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CHAPTER X. PROLONGATION OF VECTOR FIELDS AND CONNECTIONS

Back

This section is devoted to systematic investigation of the natural operators

transforming vector _elds into vector _elds or general connections into general

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

_elds and connections. We _rst determine all natural operators transforming

vector _elds on a manifold M into vector _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 _elds to the bundle

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

di_erences between a product-preserving functor and a non-product-preserving

one in this case. For the prolongations of projectable vector _elds to the r-jet

prolongation of a _bered manifold, which play an important role in the variational

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

constant, is the ow operator.

Using the ow-natural equivalence we construct a natural operator transforming

general connections on Y ! M into general connections on TAY ! TAM

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

_rst-order natural operators transforming connections on Y ! M into connections

on TY ! TM. This clari_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 ! M to FY ! M, where F is a functor

de_ned on local isomorphisms of _bered manifolds. If we apply the idea of the

ow prolongation of vector _elds, we see that such a construction depends on an

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

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

de_ned in another way in chapter VII. For the functor J1 of the _rst jet prolongation

of _bered manifolds we deduce that all natural operators transforming

a general connection on Y ! M and a linear connection on M into a general

connection on J1Y ! M form a simple 4-parameter family. In conclusion we

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

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

case of a_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 _elds to Weil bundles 351

42. Prolongations of vector _elds to Weil bundles

Let F be an arbitrary natural bundle over m-manifolds. We _rst deduce

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

operators transforming every vector _eld on a manifold M into a vector _eld on

FM. 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 TF is the ow operator

F of a natural bundle F de_ned by

FMX = @

0 F(FlXt

)

where FlX means the ow of a vector _eld X on M, cf. 6.19.

The composition TF = T _ F is another bundle functor on Mfm and the

bundle projection of T is a natural transformation TF ! F. Assume we have

a natural transformation i : TF ! TF over the identity of F. Then we can

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

proof of which consists in a standard diagram chase.

Lemma. If A: T TF is a natural operator and i : TF ! TF is a natural

transformation over the identity of F, then i _ A: T TF is also a natural

operator. _

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

TF, which is related with the natural transformations F ! F. Let 0M be the

zero vector _eld on M.

De_nition. A natural operator A: T TF is said to be an absolute operator,

if AMX = AM0M for every vector _eld X on M.

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

transforming every X 2 C1(TM) into AM0M is also natural. Hence this is an

absolute operator called associated with A.

Let LM be the Liouville vector _eld on TM, i.e. the vector _eld generated by

the one-parameter group of all homotheties of the vector bundle TM ! M. The

rule transforming every vector _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 ! T. Such a

construction can be generalized. Let '(t) be a smooth one-parameter family of

natural transformations F ! F with '(0) = id, where smoothness means that

the map ('(t))M : R _ FM ! FM is smooth for every manifold M. Then

_(M) = @

0 ('(t))M

is a vertical vector _eld on FM. The rule X 7! _(M) for every X 2 C1(TM)

is an absolute operator T TF, which is said to be generated by '(t).

42.3. Lemma. For an absolute operator A: T TF every AM0M is a vertical

vector _eld on FM.

Proof. Let J : U ! FM, U _ R _ FM, be the ow of AM0M and let Jt be

its restriction for a _xed t 2 R. Assume there exists W 2 FxM and t 2 R such

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

352 Chapter X. Prolongation of vector _elds and connections

that pMJt(W) = y 6= x, where pM : FM ! M is the bundle projection. Take

f 2 Di_(M) with the identity germ at x and f(y) 6= y, so that the restriction

of Ff to FxM is the identity. Since AM0M is a vector _eld Ff-related with

itself, we have Ff _Jt = Jt _Ff whenever both sides are de_ned. In particular,

pM(Ff)Jt(W) = fpMJt(W) = f(y) and pMJt(Ff)(W) = pMJt(W) = y,

which is a contradiction. Hence the value of AM0M at every W 2 FM is a

vertical vector. _

42.4. Order estimate. It is well known that every vector _eld X on a manifold

M with non-zero value at x 2 M can be expressed in a suitable local coordinate

system centered at x as the constant vector _eld

(1) X = @

@x1 :

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

on vector _elds. The _rst of them can be seen in the proof of the following

lemma.

Lemma. Let X and Y be two vector _elds onM with X(x) 6= 0 and jrx

X = jrx

Y .

Then there exists a local di_eomorphism f transforming X into Y such that

jr+1

x f = jr+1

x 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 = _i1

+ gi(x) with jr

0gi = 0.

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

_i1

+ gi(f1(x); : : : ; fm(x)) = @fi(x)

@x1

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

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

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

there is no derivative along the _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 _nd that the _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 TF is less than or equal to r.

Proof. Assume _rst X(x) 6= 0 and jrx

X = jrx

Y , x 2 M. Taking a local di_eomorphism

f of lemma 42.4, we have locally AMY = (TFf) _ AMX _ (Ff)􀀀1. But

TF is an (r + 1)-st order natural bundle, so that jr+1

x f = jr+1

x idM implies that

the restriction of TFf to the _ber of TFM ! M over x is the identity. Hence

AMY jFxM = AMXjFxM. In the case X(x) = 0 we take any vector _eld Z with

Z(x) 6= 0 and consider the one-parameter families of vector _elds X + tZ and

Y +tZ, t 2 R. For every t 6= 0 we have AM(X +tZ)jFxM = AM(Y +tZ)jFxM

by the _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 _elds to Weil bundles 353

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

let Z be the standard _ber of TF and let q : Z ! S be the canonical projection.

Further, let V rm

= Jr

0TRm be the space of all r-jets at zero of vector _elds on Rm

and let V0 _ V rm

be the subspace of r-jets of the constant vector _elds on Rm, i.e.

of the vector _elds invariant with respect to the translations of Rm. By 18.19

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

the associated Gr+1

m -equivariant maps A: V rm

_ S ! Z satisfying q _ A = pr2.

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

Lemma. If two associated maps A1, A2 : V rm

_ S ! Z coincide on V0 _ S _

V rm

_ S, then A1 = A2.

Proof. If X is a vector _eld on Rm with X(0) 6= 0, then there is a local di_eomorphism

transforming X into the constant vector _eld 42.1.(1). Hence if the

Gr+1

m -equivariant maps A1 and A2 coincide on V0 _ S, they coincide on those

pairs in V rm

_S, the _rst component of which corresponds to an r-jet of a vector

_eld with non-zero value at the origin. But this is a dense subset in V rm

, so that

A1 = A2. _

42.7. Absolute operators T TTB. 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

! TB2 on the whole category Mf. To determine

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

transformations TB1

! TB2 on Mfm, which requires an independent proof. If

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

m on

(TBRm)0 = Nm de_ned by

(jr

0f)(jBg) = jB(f _ g)

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

bijection between the natural transformations TB1

! TB2 on Mfm and the

m-maps Nm

! Nm

2 . Hence it su_ces to deduce

Lemma. All Gr

m-maps Nm

! Nm

2 are induced by algebra homomorphisms

B1 ! B2.

Proof. Let H: Nm

! Nm

2 be a Gr

m-map. Write H = (hi(y1; : : : ; ym)) with

yi 2 N1. The equivariance of H with respect to the homotheties in i(G1

m) _ Gr

yields khi(y1; : : : ; ym) = hi(ky1; : : : ; kym), k 2 R, k 6= 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 6= i. This implies that

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

_nd hi = h(yi), where h is a linear map N1 ! N2. On the _rst axis in Rm

consider the map x 7! 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 + y2) = h(y) + h(y2). This yields h(y2) = (h(y))2 and by

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

354 Chapter X. Prolongation of vector _elds and connections

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

N1 ! N2, that is uniquely extended to a homomorphism B1 ! 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 2 AutB is tangent to a one-parameter subgroup d(t) and determines a vector

_eld D(M) tangent to (d(t))M for t = 0 on every bundle TBM. By 42.2, the

constant maps X 7! D(M) for all X 2 C1(TM) form an absolute operator

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

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

for a D 2 AutB.

