ńņš. 56 |

hj1 ...js (X1 , . . . , Xsā’1 ) = 0

cs = c1 ,

for all s = 3, . . . , r.

This implies that the restriction of every natural transformation T (r) ā’ T (r)

to each subcategory Mfm is a homothety with a coeļ¬cient km . Taking into

account the injection R ā’ Rm , x ā’ (x, 0, . . . , 0) we ļ¬nd km = k1 .

40.9. Remark. We remark that all natural tensors of type 1 on both T (r) M

1

and the so-called extended r-th order tangent bundle (J r (M, R))ā— are determined

in [Gancarzewicz, KolĀ“Ė, to appear].

ar

41. Star bundle functors

The tangent functor T is a covariant functor on the category Mf , but its

dual T ā— can be interpreted as a covariant functor on the subcategory Mfm of

local diļ¬eomorphisms of m-manifolds only. In this section we explain how to

treat functors with a similar kind of contravariant character like T ā— on the whole

category Mf .

41.1. The category of star bundles. Consider a ļ¬bered manifold Y ā’ M

and a smooth map f : N ā’ M . Let us recall that the induced ļ¬bered manifold

f ā— Y ā’ N is given by the pullback

w

fY

f ā—Y Y

u u

wM

f

N

The restrictions of the ļ¬bered morphism fY to individual ļ¬bers are diļ¬eomor-

phisms and we can write

f ā— Y = {(x, y); x ā N, y ā Yf (x) }, fY (x, y) = y.

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

346 Chapter IX. Bundle functors on manifolds

Clearly (f ā—¦ g)ā— Y ā¼ g ā— (f ā— Y ). Let us consider another ļ¬bered manifold Y ā’ M

=

over the same base, and a base-preserving ļ¬bered morphism Ļ• : Y ā’ Y . Given

a smooth map f : N ā’ M , by the pullback property there is a unique ļ¬bered

morphism f ā— Ļ• : f ā— Y ā’ f ā— Y such that

fY ā—¦ f ā— Ļ• = Ļ• ā—¦ fY .

(1)

The pullbacks appear in many well known constructions in diļ¬erential geome-

try. For example, given manifolds M , N and a smooth map f : M ā’ N , the

cotangent mapping T ā— f transforms every form Ļ ā Tf (x) N into T ā— f Ļ ā Tx M .

ā— ā—

Hence the mapping f ā— (T ā— N ) ā’ T ā— M is a morphism over the identity on M .

We know that the restriction of T ā— to manifolds of any ļ¬xed dimension and local

diļ¬eomorphisms is a bundle functor on Mfm , see 14.9, and it seems that the

construction could be functorial on the whole category Mf as well. However

the codomain of T ā— cannot be the category FM.

Deļ¬nition. The category FMā— of star bundles is deļ¬ned as follows. The ob-

jects coincide with those of FM, but morphisms Ļ• : (Y ā’ M ) ā’ (Y ā’ M )

are couples (Ļ•0 , Ļ•1 ) where Ļ•0 : M ā’ M is a smooth map and Ļ•1 : (Ļ•0 )ā— Y ā’ Y

is a ļ¬bered morphism over idM . The composition of morphisms is given by

(Ļ0 , Ļ1 ) ā—¦ (Ļ•0 , Ļ•1 ) = (Ļ0 ā—¦ Ļ•0 , Ļ•1 ā—¦ ((Ļ•0 )ā— Ļ1 )).

(2)

Using the formulas (1) and (2) one veriļ¬es easily that this is a correct deļ¬nition

of a category. The base functor B : FMā— ā’ Mf is deļ¬ned by B(Y ā’ M ) = M ,

B(Ļ•0 , Ļ•1 ) = Ļ•0 .

41.2. Star bundle functors. A star bundle functor on Mf is a covariant

functor F : Mf ā’ FMā— satisfying

(i) B ā—¦ F = IdMf , so that the bundle projections determine a natural trans-

formation p : F ā’ IdMf .

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

M

and F i = (i, Ļ•1 ) where Ļ•1 : iā— (F M ) ā’ F U is the canonical identiļ¬cation

iā— (F M ) ā¼ pā’1 (U ) ā‚ F M .

=M

(iii) Every smoothly parameterized family of mappings is transformed into a

smoothly parameterized one.

