ńņš. 16 |

(X i , X i1 i2 , . . . , X i1 ...ir ) = (X1 , X2 , . . . , Xr ).

By the general theory, the natural transformations T (r) ā’ T (r) correspond

to Gr -equivariant maps f = (f1 , f2 , . . . , fr ) : S ā’ S. Consider ļ¬rst the equiv-

m

ariance with respect to the homotheties in GL(m) ā‚ Gr . Using (3) we obtain

m

kf1 (X1 , . . . , Xs , . . . , Xr ) = f1 (kX1 , . . . , k s Xs , . . . , k r Xr )

.

.

.

k s fs (X1 , . . . , Xs , . . . , Xr ) = fs (kX1 , . . . , k s Xs , . . . , k r Xr )

(4)

.

.

.

k r fr (X1 , . . . , Xs , . . . , Xr ) = fr (kX1 , . . . , k s Xs , . . . , k r Xr ).

By the homogeneous function theorem (see 24.1), f1 is linear in X1 and in-

dependent of X2 , . . . , Xr , while fs = gs (Xs ) + hs (X1 , . . . , Xsā’1 ), where gs is

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

41. Star bundle functors 345

linear in Xs and hs is a polynomial in X1 , . . . , Xsā’1 , 2 ā¤ s ā¤ r. Further,

the equivariancy of f with respect to the whole subgroup GL(m) implies that

gs is a GL(m)-equivariant map of the s-th symmetric tensor power S s Rm into

itself. By the invariant tensor theorem (see 24.4), gs = cs Xs (or explicitly,

g i1 ...is = cs X i1 ...is ) with cs ā R.

r

Now let us use the equivariance with respect to the kernel B1 of the jet

projection Gr ā’ GL(m), i.e. ai = Ī“j . The ļ¬rst line of (3) implies

i

m j

(5) c1 X i + ai 1 j2 (c2 X j1 j2 + hj1 j2 (X1 ))+

j

+ Ā· Ā· Ā· + ai 1 ...jr (cr X j1 ...jr + hj1 ...jr (X1 , . . . , Xrā’1 )) =

j

= c1 (X i + ai 1 j2 X j1 j2 + Ā· Ā· Ā· + ai 1 ...jr X j1 ...jr ).

j j

Setting ai 1 ...js = 0 for all s > 2, we ļ¬nd c2 = c1 and hj1 j2 (X1 ) = 0. By a

j

recurrence procedure of similar type we further deduce

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

r r

and let V0 ā‚ Vm be the subspace of r-jets of the constant vector ļ¬elds on Rm , i.e.

r

of the vector ļ¬elds invariant with respect to the translations of Rm . By 18.19

and by proposition 42.5, the natural operators A : T T F are in bijection with

r+1 r

the associated Gm -equivariant maps A : Vm Ć— S ā’ Z satisfying q ā—¦ A = pr2 .

Consider the associated maps A1 , A2 of two natural operators A1 , A2 : T TF.

r

Lemma. If two associated maps A1 , A2 : Vm Ć— S ā’ Z coincide on V0 Ć— S ā‚

r

Vm Ć— S, then A1 = A2 .

Proof. If X is a vector ļ¬eld on Rm with X(0) = 0, then there is a local diļ¬eo-

morphism transforming X into the constant vector ļ¬eld 42.1.(1). Hence if the

Gr+1 -equivariant maps A1 and A2 coincide on V0 Ć— S, they coincide on those

m

r

pairs in Vm Ć— S, the ļ¬rst component of which corresponds to an r-jet of a vector

r

ļ¬eld with non-zero value at the origin. But this is a dense subset in Vm , so that

A1 = A2 .

42.7. Absolute operators T T TB . Consider a Weil functor TB . (We

denote a Weil algebra by an unusual symbol B here, since A is taken for natural

operators.) By 35.17, for any two Weil algebras B1 and B2 there is a bijection

between the set of all algebra homomorphisms Hom(B1 , B2 ) and the set of all

natural transformations TB1 ā’ TB2 on the whole category Mf . To determine

all absolute operators T T TB , we shall need the same result for the natural

transformations TB1 ā’ TB2 on Mfm , which requires an independent proof. If

B = R Ć— N is a Weil algebra of order r, we have a canonical action of Gr onm

m m

(TB R )0 = N deļ¬ned by

r

(j0 f )(jB g) = jB (f ā—¦ g)

Assume both B1 and B2 are of order r. In 14.12 we have explained a canonical

bijection between the natural transformations TB1 ā’ TB2 on Mfm and the

Gr -maps N1 ā’ N2 . Hence it suļ¬ces to deduce

m m

m

Lemma. All Gr -maps N1 ā’ N2 are induced by algebra homomorphisms

m m

m

B1 ā’ B2 .

Proof. Let H : N1 ā’ N2 be a Gr -map. Write H = (hi (y1 , . . . , ym )) with

m m

m

yi ā N1 . The equivariance of H with respect to the homotheties in i(G1 ) ā‚ Gr m m

yields khi (y1 , . . . , ym ) = hi (ky1 , . . . , kym ), k ā R, k = 0. By the homogeneous

function theorem, all hi are linear maps. Expressing the equivariance of H

with respect to the multiplication in the direction of the i-th axis in Rm , we

obtain hj (0, . . . , yi , . . . , 0) = hj (0, . . . , kyi , . . . , 0) for j = i. This implies that

hj depends on yj only. Taking into account the exchange of the axis in Rm , we

ļ¬nd hi = h(yi ), where h is a linear map N1 ā’ N2 . On the ļ¬rst axis in Rm

consider the map x ā’ x + x2 completed by the identities on the other axes.

The equivariance of H with respect to the r-jet at zero of the latter map implies

h(y) + h(y)2 = h(y + y 2 ) = h(y) + h(y 2 ). This yields h(y 2 ) = (h(y))2 and by

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

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

polarization we obtain h(y y ) = h(y)h(ĀÆ). Hence h is an algebra homomorphism

ĀÆ y

N1 ā’ N2 , that is uniquely extended to a homomorphism B1 ā’ B2 by means of

the identity of R.