Proof. By 42.3, AM0M is a vertical vector _eld. Since AM0M is Ff-related with

itself for every f 2 Di_(M), every transformation Jt of its 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 ! B satisfying

D(ab) = D(a)b + aD(b) for all a, b 2 B. The set of all derivations of B is

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

maps B ! B. We have DerB _ L(B;B) and AutB _ GL(B).

Lemma. DerB coincides with the Lie algebra of AutB.

Proof. If ht is a one-parameter subgroup in AutB, then its tangent vector belongs

to DerB, since

0 ht(ab) = @

0 ht(a)ht(b) =

􀀀 @

0 ht(a)

b + a

􀀀 @

0 ht(b)

To prove the converse, let us consider the exponential mapping L(B;B) !

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

Dk(ab) =

i=0

Di(a)Dk􀀀i(b)

holds. Hence the one-parameter group ht =

k=0

k!Dk satis_es

ht(ab) =

k=0

i=0

Di(a)Dk􀀀i(b)

k=0

i=0

i!Di(a) tk􀀀i

(k􀀀i)!Dk􀀀i(b)

k=0

k!Dk(a)

j=0

j!Dj(b)

A = ht(a)ht(b): _

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

42. Prolongations of vector _elds to Weil bundles 355

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

TTB ! TTB over the identity of TB. The functor TTB corresponds

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

which is identi_ed with B _ 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 TTB

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

the identity on the _rst factor.

Lemma. All homomorphisms of B D _= B _B into itself over the identity on

the _rst factor are of the form

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

with any c 2 B and any D 2 DerB.

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

On the other hand, consider a map h: B _ B ! B _ B of the form h(a; b) =

(a; f(a) + g(b)), where f, g : B ! 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 _nd

g(b) = g(1)b. _

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

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

the tangent vectors of M by reals is a map m: R _ TM ! TM. Applying the

functor TB, we obtain TBm: B_TBTM ! TBTM. By 35.18 we have a natural

identi_cation TTBM _= TBTM. Then TBm can be interpreted as a map B _

TTBM ! TTBM. Since the algebra multiplication in B is the TB-prolongation

of the multiplication of reals, the action of c 2 B on (a1; : : : ; am; b1; : : : ; bm) 2

TTBRm = B2m 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 2 B on

TTBM is a natural tensor afM(c) of type

􀀀1

on M. (The tensors of type

􀀀1

are sometimes called a_nors, which justi_es our notation.)

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

all natural transformations TTB ! TTB over the identity of TB, we obtain the

following system of natural operators T TTB

(2) af(c) _ TB + op(D) for all c 2 B and all D 2 DerB.

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

356 Chapter X. Prolongation of vector _elds and connections

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

Proof. The standard _bers in the sense of 42.6 are S = Nm and Z = Nm _Bm.

Let A: V rm

_ Nm ! Nm _ Bm be the associated map of a natural operator

A: T TTB and let A0 = AjV0 _Nm. Write y 2 N, (X; Y ) 2 B = R _N and

(vi) 2 V0, so that vi 2 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

m) _ Gr+1

m , one veri_es directly that V0

is a G1

m-invariant subspace in V rm

. If we study the equivariance of (fi; gi) with

respect to G1

m, we deduce in the same way as in the proof of lemma 42.7

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

where f : N ! R, g : N ! N, h: R ! N are linear maps and k 2 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) 2 N, so

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

Consider the natural transformation HA : TTB ! TTB determined by cA and

DA in the sense of lemma 42.10. Since the ow of every constant vector _eld on

Rm is formed by the translations, its TB-prolongation on TBRm = Rm _ Nm is

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

This implies that A and the associated map of HA _ TB coincide on V0 _ Nm.

Applying lemma 42.6, we prove our assertion. _

42.13. Example. In the special case of the functor Tr

1 of 1-dimensional velocities

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 yi1

; : : : ; yir

on Tr

1M. Let Xi = dxi,

Y i

1 = dyi1

; : : : ; Y i

r = dyir

be the additional coordinates on TTr

1M. The element

x+hxr+1i 2 R[x]=hxr+1i de_nes a 􀀀 natural tensor afM(x+hxr+1i) =: QM of type 1

on Tr

1M, the coordinate expression of which is QM(Xi; Y i

1 ; Y i

2 ; : : : ; Y i

r ) =

(0;Xi; Y i

1 ; : : : ; Y i

r􀀀1). We remark that this tensor was introduced in another way

by [de Le_on, Rodriguez, 88]. The reparametrization xi(t) 7! xi(kt), 0 6= k 2 R,

induces a one-parameter group of di_eomorphisms of Tr

1M that generates the

so called generalized Liouville vector _eld LM on Tr

1M with the coordinate expression

Xi = 0, Y i

s = syis

, s = 1; : : : ; r. This gives rise to an absolute operator

L: T TTr

1 . If we `translate' theorem 42.12 from the language ofWeil algebras,

we deduce that all natural operators T TTr

1 form a (2r+1)-parameter family

linearly generated by the following operators

T r

1 ; Q _ T r

1 ; : : : ; Qr _ T r

1 ; L; Q _ L; : : : ; Qr􀀀1 _ L:

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

parameter family generated by the 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 _ T and by the classical Liouville _eld on TM. (The vertical lift transforms

every section X: M ! TM into a vertical vector _eld on TM determined by the

translations in the individual _bers of TM.) The latter result was deduced by

[Sekizawa, 88a] by the method of di_erential equations and under an additional

assumption on the order of the operators.

42.14. Remark. The natural operators T TTr

k were studied from a slightly

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

that all maps AM : C1(TM) ! C1(TTr

kM) are R-linear and that every AMX,

X 2 C1(TM) is a projectable vector _eld on Tr

kM. He has determined and

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

af(c)_T r

k , for all c 2 Dr

k. It is interesting to remark that from the list 42.11.(2) we

know that for every natural operator A: T TTB every AMX is a projectable

vector _eld on TBM. The description of the absolute operators in the case of

the functor Tr

k is very simple, since all natural equivalences Tr

! Tr

k correspond

to the elements of Gr

k acting on the velocities by reparametrization. We also

remark that for r = 1 Jany_ska determined all natural operators T TT1

k by

direct evaluation, [Krupka, Jany_ska, 90].

43. The case of the second order tangent vectors

Theorem 42.12 implies that the natural operators transforming vector _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 di_erent situation. As a concrete example we discuss the second

order tangent vectors de_ned in 12.14. We _rst deduce that all natural operators

T TT(2) form a 4-parameter family. Then we comment its most signi_cant

properties which di_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 _eld

LM on every T(2)M. Clearly, X 7! LM, X 2 C1(TM) is an absolute operator

T TT(2). Further, we have a canonical inclusion TM _ T(2)M. Using

the _ber translations on T(2)M, we can extend every section X: M ! TM

into a vector _eld V (X) on T(2)M. This de_nes a second natural operator

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

f : M ! R with respect to a vector _eld X on M, we obtain, at every point

x 2 M, a linear map from (T2_

1 M)x into the reals, i.e. an element of T(2)

x M.

This determines a _rst order operator C1(TM) ! C1(T(2)M), the coordinate

form of which is

(1) Xi @

@xi

7! Xj @Xi

@xj

@xi + XiXj @2

@xi@xj

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

translations, into a vector _eld constant on each _ber, we get from (1) another

natural operator D: T TT(2). Finally, T (2) means the ow operator as usual.