Given a smooth map f : M ā’ N we shall often use the same notation F f for

the second component Ļ• in F f = (f, Ļ•). We can also view the star bundle func-

tors as rules transforming any manifold M into a ļ¬ber bundle pM : F M ā’ M and

any smooth map f : M ā’ N into a base-preserving morphism F f : f ā— (F N ) ā’

F M with F (idM ) = idF M and F (g ā—¦ f ) = F f ā—¦ f ā— (F g).

41.3. The associated maps. A star bundle functor F is said to be of order r

r r

if for every maps f , g : M ā’ N and every point x ā M , the equality jx f = jx g

implies F f |(f ā— (F N ))x = F g|(g ā— (F N ))x , where we identify the ļ¬bers (f ā— (F N ))x

and (g ā— (F N ))x .

Let us consider an r-th order star bundle functor F : Mf ā’ FMā— . For every

r r

r-jet A = jx f ā Jx (M, N )y we deļ¬ne a map F A : Fy N ā’ Fx M by

F A = F f ā—¦ (fF N |(f ā— (F N ))x )ā’1 ,

(1)

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

41. Star bundle functors 347

where fF N : f ā— (F N ) ā’ F N is the canonical map. Given another r-jet B =

r r

jy g ā Jy (N, P )z , we have

F (B ā—¦ A) = F f ā—¦ (f ā— (F g)) ā—¦ (fgā— (F P ) |(f ā— g ā— (F P ))x )ā’1 ā—¦ (gF P |(g ā— (F P ))y )ā’1 .

Applying 41.1.(1) to individual ļ¬bers, we get

(fF N |(f ā— (F N ))x )ā’1 ā—¦ F g = f ā— (F g) ā—¦ (fgā— (F P ) |(f ā— g ā— (F P ))x )ā’1

and that is why

F (B ā—¦ A) = F f ā—¦ (fF N |(f ā— F N )x )ā’1 ā—¦ F g ā—¦ (gF P |(g ā— F P )y )ā’1

(2)

= F A ā—¦ F B.

For any two manifolds M , N we deļ¬ne

FM,N : F N Ć—N J r (M, N ) ā’ F M, (q, A) ā’ F A(q).

(3)

These maps are called the associated maps to F .

Proposition. The associated maps to any ļ¬nite order star bundle functor are

smooth.

Proof. This follows from the regularity and locality conditions in the way shown

in the proof of 14.4.

41.4. Description of ļ¬nite order star bundle functors. Let us consider

an r-th order star bundle functor F . We denote (Lr )op the dual category to

Lr , Sm = F0 Rm , m ā N0 , and we call the system S = {S0 , S1 , . . . } the system

of standard ļ¬bers of F , cf. 14.21. The restrictions m,n : Sn Ć— Lr m,n ā’ Sm ,

m,n (s, A) = F A(s), of the associated maps 41.3.(3) form the induced action of

(Lr )op on S. Indeed, given another jet B ā Lr (n, p) equality 41.3.(2) implies

ā—¦ A) =

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

On the other hand, let be an action of (Lr )op on a system S = {S0 , S1 , . . . }

of smooth manifolds and denote m the left actions of Gr on Sm given by

m

ā’1

m (A, s) = m,m (s, A ). We shall construct a star bundle functor L from these

r

data. We put LM := P M [Sm ; m ] for all manifolds M and similarly to 14.22 we

also get the action on morphisms. Given a map f : M ā’ N , x ā M , f (x) = y,

we deļ¬ne a map F A : Fy N ā’ Fx M ,

ā’1

F A({v, s}) = {u, ā—¦ A ā—¦ u)},

m,n (s, v

r r r

where m = dimM , n = dimN , A = jx f , v ā Py N , s ā Sn , and u ā Px M is

chosen arbitrarily. The veriļ¬cation that this is a correct deļ¬nition of smooth

maps satisfying F (B ā—¦ A) = F A ā—¦ F B is quite analogous to the considerations in

14.22 and is left to the reader. Now, we deļ¬ne Lf |(f ā— (F N ))x = F A ā—¦ fF N and

it follows directly from 41.1.(1) that L(g ā—¦ f ) = Lf ā—¦ f ā— (Lg).

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

348 Chapter IX. Bundle functors on manifolds

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

star bundle functors on Mf and the set of all smooth actions of the category

(Lr )op on systems S of smooth manifolds.