42.8. The group AutB of all algebra automorphisms of B is a closed subgroup

in GL(B), so that it is a Lie subgroup by 5.5. Every element of its Lie algebra

D ā Aut B is tangent to a one-parameter subgroup d(t) and determines a vector

ļ¬eld D(M ) tangent to (d(t))M for t = 0 on every bundle TB M . By 42.2, the

constant maps X ā’ D(M ) for all X ā C ā (T M ) form an absolute operator

op(D) : T T TB , which will be said to be generated by D.

Proposition. Every absolute operator A : T T TB is of the form A = op(D)

for a D ā Aut B.

Proof. By 42.3, AM 0M is a vertical vector ļ¬eld. Since AM 0M is F f -related with

itself for every f ā Diļ¬(M ), every transformation Jt of its ļ¬‚ow corresponds to a

natural transformation of TB into itself. By lemma 42.7 there is a one-parameter

group d(t) in AutB such that Jt = (d(t))M .

42.9. We recall that a derivation of B is a linear map D : B ā’ B satisfying

D(ab) = D(a)b + aD(b) for all a, b ā B. The set of all derivations of B is

denoted by Der B. The Lie algebra of GL(B) is the space L(B, B) of all linear

maps B ā’ B. We have Der B ā‚ L(B, B) and Aut B ā‚ GL(B).

Lemma. DerB coincides with the Lie algebra of AutB.

Proof. If ht is a one-parameter subgroup in Aut B, then its tangent vector be-

longs to Der B, since

ā‚ ā‚ ā‚ ā‚

ht (ab) = ht (a)ht (b) = ht (a) b + a ht (b) .

ā‚t 0 ā‚t 0 ā‚t 0 ā‚t 0

To prove the converse, let us consider the exponential mapping L(B, B) ā’

GL(B). For every derivation D the Leibniz formula

k

k

k

Di (a)Dkā’i (b)

D (ab) =

i

i=0

ā tk k

holds. Hence the one-parameter group ht = k=0 k! D satisļ¬es

ā k

k

tk

Di (a)Dkā’i (b)

ht (ab) = k! i

k=0 i=0

ā k

ti i tkā’i

D (a) (kā’i)! Dkā’i (b)

= i!

k=0 i=0

ļ£« ļ£¶

ā ā

tk k tj j

= k! D (a) j! D (b) = ht (a)ht (b).

ļ£ ļ£ø

j=0

k=0

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

42. Prolongations of vector ļ¬elds to Weil bundles 355

42.10. Using the theory of Weil algebras, we determine easily all natural trans-

formations T TB ā’ T TB over the identity of TB . The functor T TB corresponds

to the tensor product of algebras B ā—D of B with the algebra D of dual numbers,

which is identiļ¬ed with B Ć— B endowed with the following multiplication

(1) (a, b)(c, d) = (ac, ad + bc)

the products of the components being in B. The natural transformations of T TB

into itself over the identity of TB correspond to the endomorphisms of (1) over

the identity on the ļ¬rst factor.

Lemma. All homomorphisms of B ā— D ā¼ B Ć— B into itself over the identity on

=

the ļ¬rst factor are of the form

(2) h(a, b) = (a, cb + D(a))

with any c ā B and any D ā Der B.

Proof. On one hand, one veriļ¬es directly that every map (2) is a homomorphism.

On the other hand, consider a map h : B Ć— B ā’ B Ć— B of the form h(a, b) =

(a, f (a) + g(b)), where f , g : B ā’ B are linear maps. Then the homomorphism

condition for h requires af (c) + ag(d) + cf (a) + cg(b) = f (ac) + g(bc + ad)).

Setting b = d = 0, we obtain af (c) + cf (a) = f (ac), so that f is a derivation.

For a = d = 0 we have g(bc) = cg(b). Setting b = 1 and c = b we ļ¬nd

g(b) = g(1)b.

42.11. There is a canonical action of the elements of B on the tangent vectors

of TB M , [Morimoto, 76]. It can be introduced as follows. The multiplication of

the tangent vectors of M by reals is a map m : R Ć— T M ā’ T M . Applying the

functor TB , we obtain TB m : B Ć— TB T M ā’ TB T M . By 35.18 we have a natural

identiļ¬cation T TB M ā¼ TB T M . Then TB m can be interpreted as a map B Ć—

=

T TB M ā’ T TB M . Since the algebra multiplication in B is the TB -prolongation

of the multiplication of reals, the action of c ā B on (a1 , . . . , am , b1 , . . . , bm ) ā

T TB Rm = B 2m has the form

(1) c(a1 , . . . , am , b1 , . . . , bm ) = (a1 , . . . , am , cb1 , . . . , cbm ).

In particular this implies that for every manifold M the action of c ā B on

T TB M is a natural tensor afM (c) of type 1 on M . (The tensors of type 1

1 1

are sometimes called aļ¬nors, which justiļ¬es our notation.)

By lemma 42.1 and 42.10, if we compose the ļ¬‚ow operator TB of TB with

all natural transformations T TB ā’ T TB over the identity of TB , we obtain the

following system of natural operators T T TB

af(c) ā—¦ TB + op(D) for all c ā B and all D ā Der B.

(2)

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

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

42.12. Theorem. All natural operators T T TB are of the form 42.11.(2).

Proof. The standard ļ¬bers in the sense of 42.6 are S = N m and Z = N m Ć— B m .

Let A : Vm Ć— N m ā’ N m Ć— B m be the associated map of a natural operator

r

T TB and let A0 = A|V0 Ć— N m . Write y ā N , (X, Y ) ā B = R Ć— N and

A: T

(vi ) ā V0 , so that vi ā R. Then the coordinate expression of A0 has the form

yi = yi and

Xi = fi (vi , yi ), Yi = gi (vi , yi )

Taking into account the inclusion i(G1 ) ā‚ Gr+1 , one veriļ¬es directly that V0

m m

1 r

is a Gm -invariant subspace in Vm . If we study the equivariance of (fi , gi ) with

respect to G1 , we deduce in the same way as in the proof of lemma 42.7

m

(1) Xi = f (yi ) + kvi , Yi = g(yi ) + h(vi )

where f : N ā’ R, g : N ā’ N , h : R ā’ N are linear maps and k ā R.