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

358 Chapter X. Prolongation of vector _elds and connections

43.2. Proposition. All natural operators T TT(2) form the 4-parameter

family

(1) k1T (2) + k2V + k3L + k4D, k1, k2, k3, k4 2 R.

Proof. By proposition 42.5, every natural operator A: T TT(2) has order

_ 2. Let V 2m

= J2

0 (TRm), S = T(2)

0 Rm, Z = (TT(2))0Rm and q : Z ! S

be the canonical projection. We have to determine all G3

m-equivariant maps

f : V 2m

_ S ! Z satisfying q _ f = pr2. The action of G3

m on V 2m

(2) _Xi = ai

jXj ; _X i

j = ai

kl~akj

Xl + ai

kXk

l ~al

while for Xi

jk we shall need the action

(3) _X i

jk = Xi

jk + ai

jklXl

of the kernel K3 of the jet projection G3

! G2

m only. The action of G2

m on S is

(4) _ui = ai

juj + ai

jkujk; _uij = ai

kaj

l ukl;

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

and (4) implies

(5)

_ Y i =ai

jY j

i =ai

jkujY k + ai

jUj + ai

jklY lujk + ai

jkUjk

ij =ai

kmaj

l uklY m + ai

kaj

lmuklY m + ai

kaj

lUkl:

Using (4) we _nd the following coordinate expression of the ow operator T (2)

(6) Xi @

@xi +

􀀀

juj + Xi

jkujk_ @

@ui +

Xik

ukj + Xj

kuik

@uij :

Consider the _rst series of components

Y i = fi(Xj ;Xk

l ;Xm

np; uq; urs)

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

K3 reads

fi(Xj ;Xk

l ;Xm

np; uq; urs) = fi(Xj ;Xk

l ;Xm

np + am

nptXt; uq; urs):

This implies that fi are independent of Xi

jk. Then the equivariance with respect

to the subgroup ai

j = _ij

yields

fi(Xj ;Xk

l ; um; unp) = fi(Xj ;Xk

l + ak

lqXq; um + amr

surs; unp):

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

43. The case of the second order tangent vectors 359

This gives fi = fi(Xj ; ukl). Using the homotheties in i(G1

m) _ G3

m, we obtain

fi = fi(Xj ). Example 24.14 then implies

(7) Y i = kXi:

Consider further the di_erence A 􀀀 kT (2) with k taken from (7) and denote

by hi, hij its components. We evaluate easily

(8) ai

kaj

l hkl(Xm;Xn

p ;Xq

rs; ut; uuv) = hij(_X m; _X n

p ; _X q

rs; _ut; _uuv):

Quite similarly as in the _rst step we deduce hij = hij(Xk; ulm). By homogeneity

and the invariant tensor theorem, we then obtain

(9) hij = cuij + aXiXj :

For hi, we _nd

(10) ai

jhj(Xk;Xlm;Xn

pq; ur; ust) + cai

jkujk + aai

jkXjXk =

= hi(_X k; _X

; _X n

pq; _ur; _ust):

By (3), hi is independent of Xi

jk. Then the homogeneity condition implies

(11) hi = fi

j (Xk

l )Xj + gij

(Xk

l )uj :

For Xi = 0, the equivariance of (11) with respect to the subgroup ai

j = _ij

reads

(12) gij

(Xk

l )uj + cai

jkujk = gij

(Xk

l )(uj + aj

klukl):

Hence gij

(Xk

l ) = c_ij

. The remaining equivariance condition is

(13) fi

j (Xk

l )Xj + aai

jkXjXk = fi

j (Xk

l + ak

lmXm)Xj :

This implies that all the _rst order partial derivatives of fi

j (Xk

l ) are constant, so

that fi

j are at most linear in Xk

l . By the invariant tensor theorem, fi

j (Xk

l )Xj =

eXjXi

j + bXi. Then (13) yields e = a, i.e.

(14) hi = cui + bXi + aXjXi

j :

This gives the coordinate expression of (1). _

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

the form H _ TB, where H is a natural transformation TTB ! TTB over the

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

H: TT(2) ! TT(2) over the identity of T(2) form the following 3-parameter

family

Y i =k1Y i;

Ui =k1Ui + k2Y i + k3ui;

Uij =k1Uij + k3uij ;

see [Doupovec, 90]. Hence the operators of the form H_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 _elds and connections

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

di_erence between a natural operator T TTB 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 _eld on a

manifoldM into a vector _eld on T(2)M tangent to the subbundle TM _ T(2)M,

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

natural operator D: T TT(2) is not compatible with the natural inclusion

TM _ T(2)M.

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

of determining all natural operators T TT(r), r 2 N. All such operators

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

Liouville vector _eld of T(r) and by the analogies of the operator D from 43.1

de_ned by f 7! |X _{_z_X}

k-times

f, k = 1; : : : ; r.

44. Induced vector _elds on jet bundles

44.1. Let F be a bundle functor on FMm;n. The idea of the ow prolongation

of vector _elds can be applied to the projectable vector _elds on every object

p: Y ! M of FMm;n. The ow Fl_

t of a projectable vector _eld _ on Y is

formed by the local isomorphisms of Y and we de_ne the ow operator F of F

FY _ = @

0 F(Fl_

t ):

The general concept of a natural operator A transforming every projectable

vector _eld on Y 2 ObFMm;n into a vector _eld on FY was introduced in

section 18. We shall denote such an operator briey by A: Tproj TF.

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

of every natural operator Tproj TF is _ r.

Proof. This is quite similar to 42.5, see [Kol_a_r, Slov_ak, 90] for the details. _

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

of _bered manifolds. We remark that a simple evaluation leads to the following

coordinate formula for J 1_

J 1_ = _i @

@xi + _p @

@yp +

@_p

@xi + @_p

@yq yq

􀀀 @_j

@xi yp

@yp

provided _ = _i(x) @

@xi + _p(x; y) @

@yp , see [Krupka, 84]. To evaluate J r_, we

have to iterate this formula and use the canonical inclusion Jr(Y ! M) ,!

J1(Jr􀀀1(Y ! M) ! M).

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

44. Induced vector _elds on jet bundles 361

Proposition. Every natural operator A: Tproj TJr is a constant multiple of

the ow operator J r.

Proof. Let V r be the space of all r-jets of the projectable vector _elds on

Rn+m ! Rm with source 0 2 Rm+n, let V 0 _ V r be the space of all r-jets

of the constant vector _elds and V0 _ V 0 be the subset of all vector _elds with

zero component in Rn. Further, let Sr or Zr be the _ber of Jr(Rm+n ! Rm)

or TJr(Rm+n ! Rm) over 0 2 Rm+n, respectively. By lemma 44.2 and by the

general theory, we have to determine all Gr+1

m;n-maps A: V r _ Sr ! Zr over the

identity of Sr. Analogously to section 42, every projectable vector _eld on Y

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

the vector _eld @

@x1 . Hence A is determined by its restriction A0 to V0 _ Sr.

However, in the _rst part of the proof we have to consider the restriction A0 of

A to V 0 _ Sr for technical reasons.

Having the canonical coordinates xi and yp on Rm+n, let Xi, Y p be the

induced coordinates on V 0, let yp_, 1 _ j_j _ r, be the induced coordinates on

Sr and Zi = dxi, Zp = dyp, Zp_ = dyp_ be the additional coordinates on Zr. The

restriction A0 is given by some functions

Zi = fi(Xj ; Y q; ys_ )

Zp = fp(Xi; Y q; ys_ )

Zp_ = fp_ (Xi; Y q; ys_ ):

Let us denote by gi, gp, gp_ the restrictions of the corresponding f's to V0 _ Sr.

