we de¬ned the curvature R of a (general) connection “ by means of the

Fr¨licher-Nijenhuis bracket, 2R = [¦, ¦]. It holds R[X1 , X2 ] = ¦([χX1 , χX2 ])

o

for all vector ¬elds X1 , X2 on Y . In other words, given two vectors A1 , A2 ∈

Ty Y , we extend them to arbitrary vector ¬elds X1 and X2 on Y and we have

R(A1 , A2 ) = ¦([χX1 , χX2 ](y)). Clearly, we can take for X1 and X2 projectable

vector ¬elds over some vector ¬elds ξ1 , ξ2 on M . Then χXi = γξi , i = 1, 2. This

implies that R can be interpreted as a map R(y, ξ1 , ξ2 ) = ¦([γξ1 , γξ2 ](y)). Such

a map is identi¬ed with a section Y ’ V Y — Λ2 T — M . Obviously, the latter

formula can be rewritten as

R(y, ξ1 , ξ2 ) = [γξ1 , γξ2 ](y) ’ γ([ξ1 , ξ2 ])(y).

This relation is usually expressed by saying that the curvature is the obstruction

against lifting the bracket of vector ¬elds.

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