Setting vi = 0 in (1), we obtain the coordinate expression of the absolute

operator associated with A in the sense of 42.2. By proposition 42.8 and lemmas

42.3 and 42.9, f = 0 and g is a derivation in N , which is uniquely extended into

a derivation DA in B by requiring DA (1) = 0. On the other hand, h(1) ā N , so

that cA = k + h(1) is an element of B.

Consider the natural transformation HA : T TB ā’ T TB determined by cA and

DA in the sense of lemma 42.10. Since the ļ¬‚ow of every constant vector ļ¬eld on

Rm is formed by the translations, its TB -prolongation on TB Rm = Rm Ć— N m is

formed by the products of the translations on Rm and the identity map on N m .

This implies that A and the associated map of HA ā—¦ TB coincide on V0 Ć— N m .

Applying lemma 42.6, we prove our assertion.

r

42.13. Example. In the special case of the functor T1 of 1-dimensional veloc-

ities of arbitrary order r, which is used in the geometric approach to higher

order mechanics, we interpret our result in a direct geometric way. Given

some local coordinates xi on M , the r-th order Taylor expansion of a curve

xi (t) determines the induced coordinates y1 , . . . , yr on T1 M . Let X i = dxi ,

i i r

Y1i = dy1 , . . . , Yri = dyr be the additional coordinates on T T1 M . The element

i i r

x+ xr+1 ā R[x]/ xr+1 deļ¬nes a natural tensor afM (x+ xr+1 ) =: QM of type

1 r i i i i

1 on T1 M , the coordinate expression of which is QM (X , Y1 , Y2 , . . . , Yr ) =

(0, X i , Y1i , . . . , Yrā’1 ). We remark that this tensor was introduced in another way

i

by [de LeĀ“n, Rodriguez, 88]. The reparametrization xi (t) ā’ xi (kt), 0 = k ā R,

o

r

induces a one-parameter group of diļ¬eomorphisms of T1 M that generates the

r

so called generalized Liouville vector ļ¬eld LM on T1 M with the coordinate ex-

pression X i = 0, Ysi = sys , s = 1, . . . , r. This gives rise to an absolute operator

i

r

L: T T T1 . If we ā˜translateā™ theorem 42.12 from the language of Weil algebras,

r

we deduce that all natural operators T T T1 form a (2r + 1)-parameter family

linearly generated by the following operators

T1r , Q ā—¦ T1r , . . . , Qr ā—¦ T1r , L, Q ā—¦ L, . . . , Qrā’1 ā—¦ L.

For r = 1, i.e. if we have the classical tangent functor T , we obtain a 3-

parameter family generated by the ļ¬‚ow operator T , by the so-called vertical lift

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

43. The case of the second order tangent vectors 357

Q ā—¦ T and by the classical Liouville ļ¬eld on T M . (The vertical lift transforms

every section X : M ā’ T M into a vertical vector ļ¬eld on T M determined by the

translations in the individual ļ¬bers of T M .) The latter result was deduced by

[Sekizawa, 88a] by the method of diļ¬erential equations and under an additional

assumption on the order of the operators.

r

42.14. Remark. The natural operators T T Tk were studied from a slightly

diļ¬erent point of view by [Gancarzewicz, 83a]. He has assumed in addition

that all maps AM : C ā (T M ) ā’ C ā (T Tk M ) are R-linear and that every AM X,

r

X ā C ā (T M ) is a projectable vector ļ¬eld on Tk M . He has determined and

r

described geometrically all such operators. Of course, they are of the form

af(c)ā—¦Tkr , for all c ā Dr . It is interesting to remark that from the list 42.11.(2) we

k

know that for every natural operator A : T T TB every AM X is a projectable

vector ļ¬eld on TB M . The description of the absolute operators in the case of

r r r

the functor Tk is very simple, since all natural equivalences Tk ā’ Tk correspond

to the elements of Gr acting on the velocities by reparametrization. We also

k

1

remark that for r = 1 JanyĖka determined all natural operators T

s T Tk by

direct evaluation, [Krupka, JanyĖka, 90].

s

43. The case of the second order tangent vectors

Theorem 42.12 implies that the natural operators transforming vector ļ¬elds

to product preserving bundle functors have several nice properties. Some of

them are caused by the functorial character of the Weil algebras in question. It

is useful to clarify that for the non-product-preserving functors on Mf one can

meet a quite diļ¬erent situation. As a concrete example we discuss the second

order tangent vectors deļ¬ned in 12.14. We ļ¬rst deduce that all natural operators

T T (2) form a 4-parameter family. Then we comment its most signiļ¬cant

T

properties which diļ¬er from the product-preserving case.

43.1. Since T (2) is a functor with values in the category of vector bundles, the

multiplication of vectors by real numbers determines the Liouville vector ļ¬eld

LM on every T (2) M . Clearly, X ā’ LM , X ā C ā (T M ) is an absolute operator

T T (2) . Further, we have a canonical inclusion T M ā‚ T (2) M . Using

T

the ļ¬ber translations on T (2) M , we can extend every section X : M ā’ T M

into a vector ļ¬eld V (X) on T (2) M . This deļ¬nes a second natural operator

T T (2) . Moreover, if we iterate the derivative X(Xf ) of a function

V:T

f : M ā’ R with respect to a vector ļ¬eld X on M , we obtain, at every point

(2)

2ā—

x ā M , a linear map from (T1 M )x into the reals, i.e. an element of Tx M .

This determines a ļ¬rst order operator C ā (T M ) ā’ C ā (T (2) M ), the coordinate

form of which is

i 2

X i ā‚xi ā’ X j ā‚Xj + X i X j ā‚xā‚ā‚xj

ā‚ ā‚

(1) ā‚xi i

ā‚x

Since every section of the vector bundle T (2) can be extended, by means of ļ¬ber

translations, into a vector ļ¬eld constant on each ļ¬ber, we get from (1) another

T T (2) . Finally, T (2) means the ļ¬‚ow operator as usual.

natural operator D : T

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

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

T T (2) form the 4-parameter

43.2. Proposition. All natural operators T

family

k1 T (2) + k2 V + k3 L + k4 D, k1 , k2 , k3 , k4 ā R.