The ows of constant vector _elds are formed by translations, so that their r-jet

prolongations are the induced translations of Jr(Rm+n ! Rm) identical on the

standard _ber. Therefore J r @

@x1 = @

@x1 and it su_ces to prove

gi = kXi; gp = 0; gp_ = 0:

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

of i(G1

_ G1

n) _ Gr+1

m;n on all quantities is tensorial. Consider the case

r = 1. Using the equivariance with respect to the homotheties in i(G1

n), we

obtain fi(Xj ; Y p; yq

l ) = fi(Xj ; kY p; kyq

l ), so that fi depends on Xi only. Then

the equivariance of fi with respect to i(G1

m) yields fi = kXi by 24.7. The equivariance

of fp with respect to the homotheties in i(G1

n) gives kfp(Xi; Y q; ys

j ) =

fp(Xi; kY q; kys

j ). This kind of homogeneity implies fp = hpq

(Xi)Y q+hpj

q (Xi)yq

with some smooth functions hpq

, hpj

q . Using the homotheties in i(G1

m), we further

obtain hpq

(kX) = hpq

(X) and hpj

q (kX) = khpj

q (X). Hence hpq

= const

and hpj

q is linear in Xi. Then the generalized invariant tensor theorem yields

fp = aY p + byp

i Xi, a, b 2 R. Applying the same procedure to fp

i , we _nd

i = cyp

i , c 2 R.

Consider the injection G2

n ,! G2

m;n determined by the products with the

identities on Rm. The action of an element (apq

; ar

st) of the latter subgroup is

given by

_yp

i = apq

i (2)

_ Zp

i = ap

qtyq

i Zt + apq

i (3)

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

362 Chapter X. Prolongation of vector _elds and connections

and V0 is an invariant subspace. In particular, (3) with apq

= _p

q gives an equivariance

condition

cyp

i = bap

qtyq

i yt

jXj + cyp

i :

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

respect to the inclusion of G1

m;n into G2

m;n. The equivariance of fp

i with respect

to an element (_ij

; _p

q ; ap

i ) 2 G1

m;n means cyp

i = c(yp

i + ap

i ). Hence c = 0, which

completes the proof for r = 1.

For r _ 2 it su_ces to discuss the g's only. Using the homotheties in i(G1

n),

we _nd that gp

i1___is

(Xj ; yq

_), 1 _ j_j _ r, is linear in yq

_. The homotheties in

i(G1

m) and the generalized invariant tensor theorem then yield

(4) gp

i1___is

= Wp

i1___is

+ csyp

i1___isis+1___ir

Xis+1 : : :Xir

where Wp

i1___is

do not depend on yp

i1___ir

, s = 1; : : : ; r 􀀀 1, and

i1___ir

= cryp

i1___ir

(5)

gp = b1yp

i Xi + _ _ _ + bryp

i1___ir

(6) Xi1 : : :Xir :

Similarly to the _rst order case, we have an inclusion Gr+1

n ,! Gr+1

m;n determined

by the products of di_eomorphisms on Rn with the identity of Rm. One _nds

easily the following transformation law

(7) _yp

i1___is

= apq

i1___is

+ Fp

i1___is

+ apq

1___qsyq1

i1 : : : yqs

where Fp

i1___is

is a polynomial expression linear in ap

_ with 2 _ j_j _ s 􀀀 1 and

independent of yp

i1___is

. This implies

(8) _ Zp

i1___is

= apq

i1___is

+ Gp

i1___is

+ apq

1___qsqs+1yq1

i1 : : : yqs

Zqs+1

where Gp

i1___is

is a polynomial expression linear in ap

_ with 2 _ j_j _ s and linear

in Zp_, 0 _ j_j _ s 􀀀 1.

We deduce that every gp

i1___is

, 0 _ s _ r􀀀1 , is independent of yp

i1___ir

. On the

kernel of the jet projection Gr+1

! Gr

n, (8) for r = s gives

0 = apq

1___qrqr+1yq1

i1 : : : yqr

gqr+1:

Hence gp = 0. On the kernel of the jet projection Gr

! Gr􀀀1

n , (8) with s =

1; : : : ; r 􀀀 1, implies

0 = csapq

1:::qryq1

i1 : : : yqr

Xis+1 : : :Xir ;

i.e. cs = 0. By projectability, gi and gp_, 0 _ j_j _ r 􀀀 1, correspond to a

m;n-equivariant map V0 _ Sr􀀀1 ! Zr􀀀1. By the induction hypothesis, gp_ = 0

for all 0 _ j_j _ r 􀀀 1. Then on the kernel of the jet projection Gr+1

! Gr􀀀1

(8) gives 0 = crapq

1:::qryq1

i1 : : : yqr

, i.e. gp

i1___ir

= 0. _

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

45. Prolongations of connections to FY ! M 363

44.4. Bundles of contact elements. Consider the bundle functor Krn

Mfm of the n-dimensional contact elements of order r de_ned in 12.15.

Proposition. Every natural operator A: T TKrn

is a constant multiple of

the ow operator Krn

Proof. It su_ces to discuss the case M = Rm. Consider the canonical _bration

Rm = Rn _ Rm􀀀n ! Rn. As remarked at the end of 12.16, there is an identi_-

cation of an open dense subset in Krn

Rm with Jr(Rm ! Rn). By de_nition, on

this subset it holds J r_ = Krn

_ for every projectable vector _eld _ on Rm ! Rn.

Since the operator A commutes with the action of all di_eomorphisms preserving

_bration Rm ! Rn, the restriction of A to @

@x1 is a constant multiple of Krn

( @

@x1 )

by proposition 44.3. But every vector _eld on Rm can be locally transformed

into @

@x1 in a neighborhood of any point where it does not vanish. _

We _nd it interesting that we have _nished our investigation of the basic

properties of the natural operators T TF for di_erent bundle functors on

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

the only natural operators T TF.

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

T T(TT_) for manifolds of dimension at least two. Let T _ be the ow operator

of the cotangent bundle, LM : T_M ! TT_M be the vector _eld generated by

the homotheties of the vector bundle T_M and !M : TM _M T_M ! R be the

evaluation map. Then all natural operators T TT_ are of the form f(!)T _ +

g(!)L, where f, g 2 C1(R;R) are any smooth functions of one variable. In

the case F = TT_ 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 FY ! M

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

every general connection on Y ! M into a general connection on V Y ! M.

However, one meets a quite di_erent situation when replacing _bered manifold

V Y ! M e.g. by the _rst jet prolongation J1Y ! 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 J1Y ! M. Our

_rst goal is to clarify this di_erence from the conceptual point of view.

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

_bered manifold p: Y ! M into another manifold determine the same (r; s)-jet

jr;s

y f = jr;s

y g at y 2 Y , s _ r, if jr

yf = jr

yg and the restrictions of f and g to the

_ber Yp(y) satisfy js

y(fjYp(y)) = js

y(gjYp(y)), see 12.19.

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

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

jr;s

y f = jr;s

y g implies (Ff)j(FY )y = (Fg)j(FY )y:

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

364 Chapter X. Prolongation of vector _elds and connections

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

the _rst jet prolongation J1 has order (1; 1).

45.3. Denote by Jr;sTY the space of all (r; s)-jets of the projectable vector

_elds on Y ! M. This is a vector bundle over Y . Let F be a bundle functor on

FMm;n and F denote its ow operator Tproj TF.

Proposition. If the order of F is (r; s) and _ is a projectable vector _eld on Y ,

then the value (F_)(u) at every u 2 (FY )y depends only on jr;s

y _. The induced

map

FY _ Jr;sTY ! T(FY )

is smooth and linear with respect to Jr;sTY .