Proof. In the formulation of the theorem we identify naturally equivalent func-

tors. Given an r-th order star bundle functor F , we have the induced action

of (Lr )op on the system of standard ļ¬bers. So we can construct the functor L.

Analogously to 14.22, the associated maps deļ¬ne a natural equivalence between

F and L.

41.5 Remark. We clariļ¬ed in 14.24 that the actions of the category Lr on

systems of manifolds are in fact covariant functors Lr ā’ Mf . In the same way,

actions of (Lr )op correspond to covariant functors (Lr )op ā’ Mf or, equivalently,

to contravariant functors Lr ā’ Mf , which will also be denoted by Finf . Hence

we can summarize: r-th order bundle functors correspond to covariant smooth

functors Lr ā’ Mf while r-th order star bundle functors to the contravariant

ones.

41.6. Example. Consider a manifold Q and a point q ā Q. To any manifold M

Ī±

we associate the ļ¬bered manifold F M = J r (M, Q)q ā’ M and a map f : N ā’ M

ā’

is transformed into a map F f : f ā— (F M ) ā’ F N deļ¬ned as follows. Given a point

b ā f ā— (J r (M, Q)q ), b = (x, jf (x) g), we set F f (b) = jx (g ā—¦ f ) ā J r (N, Q)q . One

r r

veriļ¬es easily that F is a star bundle functor of order r. Let us mention the

corresponding contravariant functor Lr ā’ Mf . We have Finf (m) = J0 (Rm , Q)q r

and for arbitrary jets j0 f ā Lr , j0 g ā Finf (m) it holds Finf (j0 f )(j0 g) =

r r r r

m,n

r

j0 (g ā—¦ f ).

41.7. Vector bundle functors and vector star bundle functors. Let F

be a bundle functor or a star bundle functor on Mf . By the deļ¬nition of the

induced action and by the construction of the (covariant or contravariant) func-

tor Finf : Lr ā’ Mf , the values of the functor F belong to the subcategory of

vector bundles if and only if the functor Finf takes values in the category Vect of

ļ¬nite dimensional vector spaces and linear mappings. But using the construction

of dual objects and morphisms in the category Vect, we get a duality between

covariant and contravariant functors Finf : Lr ā’ Vect. The corresponding dual-

ity between vector bundle functors and vector star bundle functors is a source

of interesting geometric objects like r-th order tangent vectors, see 12.14 and

below.

41.8. Examples. Let us continue in example 41.6. If the manifold Q happens

to be a vector space and the point q its origin, we clearly get a vector star bundle

functor. Taking Q = R we get the r-th order cotangent functor T rā— . If we set

Q = Rk , then the corresponding star bundle functor is the functor Tk of therā—

(k, r)-covelocities, cf. 12.14.

The dual vector bundle functor to T rā— is the r-th order tangent functor. The

r

dual functor to the (k, r)-covelocities is the functor Tk , see 12.14.

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

Remarks 349

Remarks

Most of the exposition concerning the bundle functors on Mf is based on

[KolĀ“Ė, SlovĀ“k, 89], but the prolongation of Lie groups was described in [KolĀ“Ė,

ar a ar

83]. The generalization to bundle functors on FMm follows [SlovĀ“k, 91].

a

The existence of the canonical ļ¬‚ow-natural transformation F T ā’ T F was

ļ¬rst deduced by A. Kock in the framework of the so called synthetic diļ¬erential

geometry, see e.g. [Kock, 81]. His unpublished note originated in a discussion

with the ļ¬rst author. Then the latter developed, with consent of the former, the

proof of that result dealing with classical manifolds only.

The description of all natural transformations with the source in a Weil bundle

by means of some special elements in the standard ļ¬ber is a generalization of

an idea from [KolĀ“Ė, 86] due to [Mikulski, 89 b]. The natural transformations

ar

(r) (r)

T ā’T were ļ¬rst classiļ¬ed in [KolĀ“Ė, VosmanskĀ“, 89].

ar a

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

350

CHAPTER X.