(1)

T T (2) has order

Proof. By proposition 42.5, every natural operator A : T

(2)

ā¤ 2. Let Vm = J0 (T Rm ), S = T0 Rm , Z = (T T (2) )0 Rm and q : Z ā’ S

2 2

be the canonical projection. We have to determine all G3 -equivariant maps

m

f : Vm Ć— S ā’ Z satisfying q ā—¦ f = pr2 . The action of G3 on Vm is

2 2

m

ĀÆ ĀÆi

X i = ai X j , Xj = ai ak X l + ai Xlk al

(2) kl Ėj Ėj

j k

i

while for Xjk we shall need the action

ĀÆi

Xjk = Xjk + ai X l

i

(3) jkl

of the kernel K3 of the jet projection G3 ā’ G2 only. The action of G2 on S is

m m m

uij = ai aj ukl ,

ui = ai uj + ai ujk ,

(4) ĀÆ ĀÆ

j jk kl

see 40.8.(2). The induced coordinates on Z are Y i = dxi , U i = dui , U ij = duij ,

and (4) implies

ĀÆ

Y i =ai Y j

j

ĀÆ

U i =ai uj Y k + ai U j + ai Y l ujk + ai U jk

(5) jk j jkl jk

U ij =ai aj ukl Y m + ai aj ukl Y m + ai aj U kl .

ĀÆ km k k

l lm l

Using (4) we ļ¬nd the following coordinate expression of the ļ¬‚ow operator T (2)

j

X i ā‚xi + Xj uj + Xjk ujk

i i

+ Xk ukj + Xk uik

i

ā‚ ā‚ ā‚

(6) ā‚uij .

ā‚ui

Consider the ļ¬rst series of components

Y i = f i (X j , Xlk , Xnp , uq , urs )

m

of the associated map of A. The equivariance of f i with respect to the kernel

K3 reads

f i (X j , Xlk , Xnp , uq , urs ) = f i (X j , Xlk , Xnp + am X t , uq , urs ).

m m

npt

This implies that f i are independent of Xjk . Then the equivariance with respect

i

to the subgroup ai = Ī“j yields

i

j

f i (X j , Xlk , um , unp ) = f i (X j , Xlk + ak X q , um + am urs , unp ).

lq rs

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

43. The case of the second order tangent vectors 359

This gives f i = f i (X j , ukl ). Using the homotheties in i(G1 ) ā‚ G3 , we obtain

m m

f i = f i (X j ). Example 24.14 then implies

Y i = kX i .

(7)

Consider further the diļ¬erence A ā’ kT (2) with k taken from (7) and denote

by hi , hij its components. We evaluate easily

ai aj hkl (X m , Xp , Xrs , ut , uuv ) = hij (X m , Xp , Xrs , ut , uuv ).

ĀÆ ĀÆn ĀÆq ĀÆ ĀÆ

n q

(8) kl

Quite similarly as in the ļ¬rst step we deduce hij = hij (X k , ulm ). By homogene-

ity and the invariant tensor theorem, we then obtain

hij = cuij + aX i X j .

(9)

For hi , we ļ¬nd

(10) ai hj (X k , Xm , Xpq , ur , ust ) + cai ujk + aai X j X k =

l n

j jk jk

ĀÆ ĀÆl ĀÆn ĀÆ ĀÆ

= hi (X k , Xm , Xpq , ur , ust ).

By (3), hi is independent of Xjk . Then the homogeneity condition implies

i

hi = fj (Xlk )X j + gj (Xlk )uj .

i i

(11)

For X i = 0, the equivariance of (11) with respect to the subgroup ai = Ī“j reads

i

j

gj (Xlk )uj + cai ujk = gj (Xlk )(uj + aj ukl ).

i i

(12) jk kl

Hence gj (Xlk ) = cĪ“j . The remaining equivariance condition is

i i

fj (Xlk )X j + aai X j X k = fj (Xlk + ak X m )X j .

i i

(13) jk lm

This implies that all the ļ¬rst order partial derivatives of fj (Xlk ) are constant, so

i

that fj are at most linear in Xlk . By the invariant tensor theorem, fj (Xlk )X j =

i i

eX j Xj + bX i . Then (13) yields e = a, i.e.

i

hi = cui + bX i + aX j Xj .

i

(14)

This gives the coordinate expression of (1).

43.3. Remark. For a Weil functor TB , all natural operators T T TB are of

the form H ā—¦ TB , where H is a natural transformation T TB ā’ T TB over the

identity of TB . For T (2) , one evaluates easily that all natural transformations

H : T T (2) ā’ T T (2) over the identity of T (2) form the following 3-parameter

family

Y i =k1 Y i ,

U i =k1 U i + k2 Y i + k3 ui ,

U ij =k1 U ij + k3 uij ,

see [Doupovec, 90]. Hence the operators of the form H ā—¦T (2) form a 3-parameter

family only, in which the operator D is not included.

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

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

43.4. Remark. In the case of Weil bundles, theorem 42.12 implies that the

diļ¬erence between a natural operator T T TB and its associated absolute

operator is a linear operator. This is no more true for the non-product-preserving

functors, where the operator D is the simplest counter-example.

43.5. Remark. The operators T (2) , V and L transform every vector ļ¬eld on a

manifold M into a vector ļ¬eld on T (2) M tangent to the subbundle T M ā‚ T (2) M ,

but D does not. With a little surprise we can express it by saying that the

T T (2) is not compatible with the natural inclusion

natural operator D : T

T M ā‚ T (2) M .