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

directly from the linearity of the ow operator F. _

45.4. Let 􀀀 be a general connection on p: Y ! M. Considering the 􀀀-lift

􀀀_ of a vector _eld _ on M, one sees directly that jr;s

y 􀀀_ depends on jr

p(y)_

only, y 2 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 􀀀, we obtain a

map fF􀀀: FY _ JrTM ! TFY linear in JrTM. Let _: TM ! JrTM be

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

0 : JrTM ! TM. By linearity, the composition

(1) fF􀀀 _ (idFY _ _): FY _ TM ! TFY

is a lifting map of a general connection on FY ! M.

De_nition. The general connection F(􀀀; _) on FY ! M with lifting map (1)

is called the F-prolongation of 􀀀 with respect to _.

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

connection 􀀀 on Y ! M induces in such a way a connection V􀀀 on V Y ! M,

which was already mentioned in remark 31.4.

45.5. We show that the construction of F(􀀀; _) behaves well with respect to

morphisms of connections. Given an FM-morphism f : Y ! _ Y over f0 : M !

and two general connections 􀀀 on p: Y ! M and _

􀀀

on _p: _ Y ! _M , one sees

easily that 􀀀 and _

􀀀

are f-related in the sense of 8.15 if and only if the following

diagram commutes

TY w

T _ Y

Y _ TM w

f _ Tf0

􀀀

_ Y _ T _M

_􀀀

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

r-th order linear connections _: TM ! JrTM and __ : T _M ! JrT _M are called

f0-related, if for every z 2 TxM it holds

__(Tf0(z)) _ (jrx

f0) = (jr

zTf0) _ _(z):

Let F be as in 45.4.

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

45. Prolongations of connections to FY ! M 365

Proposition. If 􀀀 and _

􀀀

are f-related and _ and __ are f0-related, then F(􀀀; _)

and F(_

􀀀

; __) are Ff-related.

Proof. The lifting map of F(􀀀; _) can be determined as follows. For every

X 2 TxM we take a vector _eld _ on M such that jrx

_ = _(X) and we construct

its 􀀀-lift 􀀀_. Then F(􀀀; _)(u) is the value of the ow prolongation F(􀀀_) at

u 2 FxY . Let __(Tf0(X)) = jr_x

__, _x = f0(x). If _ and __ are f0-related, the

vector _elds _ and __ are f0-related up to order r at x. Since 􀀀 and _

􀀀

are frelated,

the restriction of F(􀀀_) over x and the restriction of F(_

􀀀

_) over _x are

Ff-related. _

45.6. In many concrete cases, the connection F(􀀀; _) 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 _eld _ on Y 2 ObC is called a C-_eld, if its ow is formed by

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

a projectable vector _eld _ on a principal _ber bundle is a PB(G)-_eld

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

one deduces easily that a projectable vector _eld _ on a vector bundle E is a

VB-_eld if and only if _ is a linear morphism E ! TE, see 6.11. A connection 􀀀

on (p: Y ! M) 2 ObC is called a C-connection, if 􀀀_ is a C-_eld for every vector

_eld _ 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 _ber Yx, i.e. a

section _ : Yx ! TY such that Tp _ _ is a constant map, is said to be a C-family,

if there exists a C-_eld _ on Y such that _ is the restriction of _ to Yx. We shall

say that the category C is in_nitesimally regular, if any projectable vector _eld

on a C-object the restriction of which to each _ber is a C-family is a C-_eld.

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

_nitesimally regular category D over FMand 􀀀 is a C-connection, then F(􀀀; _)

is a D-connection for every _.

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

F(􀀀; _)-lift of every vector X 2 TM is a D-family. Since D is in_nitesimally

regular, the F(􀀀; _)-lift of every vector _eld on TM is a D-_eld. _

45.7. In the special case F = J1 we determine all natural operators transforming

a general connection on Y ! M and a _rst order linear connection

_ on M into a general connection on J1Y ! M. Taking into account the

rigidity of the symmetric linear connections on M deduced in 25.3, we _rst assume

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

J1 _ Q_P1B J1(J1 ! B).

On one hand, 􀀀 and _ induce the J1-prolongation J 1(􀀀; _) of 􀀀 with respect

to _. On the other hand, since J1Y is an a_ne bundle with associated vector

bundle V Y T_M, the section 􀀀: Y ! J1Y determines an identi_cation

I􀀀 : J1Y _= V Y T_M. The vertical prolongation V􀀀 of 􀀀 is linear over Y , see

31.1.(3), so that we can construct the tensor product V􀀀 __ with the dual

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

366 Chapter X. Prolongation of vector _elds and connections

connection __ on T_M, see 47.14 and 47.15. The identi_cation I􀀀 transforms

V􀀀 __ into another connection P(􀀀; _) on J1Y ! M.

45.8. Proposition. All natural operators J1 _ Q_P1B J1(J1 ! B) form

the one-parameter family

(1) tP + (1 􀀀 t)J 1; t 2 R:

Proof. In usual local coordinates, let

(2) dyp = Fp

i (x; y)dxi

be the equations of 􀀀 and

(3) d_i = _i

jk(x)_jdxk

be the equations of _. By direct evaluation, one _nds the equations of J 1(􀀀; _)

in the form (2) and

(4) dyp

i =

@Fp

@xi + @Fp

@yq yq

i + _kj

i(Fp

􀀀 yp

dxj

while the equations of P(􀀀; _) have the form (2) and

(5) dyp

i =

@Fp

@yq (yq

􀀀 Fq

i ) + @Fp

@xj + @Fp

@yq Fq

􀀀 _k

ij(yp

􀀀 Fp

k )

dxj :

First we discuss the operators of _rst order in 􀀀 and of order zero in _.

Let S1 = J1

0 (J1(Rn+m ! Rm) ! Rn+m) be the standard _ber from 27.3,

S0 = J1

0 (Rm+n ! Rm), _ = (Q_P1Rm)0 and Z = J1

0 (J1(Rm+n ! Rm) ! Rm).

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

m;n-maps

S1 _ _ _ S0 ! Z over the identity of S0. The canonical coordinates on S1 are

i , yp

iq, yp

ij and the action of G2

m;n is given by 27.3.(1)-(3). On S0 we have the

well known coordinates Y p

i and the action

(6) _ Y p

i = apq

Y q

j ~aj

i + ap

j ~aj

i :

The standard coordinates on _ are _i

jk = _i

kj and the action is

(7) __i

jk = ail

mn~amj

~ank

+ ai

lm~al

j~amk:

The induced coordinates on Z are zp

i , Zp

ij and one evaluates easily that the

action on both zp

i and Zp

i has form (6), while

(8)

_ Zp

ij = apq

kl~aki

~al

j + apq

rzq

kZr

l ~aki

~al

j + ap

qkZq

l ~aki

~al

+ ap

qlzq

k~aki

~al

j + apq

k~ak

ij + ap

k~ak

ij + ap

kl~aki

~al

j :

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

45. Prolongations of connections to FY ! M 367

Write Y = (Y p

i ), y = (yp

i ), y1 = (yp

iq), y2 = (yp

ij ), _ = (_i

jk). Then the

coordinate form of a map f : S1 ___S0 ! Z over the identity of S0 is zp

i = Y p

and

(9)

i = fp

i (Y; y; y1; y2; _)

ij = fp

ij(Y; y; y1; y2; _):

The equivariance of fp

i with respect to the homotheties in i(G1

m) yields

kfp

i = fp

i (kY; ky; ky1; k2y2; k_)

so that fp

i is linear in Y , y, y1, _ and independent of y2. The homotheties in

i(G1

n) give that fp

i is independent of y1 and _. By the generalized invariant

tensor theorem 27.1, the equivariance with respect to i(G1

_ G1

n) implies

i = aY p

i + byp

i :

Then the equivariance with respect to the subgroup K characterized by ai

j = _ij

apq

= _p

q yields

b = 1 􀀀 a:

For fp

ij the homotheties in i(G1

m) and i(G1

n) give

k2fp

ij = fp

ij(kY; ky; ky1; k2y2; k_)

kfp

ij = fp

ij(kY; ky; y1; ky2; _)

so that fp

ij is linear in y2 and bilinear in the pairs (Y; y1),(y; y1), (Y; _), (y; _).