PROLONGATION OF VECTOR FIELDS

AND CONNECTIONS

This section is devoted to systematic investigation of the natural operators

transforming vector ļ¬elds into vector ļ¬elds or general connections into general

connections. For the sake of simplicity we also speak on the prolongations of vec-

tor ļ¬elds and connections. We ļ¬rst determine all natural operators transforming

vector ļ¬elds on a manifold M into vector ļ¬elds on a Weil bundle over M . In the

formulation of the result as well as in the proof we use heavily the technique of

Weil algebras. Then we study the prolongations of vector ļ¬elds to the bundle

of second order tangent vectors. We like to comment the interesting general

diļ¬erences between a product-preserving functor and a non-product-preserving

one in this case. For the prolongations of projectable vector ļ¬elds to the r-jet

prolongation of a ļ¬bered manifold, which play an important role in the varia-

tional calculus, we prove that the unique natural operator, up to a multiplicative

constant, is the ļ¬‚ow operator.

Using the ļ¬‚ow-natural equivalence we construct a natural operator transform-

ing general connections on Y ā’ M into general connections on TA Y ā’ TA M

for every Weil algebra A. In the case of the tangent functor we determine all

ļ¬rst-order natural operators transforming connections on Y ā’ M into connec-

tions on T Y ā’ T M . This clariļ¬es that the above mentioned operator is not the

unique natural operator in general. Another class of problems is to study the

prolongations of connections from Y ā’ M to F Y ā’ M , where F is a functor

deļ¬ned on local isomorphisms of ļ¬bered manifolds. If we apply the idea of the

ļ¬‚ow prolongation of vector ļ¬elds, we see that such a construction depends on an

r-th order linear connection on the base manifold, provided r means the horizon-

tal order of F . In the case of the vertical tangent functor we obtain the operator

deļ¬ned in another way in chapter VII. For the functor J 1 of the ļ¬rst jet prolon-

gation of ļ¬bered manifolds we deduce that all natural operators transforming

a general connection on Y ā’ M and a linear connection on M into a general

connection on J 1 Y ā’ M form a simple 4-parameter family. In conclusion we

study the prolongation of general connections from Y ā’ M to V Y ā’ Y . From

the general point of view it is interesting that such an operator exists only in the

case of aļ¬ne bundles (with vector bundles as a special sub case). But we can

consider arbitrary connections on them (i.e. arbitrary nonlinear connections in

the vector bundle case).

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

42. Prolongations of vector ļ¬elds to Weil bundles 351

42. Prolongations of vector ļ¬elds to Weil bundles

Let F be an arbitrary natural bundle over m-manifolds. We ļ¬rst deduce

some general properties of the natural operators A : T T F , i.e. of the natural

operators transforming every vector ļ¬eld on a manifold M into a vector ļ¬eld on

F M . Starting from 42.7 we shall discuss the case that F is a Weil functor.

42.1. One general example of a natural operator T T F is the ļ¬‚ow operator

F of a natural bundle F deļ¬ned by

F (FlX )

ā‚

FM X = t

ā‚t 0

where FlX means the ļ¬‚ow of a vector ļ¬eld X on M , cf. 6.19.

The composition T F = T ā—¦ F is another bundle functor on Mfm and the

bundle projection of T is a natural transformation T F ā’ F . Assume we have

a natural transformation i : T F ā’ T F over the identity of F . Then we can

construct further natural operators T T F by using the following lemma, the

proof of which consists in a standard diagram chase.

T F is a natural operator and i : T F ā’ T F is a natural

Lemma. If A : T

transformation over the identity of F , then i ā—¦ A : T T F is also a natural

operator.

42.2. Absolute operators. This is another class of natural operators T

T F , which is related with the natural transformations F ā’ F . Let 0M be the

zero vector ļ¬eld on M .

Deļ¬nition. A natural operator A : T T F is said to be an absolute operator,

if AM X = AM 0M for every vector ļ¬eld X on M .

It is easy to check that, for every natural operator A : T T F , the operator

ā

transforming every X ā C (T M ) into AM 0M is also natural. Hence this is an

absolute operator called associated with A.

Let LM be the Liouville vector ļ¬eld on T M , i.e. the vector ļ¬eld generated by

the one-parameter group of all homotheties of the vector bundle T M ā’ M . The

rule transforming every vector ļ¬eld on M into LM is the simplest example of

an absolute operator in the case F = T . The naturality of this operator follows

from the fact that every homothety is a natural transformation T ā’ T . Such a

construction can be generalized. Let Ļ•(t) be a smooth one-parameter family of

natural transformations F ā’ F with Ļ•(0) = id, where smoothness means that

the map (Ļ•(t))M : R Ć— F M ā’ F M is smooth for every manifold M . Then

ā‚

Ī¦(M ) = (Ļ•(t))M

ā‚t 0

is a vertical vector ļ¬eld on F M . The rule X ā’ Ī¦(M ) for every X ā C ā (T M )

is an absolute operator T T F , which is said to be generated by Ļ•(t).