43.6. Remark. Recently [Mikulski, to appear b], has solved the general prob-

T T (r) , r ā N. All such operators

lem of determining all natural operators T

form an (r + 2)-parameter family linearly generated by the ļ¬‚ow operator, by the

Liouville vector ļ¬eld of T (r) and by the analogies of the operator D from 43.1

deļ¬ned by f ā’ X Ā· Ā· Ā· X f , k = 1, . . . , r.

k-times

44. Induced vector ļ¬elds on jet bundles

44.1. Let F be a bundle functor on FMm,n . The idea of the ļ¬‚ow prolongation

of vector ļ¬elds can be applied to the projectable vector ļ¬elds on every object

p : Y ā’ M of FMm,n . The ļ¬‚ow FlĪ· of a projectable vector ļ¬eld Ī· on Y is

t

formed by the local isomorphisms of Y and we deļ¬ne the ļ¬‚ow operator F of F

by

FY Ī· = ā‚t 0 F (FlĪ· ).

ā‚

t

The general concept of a natural operator A transforming every projectable

vector ļ¬eld on Y ā ObFMm,n into a vector ļ¬eld on F Y was introduced in

section 18. We shall denote such an operator brieļ¬‚y by A : Tproj TF.

44.2. Lemma. If F is an r-th order bundle functor on FMm,n , then the order

T F is ā¤ r.

of every natural operator Tproj

Proof. This is quite similar to 42.5, see [KolĀ“Ė, SlovĀ“k, 90] for the details.

ar a

44.3. We shall discuss the case F is the functor J r of the r-th jet prolongation

of ļ¬bered manifolds. We remark that a simple evaluation leads to the following

coordinate formula for J 1 Ī·

ā‚Ī· p ā‚Ī· p q ā‚Ī· j p

J 1 Ī· = Ī· i ā‚xi + Ī· p ā‚yp +

ā‚ ā‚ ā‚

ā’

+ ā‚y q yi ā‚xi yj p

ā‚xi ā‚yi

ā‚ ā‚

provided Ī· = Ī· i (x) ā‚xi + Ī· p (x, y) ā‚yp , see [Krupka, 84]. To evaluate J r Ī·, we

have to iterate this formula and use the canonical inclusion J r (Y ā’ M ) ā’

J 1 (J rā’1 (Y ā’ M ) ā’ M ).

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

44. Induced vector ļ¬elds on jet bundles 361

T J r is a constant multiple of

Proposition. Every natural operator A : Tproj

the ļ¬‚ow operator J r .

Proof. Let V r be the space of all r-jets of the projectable vector ļ¬elds on

Rn+m ā’ Rm with source 0 ā Rm+n , let V 0 ā‚ V r be the space of all r-jets

of the constant vector ļ¬elds and V0 ā‚ V 0 be the subset of all vector ļ¬elds with

zero component in Rn . Further, let S r or Z r be the ļ¬ber of J r (Rm+n ā’ Rm )

or T J r (Rm+n ā’ Rm ) over 0 ā Rm+n , respectively. By lemma 44.2 and by the

general theory, we have to determine all Gr+1 -maps A : V r Ć— S r ā’ Z r over the

m,n

r

identity of S . Analogously to section 42, every projectable vector ļ¬eld on Y

with non-zero projection to the base manifold can locally be transformed into

ā‚

the vector ļ¬eld ā‚x1 . Hence A is determined by its restriction A0 to V0 Ć— S r .

However, in the ļ¬rst part of the proof we have to consider the restriction A0 of

A to V 0 Ć— S r for technical reasons.

Having the canonical coordinates xi and y p on Rm+n , let X i , Y p be the

induced coordinates on V 0 , let yĪ± , 1 ā¤ |Ī±| ā¤ r, be the induced coordinates on

p

S r and Z i = dxi , Z p = dy p , ZĪ± = dyĪ± be the additional coordinates on Z r . The

p p

restriction A0 is given by some functions

Z i = f i (X j , Y q , yĪ² )

s

Z p = f p (X i , Y q , yĪ² )

s

ZĪ± = fĪ± (X i , Y q , yĪ² ).

p p s

Let us denote by g i , g p , gĪ± the restrictions of the corresponding f ā™s to V0 Ć— S r .

p

The ļ¬‚ows of constant vector ļ¬elds are formed by translations, so that their r-jet

prolongations are the induced translations of J r (Rm+n ā’ Rm ) identical on the

ā‚ ā‚

standard ļ¬ber. Therefore J r ā‚x1 = ā‚x1 and it suļ¬ces to prove

g i = kX i , g p = 0, p

gĪ± = 0.

We shall proceed by induction on the order r. It is easy to see that the ac-

tion of i(G1 Ć— G1 ) ā‚ Gr+1 on all quantities is tensorial. Consider the case

m n m,n

r = 1. Using the equivariance with respect to the homotheties in i(G1 ), we n

q q

obtain f i (X j , Y p , yl ) = f i (X j , kY p , kyl ), so that f i depends on X i only. Then

the equivariance of f i with respect to i(G1 ) yields f i = kX i by 24.7. The equiv-

m

ariance of f with respect to the homotheties in i(G1 ) gives kf p (X i , Y q , yj ) =

p s

n

q

f p (X i , kY q , kyj ). This kind of homogeneity implies f p = hp (X i )Y q + hpj (X i )yj

s

q q

with some smooth functions hp , hpj . Using the homotheties in i(G1 ), we fur-

q q m

ther obtain hp (kX) = hp (X) and hpj (kX) = khpj (X). Hence hp = const

q q q q q

pj i

and hq is linear in X . Then the generalized invariant tensor theorem yields

f p = aY p + byi X i , a, b ā R. Applying the same procedure to fip , we ļ¬nd

p

fip = cyi , c ā R.

p

Consider the injection G2 ā’ G2 m,n determined by the products with the

n

identities on R . The action of an element (ap , ar ) of the latter subgroup is

m

q st

given by

ĀÆp q

yi = ap yi

(2) q

Zi = ap yi Z t + ap Zi

ĀÆp q q

(3) qt q

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

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

and V0 is an invariant subspace. In particular, (3) with ap = Ī“q gives an equiv-

p

q

ariance condition

cyi = bap yi yj X j + cyi .

p qt p

qt

This yields b = 0, so that g p = 0. Further, the subspace V0 is invariant with the

respect to the inclusion of G1 into G2 . The equivariance of fip with respect

m,n m,n

to an element (Ī“j , Ī“q , ap ) ā G1 means cyi = c(yi + ap ). Hence c = 0, which

p p

ip

m,n

i i

completes the proof for r = 1.