Considering equivariance with respect to i(G1

_G1

n), we obtain fp

ij in the form

of a 16-parameter family

ij = k1yp

ij + k2yp

ji + k3Y p

i yq

qj + k4Y p

j yq

qi + k5Y q

i yp

qj + k6Y q

j yp

+ k7yp

i yq

qj + k8yp

j yq

qi + k9yq

i yp

qj + k10yq

j yp

qi + k11Y p

k _k

+ k12Y p

i _kk

j + k13Y p

j _kk

i + k14yp

k_k

ij + k15yp

i _kk

j + k16yp

j_kki:

Evaluating the equivariance with respect to K, we _nd a = 0 and such relations

among k1; : : : ; k16, which correspond to (1).

Furthermore, 23.7 implies that every natural operator of our type has _nite

order. Having a natural operator of order r in 􀀀 and of order s in _, we shall

deduce r = 1 and s = 0, which corresponds to the above case. Let _ and be

multi indices in xi and _ be a multi index in yp. The associated map of our

operator has the form zp

i = Y p

i and

i = fp

i (Y; y__;_); Zp

ij = fp

ij(Y; y__;_)

where j_j + j_j _ r, jj _ s. Using the homotheties in i(G1

m), we obtain

kfp

i = fp

i (kY; k1+j_jy__; k1+jj_):

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

368 Chapter X. Prolongation of vector _elds and connections

Hence fp

i is linear in Y , y_ and _, and is independent of the variables with

j_j > 0 or jj > 0. The homotheties in i(G1

n) then imply that fp

i is independent

of y_ with j_j > 1. For fp

ij , the homotheties in i(G1

m) yield

(10) k2fp

ij = fp

ij(kY; k1+j_jy__; k1+jj_)

so that fp

ij is a polynomial independent of the variables with j_j > 1 or jj > 1.

The homotheties in i(G1

n) imply

(11) kfp

ij = fp

ij(kY; k1􀀀j_jy__;_)

for j_j _ 1, jj _ 1. Combining (10) with (11) we deduce that fp

ij is independent

of y__ for j_j + j_j > 1 and _ for jj > 0. _

45.9. Using a similar procedure as in 45.8 one can prove that the use of a

linear connection on the base manifold for a natural construction of an induced

connection on J1Y ! M is unavoidable. In other words, the following assertion

holds, a complete proof of which can be found in [Kol_a_r, 87a].

Proposition. There is no natural operator J1 J1(J1 ! B).

45.10. If we admit an arbitrary linear connection _ on the base manifold in

the above problem, the natural operators QP1 QP1 from proposition 25.2

must appear in the result. By proposition 25.2, all natural operators QP1

T T_ T_ form a 3-parameter family

N(_) = k1S + k2I ^ S + k3 ^ S I:

By 12.16, J1(J1Y ! M) is an a_ne bundle with associated vector bundle

V J1Y T_M. We construct some natural `di_erence tensors' for this case.

Consider the exact sequence of vector bundles over J1Y established in 12.16

0 􀀀! V Y

J1Y T_M 􀀀! V J1Y

V _

􀀀􀀀! V Y 􀀀! 0

where

J1Y denotes the tensor product of the pullbacks over J1Y . The connection

􀀀 determines a map _(􀀀): J1Y ! V Y T_M transforming every

u 2 J1Y into the di_erence u 􀀀 􀀀(_u) 2 V Y T_M. Hence for every k1,

k2, k3 we can extend the evaluation map TM _ T_M ! R into a contraction

h_(􀀀);N(_)i : J1Y ! V Y

J1Y T_MT_M _ V J1Y T_M. By the procedure

used in 45.8 one can prove the following assertion, see [Kol_a_r, 87a].

Proposition. All natural operators transforming a connection 􀀀 on Y into a

connection on J1Y ! M by means of a linear connection _ on the base manifold

form the 4-parameter family

tP (􀀀; ~_) + (1 􀀀 t)J 1(􀀀; _) + h_(􀀀);N(_)i

t, k1, k2, k3 2 R, where ~_ means the conjugate connection of _.

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

46. The cases FY ! FM and FY ! Y 369

46. The cases FY ! FM and FY ! Y

46.1. We _rst describe a geometrical construction transforming every connection

􀀀 on a _bered manifold p: Y ! M into a connection TA􀀀 on TAp: TAY !

TAM for every Weil functor TA. Consider 􀀀 in the form of the lifting map

(1) 􀀀: Y _ TM ! TY:

Such a lifting map is characterized by the condition

(2) (_Y ; Tp) _ 􀀀 = idY _TM

where _ : T ! Id is the bundle projection of the tangent functor, and by the

fact that, if we interpret (1) as the pullback map

p_TM ! TY;

this is a vector bundle morphism over Y . Let _: TAT ! TTA be the ow-natural

equivalence corresponding to the exchange homomorphism A D ! D A, see

35.17 and 39.2.

Proposition. For every general connection 􀀀: Y _ TM ! TY , the map

(3) TA􀀀 := _Y _ (TA􀀀) _ (idTAY _ _􀀀1

M ) : TAY _ TTAM ! TTAY

is a general connection on TAp: TAY ! TAM.

Proof. Applying TA to (2), we obtain

(TA_Y ; TATp) _ TA􀀀 = idTAY _TATM:

Since _ is the ow-natural equivalence, it holds _M _ TATp _ _􀀀1

Y = TTAp and

TA_Y _ _􀀀1

Y = _TAY . This yields

(_TAY ; TTAp) _ TA􀀀 = idTA_TTAM

so that TA􀀀 satis_es the analog of (2). Further, one deduces easily that _Y :

TATY ! TTAY is a vector bundle morphism over TAY . Even _􀀀1

M : TTAM !

TATM is a linear morphism over TAM, so that the pullback map (TAp)__􀀀1

M :

(TAp)_TTAM ! (TAp)_TATM is also linear. But we have a canonical identi

_cation (TAp)_TATM _= TA(p_TM). Hence the pullback form of TA􀀀 on

(TAp)_TTAM ! TTAY is a composition of three vector bundle morphisms over

TAY , so that it is linear as well. _

46.2. Remark. If we look for a possible generalization of this construction to

an arbitrary bundle functor F on Mf, we realize that we need a natural equivalence

FT ! TF with suitable properties. However, the ow-natural transformation

FT ! TF from 39.2 is a natural equivalence if and only if F preserves

products, i.e. F is a Weil functor. We remark that we do not know any natural

operator transforming general connections on Y ! M into general connections

on FY ! FM for any concrete non-product-preserving functor F on Mf.

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

370 Chapter X. Prolongation of vector _elds and connections

46.3. Remark. Slov_ak has proved in [Slov_ak, 87a] that if 􀀀 is a linear connection

on a vector bundle p: E ! M, then TA􀀀 is also a linear connection on

the induced vector bundle TAp: TAE ! TAM. Furthermore, if p: P ! M is a

principal bundle with structure group G, then TAp: TAP ! TAM is a principal

bundle with structure group TAG. Using the ideas from 37.16 one deduces directly

that for every principal connection 􀀀 on P ! M the induced connection

TA􀀀 is also principal on TAP ! TAM.