42.3. Lemma. For an absolute operator A : T T F every AM 0M is a vertical

vector ļ¬eld on F M .

Proof. Let J : U ā’ F M , U ā‚ R Ć— F M , be the ļ¬‚ow of AM 0M and let Jt be

its restriction for a ļ¬xed t ā R. Assume there exists W ā Fx M and t ā R such

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

352 Chapter X. Prolongation of vector ļ¬elds and connections

that pM Jt (W ) = y = x, where pM : F M ā’ M is the bundle projection. Take

f ā Diļ¬(M ) with the identity germ at x and f (y) = y, so that the restriction

of F f to Fx M is the identity. Since AM 0M is a vector ļ¬eld F f -related with

itself, we have F f ā—¦ Jt = Jt ā—¦ F f whenever both sides are deļ¬ned. In particular,

pM (F f )Jt (W ) = f pM Jt (W ) = f (y) and pM Jt (F f )(W ) = pM Jt (W ) = y,

which is a contradiction. Hence the value of AM 0M at every W ā F M is a

vertical vector.

42.4. Order estimate. It is well known that every vector ļ¬eld X on a manifold

M with non-zero value at x ā M can be expressed in a suitable local coordinate

system centered at x as the constant vector ļ¬eld

ā‚

(1) X= ā‚x1 .

This simple fact has several pleasant consequences for the study of natural oper-

ators on vector ļ¬elds. The ļ¬rst of them can be seen in the proof of the following

lemma.

r r

Lemma. Let X and Y be two vector ļ¬elds on M with X(x) = 0 and jx X = jx Y .

Then there exists a local diļ¬eomorphism f transforming X into Y such that

r+1 r+1

jx f = jx idM .

Proof. Take a local coordinate system centered at x such that (1) holds. Then

the coordinate functions Y i of Y have the form Y i = Ī“1 + g i (x) with j0 g i = 0.

i r

Consider the solution f = (f i (x)) of the following system of equations

ā‚f i (x)

Ī“1 + g i (f 1 (x), . . . , f m (x)) =

i

ā‚x1

determined by the initial condition f = id on the hyperplane x1 = 0. Then f

is a local diļ¬eomorphism transforming X into Y . We claim that the k-th order

partial derivatives of f at the origin vanish for all 1 < k ā¤ r + 1. Indeed, if

there is no derivative along the ļ¬rst axis, all the derivatives of order higher than

one vanish according to the initial condition, and all other cases follow directly

from the equations. By the same argument we ļ¬nd that the ļ¬rst order partial

derivatives of f at the origin coincide with the partial derivatives of the identity

map.

This lemma enables us to derive a simple estimate of the order of the natural

operators T TF.

42.5.Proposition. If F is an r-th order natural bundle, then the order of every

natural operator A : T T F is less than or equal to r.

r r

Proof. Assume ļ¬rst X(x) = 0 and jx X = jx Y , x ā M . Taking a local diļ¬eomor-

phism f of lemma 42.4, we have locally AM Y = (T F f ) ā—¦ AM X ā—¦ (F f )ā’1 . But

r+1 r+1

T F is an (r + 1)-st order natural bundle, so that jx f = jx idM implies that

the restriction of T F f to the ļ¬ber of T F M ā’ M over x is the identity. Hence

AM Y |Fx M = AM X|Fx M . In the case X(x) = 0 we take any vector ļ¬eld Z with

Z(x) = 0 and consider the one-parameter families of vector ļ¬elds X + tZ and

Y + tZ, t ā R. For every t = 0 we have AM (X + tZ)|Fx M = AM (Y + tZ)|Fx M

by the ļ¬rst part of the proof. Since A is regular, this relation holds for t = 0 as

well.

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

42. Prolongations of vector ļ¬elds to Weil bundles 353

42.6. Let S be the standard ļ¬ber of an r-th order bundle functor F on Mfm ,

let Z be the standard ļ¬ber of T F and let q : Z ā’ S be the canonical projection.

Further, let Vm = J0 T Rm be the space of all r-jets at zero of vector ļ¬elds on Rm

ńņš. 56 |