For r ā„ 2 it suļ¬ces to discuss the gā™s only. Using the homotheties in i(G1 ),

n

p q q

j

we ļ¬nd that gi1 Ā·Ā·Ā·is (X , yĪ² ), 1 ā¤ |Ī²| ā¤ r, is linear in yĪ² . The homotheties in

i(G1 ) and the generalized invariant tensor theorem then yield

m

gi1 Ā·Ā·Ā·is = Wip Ā·Ā·Ā·is + cs yi1 Ā·Ā·Ā·is is+1 Ā·Ā·Ā·ir X is+1 . . . X ir

p p

(4) 1

where Wip Ā·Ā·Ā·is do not depend on yi1 Ā·Ā·Ā·ir , s = 1, . . . , r ā’ 1, and

p

1

p p

(5) gi1 Ā·Ā·Ā·ir = cr yi1 Ā·Ā·Ā·ir

p p

g p = b1 yi X i + Ā· Ā· Ā· + br yi1 Ā·Ā·Ā·ir X i1 . . . X ir .

(6)

Similarly to the ļ¬rst order case, we have an inclusion Gr+1 ā’ Gr+1 determined

n m,n

by the products of diļ¬eomorphisms on Rn with the identity of Rm . One ļ¬nds

easily the following transformation law

yi1 Ā·Ā·Ā·is = ap yi1 Ā·Ā·Ā·is + Fip Ā·Ā·Ā·is + ap1 Ā·Ā·Ā·qs yi1 . . . yis

ĀÆp q q1 qs

(7) q q

1

where Fip Ā·Ā·Ā·is is a polynomial expression linear in ap with 2 ā¤ |Ī±| ā¤ s ā’ 1 and

Ī±

1

p

independent of yi1 Ā·Ā·Ā·is . This implies

Zi1 Ā·Ā·Ā·is = ap Zi1 Ā·Ā·Ā·is + Gp1 Ā·Ā·Ā·is + ap1 Ā·Ā·Ā·qs qs+1 yi1 . . . yis Z qs+1

ĀÆp q q1 qs

(8) q q

i

where Gp1 Ā·Ā·Ā·is is a polynomial expression linear in ap with 2 ā¤ |Ī±| ā¤ s and linear

Ī±

i

p

in ZĪ± , 0 ā¤ |Ī±| ā¤ s ā’ 1.

p p

We deduce that every gi1 Ā·Ā·Ā·is , 0 ā¤ s ā¤ r ā’ 1 , is independent of yi1 Ā·Ā·Ā·ir . On the

kernel of the jet projection Gr+1 ā’ Gr , (8) for r = s gives

n n

q1 qr

0 = ap1 Ā·Ā·Ā·qr qr+1 yi1 . . . yir g qr+1 .

q

Hence g p = 0. On the kernel of the jet projection Gr ā’ Grā’1 , (8) with s =

n n

1, . . . , r ā’ 1, implies

q1 qr

0 = cs ap1 ...qr yi1 . . . yir X is+1 . . . X ir ,

q

i.e. cs = 0. By projectability, g i and gĪ± , 0 ā¤ |Ī±| ā¤ r ā’ 1, correspond to a

p

Gr -equivariant map V0 Ć— S rā’1 ā’ Z rā’1 . By the induction hypothesis, gĪ± = 0

p

m,n

for all 0 ā¤ |Ī±| ā¤ r ā’ 1. Then on the kernel of the jet projection Gr+1 ā’ Grā’1

n n

q1 qr p

(8) gives 0 = cr ap1 ...qr yi1 . . . yir , i.e. gi1 Ā·Ā·Ā·ir = 0.

q

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

45. Prolongations of connections to F Y ā’ M 363

r

44.4. Bundles of contact elements. Consider the bundle functor Kn on

Mfm of the n-dimensional contact elements of order r deļ¬ned in 12.15.

r

Proposition. Every natural operator A : T T Kn is a constant multiple of

r

the ļ¬‚ow operator Kn .

Proof. It suļ¬ces to discuss the case M = Rm . Consider the canonical ļ¬bration

Rm = Rn Ć— Rmā’n ā’ Rn . As remarked at the end of 12.16, there is an identiļ¬-

cation of an open dense subset in Kn Rm with J r (Rm ā’ Rn ). By deļ¬nition, on

r

this subset it holds J r Ī¾ = Kn Ī¾ for every projectable vector ļ¬eld Ī¾ on Rm ā’ Rn .

r

Since the operator A commutes with the action of all diļ¬eomorphisms preserving

ā‚ ā‚

ļ¬bration Rm ā’ Rn , the restriction of A to ā‚x1 is a constant multiple of Kn ( ā‚x1 )

r

by proposition 44.3. But every vector ļ¬eld on Rm can be locally transformed

ā‚

into ā‚x1 in a neighborhood of any point where it does not vanish.

We ļ¬nd it interesting that we have ļ¬nished our investigation of the basic

properties of the natural operators T T F for diļ¬erent bundle functors on

Mfm by an example in which the constant multiples of the ļ¬‚ow operator are

the only natural operators T TF.

T T ā— and

44.5. Remark. [Kobak, 91] determined all natural operators T

T (T T ā— ) for manifolds of dimension at least two. Let T ā— be the ļ¬‚ow operator

T

of the cotangent bundle, LM : T ā— M ā’ T T ā— M be the vector ļ¬eld generated by

the homotheties of the vector bundle T ā— M and ĻM : T M Ć—M T ā— M ā’ R be the

T T ā— are of the form f (Ļ)T ā— +

evaluation map. Then all natural operators T

g(Ļ)L, where f , g ā C ā (R, R) are any smooth functions of one variable. In

the case F = T T ā— the result is of similar character, but the complete list is

somewhat longer, so that we refer the reader to the above mentioned paper.