46.4. We deduce one geometric property of the connection TA􀀀. If we consider

a general connection 􀀀 on Y ! M in the form 􀀀: Y _ TM ! TY , the 􀀀-lift 􀀀_

of a vector _eld _ : M ! TM is given by

(1) (􀀀_)(y) = 􀀀(y; _(p(y))), i.e. 􀀀_ = 􀀀 _ (idY ; _ _ p).

On one hand, 􀀀_ is a vector _eld on Y and we can construct its ow prolongation

TA(􀀀_) = _Y _ TA(􀀀_). On the other hand, the ow prolongation TA_ = _M _

TA_ of _ is a vector _eld on TAM and we construct its TA􀀀-lift (TA􀀀)(TA_).

The following assertion is based on the fact that we have used a ow-natural

equivalence in the de_nition of TA􀀀.

Proposition. For every vector _eld _ on M, we have (TA􀀀)(TA_) = TA(􀀀_).

Proof. By (1), we have TA􀀀(TA_) = TA􀀀 _ (idTAY ; TA_ _ TAp) = _Y _ TA􀀀 _

(idTAY ; _􀀀1

_ _M _ TA_ _ TAp) = _Y _ TA(􀀀 _ (idY ; _ _ p)) = TA(􀀀_). _

We remark that several further geometric properties of TA􀀀 are deduced in

[Slov_ak, 87a].

46.5. Let _

􀀀

be another connection on another _bered manifold _ Y and let

f : Y ! _ Y be a connection morphism of 􀀀 into _􀀀, i.e. the following diagram

commutes

(1)

TY w

T _ Y

Y _ TBY w

f _ TBf

􀀀

_ Y _ TB _ Y

_􀀀

Proposition. If f : Y ! _ Y is a connection morphism of 􀀀 into _􀀀

, then TAf :

TAY ! TA _ Y is a connection morphism of TA􀀀 into TA_

􀀀

Proof. Applying TA to (1), we obtain TATf _ TA􀀀 = (TA_􀀀) _ (TAf _ TATBf).

From 46.1.(3) we then deduce directly TTAf _TA􀀀 = TA_􀀀

_(TAf _TTABf). _

46.6. The problem of _nding all natural operators transforming connections on

Y ! M into connections on TAY ! TAM seems to be much more complicated

than e.g. the problem of _nding all natural operators T TTA discussed in

section 42. We shall clarify the situation in the case that TA is the classical

tangent functor T and we restrict ourselves to the _rst order natural operators.

Let T be the operator from proposition 46.1 in the case TA = T. Hence

T transforms every element of C1(J1Y ) into C1(J1(TY ! TBY )), where

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

46. The cases FY ! FM and FY ! Y 371

J1 and J1(T ! TB) are considered as bundle functors on FMm;n. Further we

construct a natural `di_erence tensor _eld' [CY 􀀀] for connections on TY ! TBY

from the curvature of a connection 􀀀 on Y . Write BY = M. In general, the

di_erence of two connections on Y is a section of V Y T_M, which can be

interpreted as a map Y _ TM ! V Y . In the case of TY ! TM we have TY _

TTM ! V (TY ! TM). To de_ne the operator [C], consider both canonical

projections pTM, TpM : TTM ! TM. If we compose (pTM; TpM) : TTM !

TM _ TM with the antisymmetric tensor power and take the _bered product

of the result with the bundle projection TY ! Y , we obtain a map _Y : TY _

TTM ! Y _ _2TM. Since CY 􀀀: Y _ _2TM ! V Y , the values of CY 􀀀 _ _Y

lie in V Y . Every vector A 2 V Y is identi_ed with a vector i(A) 2 V (V Y ! Y )

tangent to the curve of the scalar multiples of A. Then we construct [CY 􀀀](U;Z),

U 2 TY , Z 2 TTM by translating i(CY 􀀀(_Y (U;Z))) to the point U in the same

_ber of V (TY ! TM). This yields a map [CY 􀀀]: TY _TTM ! V (TY ! TM)

of the required type.

46.7. Proposition. All _rst order natural operators J1 J1(T ! TB) form

the following one-parameter family

T + k[C], k 2 R.

Proof. Let

(1) dyp = Fp

i (x; y)dxi

be the equations of 􀀀. Evaluating 46.1.(3) in the case TA = T, one _nds that

the equations of T 􀀀 are (1) and

(2) d_p =

@Fp

@xj _j + @Fp

@yq _q

dxi + Fp

i (x; y)d_i

where _i = dxi, _p = dyp are the induced coordinates on TY . The equations of

[CY 􀀀]

(3) dyp = 0; d_p =

@Fp

@xj + @Fp

@yq Fq

_j ^ dxi

follow directly from the de_nition.

Let S1 = J1

0 (J1(Rm+n ! Rm) ! Rm+n), Q = T0(Rm+n), Z = J1

0 (TRm+n

! TRm) be the standard _bers in question and q : Z ! Q be the canonical projection.

According to 18.19, the _rst order natural operators A: J1 J1(T !

TB) are in bijection with the G2

m;n-maps A: S1 _Q ! Z satisfying q _A = pr2.

The canonical coordinates yp

i , yp

iq, yp

ij on S1 and the action of G2

m;n on S1 are

described in 27.3. It will be useful to replace yp

ij by Sp

ij and Rp

ij in the same way

as in 28.2. One sees directly that the action of G2

m;n on Q with coordinates _i,

_p is

(4) __i = ai

j_j ; __p = ap

i _i + apq

_q:

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

372 Chapter X. Prolongation of vector _elds and connections

The coordinates on Z are _i, _p and the quantities Ap

i , Bp

i , Cp

i , Dp

i determined

(5) dyp = Ap

i dxi + Bp

i d_i; d_p = Cp

i dxi + Dp

i d_i:

A direct calculation yields that the action of G2

m;n on Z is (4) and

(6)

A_p

i = apq

j~aj

􀀀 ~aq

i + Bq

j ~aj

ikakl

i = apq

j ~aj

_ Cp

i = apq

􀀀

􀀀~aq

__j 􀀀 ~aq

__j _ Ari

􀀀 ~aq

ir __r 􀀀 ~aq

rs__r _ Asi

+ Cq

j ~aj

i + Dq

j ~aj

__k_

i = apq

􀀀

􀀀~aq

jraj

k_kar

sBs

l ~al

􀀀 ~aq

rsar

k_kas

uBu

j ~aj

􀀀 ~aq

rsart

_tas

uBu

j ~aj

i + Dq

j ~aj

Write _ = (_i), _ = (_p), y = (yp

i ), y1 = (yp

iq), S = (Sp

ij ), R = (Rp

ij ).

I. Consider _rst the coordinate functions Bp

i (_; _; y; y1; S;R) of A. The common

kernel L of _2

1 : G2

m;n

! G1

m;n and of the projection G2

m;n

! G2

_ G2

described in 28.2 is characterized by ai

j = _ij

, apq

= _p

q , ap

i = 0, ai

jk = 0, apq

r = 0.

The equivariance of Bp

i with respect to L implies that Bp

i are independent of y1

and S. Then the homotheties in i(G1

n) _ G2

m;n yield a homogeneity condition

kBp

i = Bp

i (_; k_; ky; kR):

Therefore we have

i = fp

iq(_)_q + fpj

iq (_)yq

j + fpjk

iq (_)Rq

with some smooth functions of _. Now the homotheties in i(G1

m) give

k􀀀1Bp

i = fp

iq(k_)_q + fpj

iq (k_)k􀀀1yq

j + fpjk

iq (k_)k􀀀2Rq

jk:

Hence it holds a) fp

iq(_) = kfp

iq(k_), b) fpj

iq (_) = fpj

iq (k_), c) kfpjk

iq (_) = fpjk

iq (k_).

