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

r

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 T F are in bijection with

r+1 r

the associated Gm -equivariant maps A : Vm — S ’ Z satisfying q —¦ 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 — S ’ Z coincide on V0 — S ‚

r

Vm — S, then A1 = A2 .

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

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

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

m

r

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

r

¬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 ’ 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 ’ 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 onm

m m

(TB R )0 = N de¬ned by

r

(j0 f )(jB g) = 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

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

m m

m

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

m m

m

B1 ’ B2 .

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

m m

m

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

yields khi (y1 , . . . , ym ) = hi (ky1 , . . . , kym ), k ∈ 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

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

consider the map x ’ 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

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354 Chapter X. Prolongation of vector ¬elds and connections

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

¯ y

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 ∈ Aut B 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 TB M . By 42.2, the

constant maps X ’ D(M ) for all X ∈ C ∞ (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 ∈ Aut B.

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

itself for every f ∈ 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 ∈ 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 ’ B. We have Der B ‚ L(B, B) and Aut B ‚ 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

‚ ‚ ‚ ‚

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

‚t 0 ‚t 0 ‚t 0 ‚t 0

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

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

k

k

k

Di (a)Dk’i (b)

D (ab) =

i

i=0

∞ tk k

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

∞ k

k

tk

Di (a)Dk’i (b)

ht (ab) = k! i

k=0 i=0

∞ k

ti i tk’i

D (a) (k’i)! Dk’i (b)

= i!

k=0 i=0

«

∞ ∞

tk k tj j

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

j=0

k=0

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 trans-

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

to the tensor product of algebras B —D 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 T TB

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 ∈ B and any D ∈ Der B.

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 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 — T M ’ T M . Applying the

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

identi¬cation T TB M ∼ TB T M . Then TB m can be interpreted as a map B —

=

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

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

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 ∈ 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 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 T TB ’ T TB over the identity of TB , we obtain the

following system of natural operators T T TB

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

(2)

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 T TB are of the form 42.11.(2).

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

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

r

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

A: T

(vi ) ∈ V0 , so that vi ∈ 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 ) ‚ Gr+1 , one veri¬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 ’ R, g : N ’ N , h : R ’ N are linear maps and k ∈ 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) ∈ N , so

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

Consider the natural transformation HA : T TB ’ T TB 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 TB Rm = Rm — 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 —¦ TB coincide on V0 — 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 ∈ R[x]/ xr+1 de¬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 , . . . , Yr’1 ). We remark that this tensor was introduced in another way

i

by [de Le´n, Rodriguez, 88]. The reparametrization xi (t) ’ xi (kt), 0 = k ∈ R,

o

r

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

r

so called generalized Liouville vector ¬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 ˜translate™ 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 —¦ T1r , . . . , Qr —¦ T1r , 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 T M . (The vertical lift transforms

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

translations in the individual ¬bers of T M .) 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.

r

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

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

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

r

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

r

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

af(c)—¦Tkr , for all c ∈ 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 ¬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 ’ Tk correspond

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

k

1

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

s T Tk by

direct evaluation, [Krupka, Janyˇka, 90].

s

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

T

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 ’ LM , X ∈ C ∞ (T M ) is an absolute operator

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

T

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

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

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

V:T

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

(2)

2—

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

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

form of which is

i 2

X i ‚xi ’ X j ‚Xj + X i X j ‚x‚‚xj

‚ ‚

(1) ‚xi i

‚x

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

T T (2) . Finally, T (2) means the ¬‚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 ¬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 ∈ R.

(1)

T T (2) has order

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

(2)

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

2 2

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

m

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

2 2

m

¯ ¯i

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

(2) kl ˜j ˜j

j k

i

while for Xjk we shall need the action

¯i

Xjk = Xjk + ai X l

i

(3) jkl

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

m m m

uij = ai aj ukl ,

ui = ai uj + ai ujk ,

(4) ¯ ¯

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

¯

Y i =ai Y j

j

¯

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 .

¯ km k k

l lm l

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

j

X i ‚xi + Xj uj + Xjk ujk

i i

+ Xk ukj + Xk uik

i

‚ ‚ ‚

(6) ‚uij .

‚ui

Consider the ¬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