45. Prolongations of connections to F Y ā’ M

45.1. In 31.1 we deduced that there is exactly one natural operator transforming

every general connection on Y ā’ M into a general connection on V Y ā’ M .

However, one meets a quite diļ¬erent situation when replacing ļ¬bered manifold

V Y ā’ M e.g. by the ļ¬rst jet prolongation J 1 Y ā’ M of Y . Pohl has observed in

the vector bundle case, [Pohl, 66], that one needs an auxiliary linear connection

on the base manifold M to construct an induced connection on J 1 Y ā’ M . Our

ļ¬rst goal is to clarify this diļ¬erence from the conceptual point of view.

45.2. Bundle functors of order (r, s). We recall that two maps f , g of a

ļ¬bered manifold p : Y ā’ M into another manifold determine the same (r, s)-jet

r,s r,s r r

jy f = jy g at y ā Y , s ā„ r, if jy f = jy g and the restrictions of f and g to the

s s

ļ¬ber Yp(y) satisfy jy (f |Yp(y) ) = jy (g|Yp(y) ), see 12.19.

Deļ¬nition. A bundle functor on a category C over FM is said to be of order

ĀÆ

(r, s), if for any two C-morphisms f , g of Y into Y

r,s r,s

jy f = jy g implies (F f )|(F Y )y = (F g)|(F Y )y .

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

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

For example, the order of the vertical functor V is (0, 1), while the functor of

the ļ¬rst jet prolongation J 1 has order (1, 1).

45.3. Denote by J r,s T Y the space of all (r, s)-jets of the projectable vector

ļ¬elds on Y ā’ M . This is a vector bundle over Y . Let F be a bundle functor on

FMm,n and F denote its ļ¬‚ow operator Tproj TF.

Proposition. If the order of F is (r, s) and Ī· is a projectable vector ļ¬eld on Y ,

r,s

then the value (FĪ·)(u) at every u ā (F Y )y depends only on jy Ī·. The induced

map

F Y ā• J r,s T Y ā’ T (F Y )

is smooth and linear with respect to J r,s T Y .

Proof. Smoothness can be proved in the same way as in 14.14. Linearity follows

directly from the linearity of the ļ¬‚ow operator F.

45.4. Let Ī“ be a general connection on p : Y ā’ M . Considering the Ī“-lift

r,s r

Ī“Ī¾ of a vector ļ¬eld Ī¾ on M , one sees directly that jy Ī“Ī¾ depends on jp(y) Ī¾

only, y ā Y . Let F be a bundle functor on FMm,n of order (r, s). If we

combine the map of proposition 45.3 with the lifting map of Ī“, we obtain a

map F Ī“ : F Y ā• J r T M ā’ T F Y linear in J r T M . Let Ī : T M ā’ J r T M be

an r-th order linear connection on M , i.e. a linear splitting of the projection

Ļ0 : J r T M ā’ T M . By linearity, the composition

r

F Ī“ ā—¦ (idF Y ā• Ī) : F Y ā• T M ā’ T F Y

(1)

is a lifting map of a general connection on F Y ā’ M .

Deļ¬nition. The general connection F(Ī“, Ī) on F Y ā’ M with lifting map (1)

is called the F -prolongation of Ī“ with respect to Ī.

If the order of F is (0, s), we need no connection Ī on M . In particular, every

connection Ī“ on Y ā’ M induces in such a way a connection VĪ“ on V Y ā’ M ,

which was already mentioned in remark 31.4.

45.5. We show that the construction of F(Ī“, Ī) behaves well with respect to

ĀÆ

morphisms of connections. Given an FM-morphism f : Y ā’ Y over f0 : M ā’

ĀÆ ĀÆ ĀÆĀÆ ĀÆ

M and two general connections Ī“ on p : Y ā’ M and Ī“ on p : Y ā’ M , one sees

ĀÆ

easily that Ī“ and Ī“ are f -related in the sense of 8.15 if and only if the following

diagram commutes

wu

u

Tf ĀÆ

TY TY

ĀÆ

Ī“ Ī“

w

f ā• T f0 ĀÆ ĀÆ

Y ā• TM Y ā• TM

ĀÆ

In such a case f is also called a connection morphism of Ī“ into Ī“. Further, two

ĀÆ ĀÆ ĀÆ

r-th order linear connections Ī : T M ā’ J T M and Ī : T M ā’ J r T M are called

r

f0 -related, if for every z ā Tx M it holds

ĀÆ r r

Ī(T f0 (z)) ā—¦ (jx f0 ) = (jz T f0 ) ā—¦ Ī(z).

Let F be as in 45.4.

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

45. Prolongations of connections to F Y ā’ M 365

ĀÆ ĀÆ

Proposition. If Ī“ and Ī“ are f -related and Ī and Ī are f0 -related, then F(Ī“, Ī)

ĀÆĀÆ

and F(Ī“, Ī) are F f -related.

Proof. The lifting map of F(Ī“, Ī) can be determined as follows. For every

r

X ā Tx M we take a vector ļ¬eld Ī¾ on M such that jx Ī¾ = Ī(X) and we construct

its Ī“-lift Ī“Ī¾. Then F(Ī“, Ī)(u) is the value of the ļ¬‚ow prolongation F(Ī“Ī¾) at

rĀÆ

ĀÆ ĀÆ

u ā Fx Y . Let Ī(T f0 (X)) = jx Ī¾, x = f0 (x). If Ī and Ī are f0 -related, the

ĀÆ

ĀÆ

ĀÆ ĀÆ

vector ļ¬elds Ī¾ and Ī¾ are f0 -related up to order r at x. Since Ī“ and Ī“ are f -

ĀÆ

related, the restriction of F(Ī“Ī¾) over x and the restriction of F(Ī“Ī¾) over x are

ĀÆ

F f -related.

45.6. In many concrete cases, the connection F(Ī“, Ī) is of special kind. We are

going to deduce a general result of this type.