If we let k ! 0 in a) and b), we obtain fp

iq = 0 and fpj

iq = const. The relation

c) yields that fpjk

iq is linear in _. The equivariance of Bp

i with respect to the

whole group i(G1

_G1

n) implies that fpj

iq and fpjk

iq correspond to the generalized

invariant tensors. By theorem 27.1 we obtain

i = c1Rp

ij_j + c2yp

with real parameters c1, c2. Consider further the equivariance of Bp

i with respect

to the subgroup K _ G2

m;n characterized by ai

j = _ij

, apq

= _p

q . This yields

c1Rp

ij_j + c2yp

i = c1Rp

ij_j + c2(yp

i + ap

i ):

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

46. The cases FY ! FM and FY ! Y 373

This relation implies c2 = 0.

II. For Ap

i we obtain in the same way as in I

i = aRp

ij_j + c3yp

i :

The equivariance with respect to subgroup K gives c3 = 1 and c1 = 0.

III. Analogously to I and II we deduce

i = bRp

ij_j + c4yp

i :

Taking into account the equivariance of Dp

i with respect to K, we _nd c4 = 1.

IV. Here it is useful to summarize. Up to now, we have deduced

(7) Ap

i = aRp

ij_j + yp

i ; Bp

i = 0; Dp

i = bRp

ij_j + yp

i :

Consider the di_erence A 􀀀 T , where T is the operator (1) and (2). Write

(8) Ep

i = Cp

􀀀 yp

ij_j 􀀀 yp

iq_q:

Using ap

ij , we _nd easily that Ep

i does not depend on Sp

ij . By (6) and (8), the

action of K on Ep

i is

(9)

􀀀a~ap

jq_jRq

ik_k + aapq

raq

j _jRr

ik_k + aapq

r_qRr

ij_j + Ep

􀀀 bRp

jk_kaj

il_l

= Ep

􀀀

_; _q + aq

j _j ; yr

j + arj

; ysk

t + as

kt + as

tuyu

k ;R

If we set Ep

i = ayp

jq_jRq

ik_k +Fp

i , then (9) implies that Fp

i is independent of y1.

The action of i(G1

_G1

n) on Fp

i (_; _; y;R) is tensorial. Hence we have the same

situation as for Bp

i in I. This implies Fp

i = kRp

ij_j + eyp

i . Using once again (9)

we obtain a = b = e = 0. Hence Ep

i = kRp

ij_j and Cp

i = yp

ij_j + yp

iq_q + kRp

ij_j .

Thus we have deduced the coordinate form of our statement. _

46.8. Prolongation of connections to FY ! Y . Given a bundle functor

F on Mf and a _bered manifold Y ! M, there are three canonical structures

of a _bered manifold on FY , namely FY ! M, FY ! FM and FY ! Y .

Unlike the _rst two cases, it seems that there should be only poor results on the

prolongation of connections to FY ! Y . We _rst present a negative result for

the case of the tangent functor T.

Proposition. There is no _rst order natural operator transforming connections

on Y ! M into connections on TY ! Y .

Proof. We shall use the notation from the proof of proposition 46.7. The equations

of a connection on TY ! Y are

d_i = Mi

jdxj + Nipdyp; d_p = Pp

i dxi + Qpq

dyq:

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

374 Chapter X. Prolongation of vector _elds and connections

One evaluates easily the action formulae __i = ai

j_j and

j = ai

kMk

l ~al

j + ai

kNk

p ~ap

􀀀 ail

~al

jkak

m_m

= ai

jNj

q ~aq

The homotheties in i(G1

n) give

Nip

= kNip

(_j ; k_q; kyrk

; ys

tl; kyu

mn):

Hence Nip

= 0. For Mi

j , the homotheties in i(G1

n) imply the independence of Mi

of _p, yp

i , yp

ij . The equivariance of Mi

j with respect to the subgroup K means

j (_j ; yp

kq) + ai

jk_k = Mi

j (_j ; yp

kq + ap

kq):

Since the expressions Mi

j on both sides are independent of ai

jk, the di_erentiation

with respect to ai

jk yields some relations among _i only. _

46.9. Prolongation of connections to V Y ! Y . We pay special attention

to this problem because of its relation to Finslerian geometry. We are going to

study the _rst order natural operators transforming connections on Y ! M into

connections on V Y ! Y , i.e. the natural operators J1 J1(V ! Id) where Id

means the identity functor. In this case it will be instructive to start from the

computational aspect of the problem.

Using the notation from 46.7, the equations of a connection on V Y ! Y are

(1) d_p = Ap

i (xj ; yq; _r)dxi + Bp

q (xj ; yr; _s)dyq:

The induced coordinates on the standard _ber Z = J1

0 (V (Rm+n ! Rm) !

Rm+n) are _p, Ap

i , Bp

i and the action of G2

m;n on Z has the form

__p = apq

(2) _q

A_p

i = ap

qj~aj

i _q + apq

j~aj

􀀀 apq

r ~ar

sasj

~aj

􀀀 apr

s~asq

_raq

j~aj

i (3)

q = apr

s ~asq

+ apr

s~asq

(4) _r:

Our problem is to _nd all G2

m;n-maps S1 _ Rn ! Z over the identity on Rn.

Consider _rst the component Bp

q (_r; ys

i ; yt

ju; yv

kl) of such a map. The homotheties

in i(G1

n) yield

q (_r; ys

i ; yt

ju; yv

kl) = Bp

q (k_r; kys

i ; yt

ju; kyv

kl)

so that Bp

q depends on yr

is only. Then the homotheties in i(G1

m) give Bp

q (yr

is) =

q (kyr

is), which implies Bp

q = const. By the invariant tensor theorem, Bp

q = k_p

q .

The invariance under the subgroup K reads

k_p

q + apq

r_r = k_p

q :

This cannot be satis_ed for any k. Thus, there is no _rst order operator J1

J1(V ! Id) natural on the category FMm;n.

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

Remarks 375

46.10. However, the obstruction is apq

r and the condition apq

r = 0 characterizes

the a_ne bundles (with vector bundles as a special case). Let us restrict ourselves

to the a_ne bundles and continue in the previous consideration. By 46.9.(3),

the action of i(G1

_ G1

n) on Ap

i (_q; yr

i ; ys

jt; yu

kl) is tensorial. Using homotheties

in i(G1

m), we _nd that Ap

i is linear in yp

i , yp

iq, but the coe_cients are smooth

functions in _p. Using homotheties in i(G1

n), we deduce that the coe_cients by

i are constant and the coe_cients by yp

iq are linear in _p. By the generalized

invariant tensor theorem, we obtain

(1) Ap

i = ayp

i + byq

qi_p + cyp

iq_q a; b; c 2 R:

The equivariance of (1) on the subgroup K implies a = 􀀀k, b = 0, c = 1. Thus

we have proved

Proposition. All _rst order operators J1 J1(V ! Id) which are natural on

the local isomorphisms of a_ne bundles form the following one-parameter family

d_p = yp

iq_qdxi + k(dyp 􀀀 yp

i dxi); k 2 R:

Remarks

Section 42 is based on [Kol_a_r, 88a]. The order estimate in 42.4 follows an idea

by [Zajtz, 88b] and the proof of lemma 42.7 was communicated by the second

author. The results of section 43 were deduced by [Doupovec, 90]. Section 44

is based on [Kol_a_r, Slov_ak, 90]. The construction of the connection F(􀀀; _)

from 45.4 was _rst presented in [Kol_a_r, 82b]. Proposition 46.7 was proved by

[Doupovec, Kol_a_r, 88]. The relation of proposition 46.10 to Finslerian geometry

was pointed out by B. Kis.

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

376

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