Let C be a category over FM, cf. 51.4. Analogously to example 1 from 18.18,

a projectable vector ļ¬eld Ī· on Y ā ObC is called a C-ļ¬eld, if its ļ¬‚ow is formed by

local C-morphisms. For example, for the category PB(G) of smooth principal G-

bundles, a projectable vector ļ¬eld Ī· on a principal ļ¬ber bundle is a PB(G)-ļ¬eld

if and only if Ī· is right-invariant. For the category VB of smooth vector bundles,

one deduces easily that a projectable vector ļ¬eld Ī· on a vector bundle E is a

VB-ļ¬eld if and only if Ī· is a linear morphism E ā’ T E, see 6.11. A connection Ī“

on (p : Y ā’ M ) ā ObC is called a C-connection, if Ī“Ī¾ is a C-ļ¬eld for every vector

ļ¬eld Ī¾ on M . Obviously, a PB(G)-connection or a VB-connection is a classical

principal or linear connection, respectively.

More generally, a projectable family of tangent vectors along a ļ¬ber Yx , i.e. a

section Ļ : Yx ā’ T Y such that T p ā—¦ Ļ is a constant map, is said to be a C-family,

if there exists a C-ļ¬eld Ī· on Y such that Ļ is the restriction of Ī· to Yx . We shall

say that the category C is inļ¬nitesimally regular, if any projectable vector ļ¬eld

on a C-object the restriction of which to each ļ¬ber is a C-family is a C-ļ¬eld.

Proposition. If F is a bundle functor of a category C over FM into an in-

ļ¬nitesimally regular category D over FM and Ī“ is a C-connection, then F(Ī“, Ī)

is a D-connection for every Ī.

Proof. By the construction that we used in the proof of proposition 45.5, the

F(Ī“, Ī)-lift of every vector X ā T M is a D-family. Since D is inļ¬nitesimally

regular, the F(Ī“, Ī)-lift of every vector ļ¬eld on T M is a D-ļ¬eld.

45.7. In the special case F = J 1 we determine all natural operators trans-

forming a general connection on Y ā’ M and a ļ¬rst order linear connection

Ī on M into a general connection on J 1 Y ā’ M . Taking into account the

rigidity of the symmetric linear connections on M deduced in 25.3, we ļ¬rst as-

sume Ī to be without torsion. Thus we are interested in the natural operators

J 1 ā• QĻ„ P 1 B J 1 (J 1 ā’ B).

On one hand, Ī“ and Ī induce the J 1 -prolongation J 1 (Ī“, Ī) of Ī“ with respect

to Ī. On the other hand, since J 1 Y is an aļ¬ne bundle with associated vec-

tor bundle V Y ā— T ā— M , the section Ī“ : Y ā’ J 1 Y determines an identiļ¬cation

IĪ“ : J 1 Y ā¼ V Y ā— T ā— M . The vertical prolongation VĪ“ of Ī“ is linear over Y , see

=

31.1.(3), so that we can construct the tensor product VĪ“ ā— Īā— with the dual

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

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

connection Īā— on T ā— M , see 47.14 and 47.15. The identiļ¬cation IĪ“ transforms

VĪ“ ā— Īā— into another connection P (Ī“, Ī) on J 1 Y ā’ M .

45.8. Proposition. All natural operators J 1 ā• QĻ„ P 1 B J 1 (J 1 ā’ B) form

the one-parameter family

tP + (1 ā’ t)J 1 , t ā R.

(1)

Proof. In usual local coordinates, let

dy p = Fip (x, y)dxi

(2)

be the equations of Ī“ and

dĪ¾ i = Īi (x)Ī¾ j dxk

(3) jk

be the equations of Ī. By direct evaluation, one ļ¬nds the equations of J 1 (Ī“, Ī)

in the form (2) and

p p

ā‚Fj ā‚Fj q

p p p

+ Īk (Fk ā’ yk ) dxj

(4) dyi = + ā‚y q yi ji

ā‚xi

while the equations of P (Ī“, Ī) have the form (2) and

p p p

ā‚Fj ā‚Fi ā‚Fi

p q

ā’ Fiq ) + q p p

ā’ Īk (yk ā’ Fk ) dxj .

(5) dyi = q (yi + q Fj ij

j

ā‚y ā‚x ā‚y

First we discuss the operators of ļ¬rst order in Ī“ and of order zero in Ī.

Let S1 = J0 (J 1 (Rn+m ā’ Rm ) ā’ Rn+m ) be the standard ļ¬ber from 27.3,

1

S0 = J0 (Rm+n ā’ Rm ), Ī = (QĻ„ P 1 Rm )0 and Z = J0 (J 1 (Rm+n ā’ Rm ) ā’ Rm ).

1 1

By using the general theory, the operators in question correspond to G2 -maps

m,n

S1 Ć— Ī Ć— S0 ā’ Z over the identity of S0 . The canonical coordinates on S1 are

p p p

yi , yiq , yij and the action of G2 is given by 27.3.(1)-(3). On S0 we have the

m,n

well known coordinates Yip and the action

Yip = ap Yjq aj + ap aj .

ĀÆ

(6) Ėi j Ėi

q

The standard coordinates on Ī are Īi = Īi and the action is

jk kj

ĀÆ

Īi = ai Īl am an + ai al am .

(7) l mn Ėj Ėk lm Ėj Ėk

jk

p p p

The induced coordinates on Z are zi , Zi , Zij and one evaluates easily that the

p p

action on both zi and Zi has form (6), while

Zij = ap Zkl ak al + ap zk Zlr ak al + ap Zlq ak al

ĀÆp q q

Ėi Ėj Ėi Ėj Ėi Ėj

q qr qk

(8)

+ ap zk ak al + ap zk ak + ap ak + ap ak al .

q q

ql Ėi Ėj Ėij k Ėij kl Ėi Ėj

q

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

45. Prolongations of connections to F Y ā’ M 367

Write Y = (Yip ), y = (yi ), y1 = (yiq ), y2 = (yij ), Ī = (Īi ). Then the

p p p

jk

coordinate form of a map f : S1 Ć— Ī Ć— S0 ā’ Z over the identity of S0 is zi = Yip

p

and

ńņš. 16 |