natural on the category FMm,n .

Proposition. The vertical prolongation V is the only natural operator J 1

J 1V .

We start the proof with ¬nding the equations of V“. If

dy p = Fip (x, y)dxi

(2)

is the coordinate form of “ and Y p = dy p are the additional coordinates on V Y ,

then (1) implies that the equations of V“ are (2) and

‚Fip q i

p

(3) dY = Y dx .

‚y q

31.2. The standard ¬ber of V on the category FMm,n is Rn . Let S1 =

J0 (J 1 (Rm — Rn ’ Rm ) ’ Rm — Rn ) and Z = J0 (V (Rm — Rn ) ’ Rm ),

1 1

0 ∈ Rm — Rn . By 18.19, the ¬rst order natural operators are in bijection with

G2 -maps S1 — Rn ’ Z over the identity of Rn . The canonical coordinates on

m,n

p p p

S1 are yi , yiq , yij and the action of G2 can be found in 27.3. The action of

m,n

2 n

Gm,n on R is

¯

Y p = ap Y q .

(1) q

The coordinates on Z are Y p , zi = ‚y p /‚xi , Yip = ‚Y p /‚xi . By standard

p

evaluation we ¬nd the following action of G2

m,n

zi = ap zj aj + ap aj

¯p q

¯

Y p = ap Y q , ˜i j ˜i

q q

(2)

Yip = ap Yjq aj + ap zj Y q aj + ap Y q aj .

¯ r

˜i ˜i ˜i

q qr qj

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

256 Chapter VII. Further applications

The coordinate form of any map S1 —Rn ’ Z over the identity of Rn is Y p = Y p

and

zi = fip (Y q , yj , ykt , ylm )

p rs u

Yip = gi (Y q , yj , ykt , ylm ).

p rs u

First we discuss fip . The equivariance with respect to base homotheties yields

kfip = fip (Y q , kyj , kykt , k 2 ylm ).

r s u

(3)

By the homogeneous function theorem, if we ¬x Y q , then fip is linear in yj , ykt

r s

u

and independent of ylm . The ¬ber homotheties then give

kfip = fip (kY q , kyj , ykt ).

rs

(4)

By (3) and (4), fip is a sum of an expression linear in yi and bilinear in Y p and

p

yiq . Since fip is GL(m) — GL(n)-equivariant, the generalized invariant tensor

p

theorem implies it has the following form

p q p

ayi + bY p yqi + cY q yqi .

(5)

The equivariance on the kernel K of the projection G2 ’ G1 — G1 yields

m,n m n

ap = aap + bY p (aq + aq yi ) + cY q (ap + ap yi ).

r r

(6) qr qr

i i qi qi

This implies a = 1, b = c = 0, which corresponds to 31.1.(2).

p

For gi , the above procedure leads to the same form (5). Then the equivariance

with respect to K yields a = b = 0, c = 1. This corresponds to 31.1.(3). Thus

we have proved that V is the only ¬rst order natural operator J 1 J 1V .

31.3. By 23.7, every natural operator A : J 1 J 1 V has a ¬nite order r. Let f =

(fip , gi ) : Sr — Rn ’ Z be the associated map of A, where S r = J0 (J 1 (Rm+n ’

p r

Rm ) ’ Rm+n ). Consider ¬rst fip (Y q , yj±β ) with the same notation as in the

r

second step of the proof of proposition 27.3. The base homotheties yield

kfip = fip (Y q , k |±|+1 yj±β ).

r

(1)

By the homogeneous function theorem, if we ¬x Y p , then fip are independent

p p

of yi±β with |±| ≥ 1 and linear in yiβ . Hence the only s-th order term is

•s = •pjβ (Y r )yjβ , |β| = s, s ≥ 2. Using ¬ber homotheties we ¬nd that •s is of

q

iq

degree s in Y p . Then the generalized invariant tensor theorem implies that •s

is of the form

p q1

as yiq1 ...qs Y q1 . . . Y qs + bs Y p yiq1 q2 ...qs Y q2 . . . Y qs .

Consider the equivariance with respect to the kernel of the jet projection Gr+1 ’

m,n

Gr . Using induction we deduce the transformation law

m,n

yiq1 ...qr = yiq1 ...qr + ap 1 ...qr yi + ap 1 ...qr ,

¯p p t

tq iq

while the lower order terms remain unchanged. By direct evaluation we ¬nd

p

ar = br = 0. The same procedure takes place for gi . Hence A is an operator

of order r ’ 1. By recurrence we conclude A is a ¬rst order operator. This

completes the proof of proposition 31.1.

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

31. Two problems on general connections 257

31.4. Remark. In [Kol´ˇ, 81a] it was clari¬ed geometrically that the vertical

ar

prolongation V“ plays an important role in the theory of the original connec-

tion “. The uniqueness of V“ proved in proposition 31.1 gives a theoretical

justi¬cation of this phenomenon.

It is remarkable that there is another construction of V“ using ¬‚ow pro-

longations of vector ¬elds, see 45.4. The equivalence of both de¬nitions is an

interesting consequence of the uniqueness of operator V.

31.5. Natural operators transforming second order di¬erential equa-

tions into general connections. We recall that a second order di¬erential

equation on a manifold M is usually de¬ned as a vector ¬eld ξ : T M ’ T T M on

T M satisfying T pM —¦ ξ = idT M , where pM : T M ’ M is the bundle projection.

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

the homotheties. If [ξ, LM ] = ξ, then ξ is said to be a spray. There is a classi-

cal bijection between sprays and linear symmetric connections, which is used in

several branches of di¬erential geometry. (We shall obtain it as a special case of

a more general construction.)

A. Dekr´t, [Dekr´t, 88], studied the problem whether an arbitrary second

e e

order di¬erential equation on M determines a general connection on T M by

means of the naturality approach. He deduced rather quickly a simple analytical

expression for all ¬rst order natural operators. Only then he looked for the

geometrical interpretation. Keeping the style of this book, we ¬rst present the

geometrical construction and then we discuss the naturality problem.

According to 9.3, the horizontal projection of a connection “ on an arbitrary

¬bered manifold Y is a vector valued 1-form on Y , which will be called the

horizontal form of “.

On the tangent bundle T M , we have the following natural tensor ¬eld VM of

type 1 . Since T M is a vector bundle, its vertical tangent bundle is identi¬ed

1

with T M • T M . For every B ∈ T T M we de¬ne

(1) VM (B) = (pT M B, T pM B).

(A general approach to natural tensor ¬elds of type 1 on an arbitrary Weil

1

bundle is explained in [Kol´ˇ, Modugno, 92].)

ar

Given a second order di¬erential equation ξ on M , the Lie derivative Lξ VM

is a vector valued 1-form on T M . Let 1T T M be the identity on T T M . The

following result gives a construction of a general connection on T M determined

by ξ.

Lemma. For every second order di¬erential equation ξ on M , 1 (1T T M ’Lξ VM )

2

is the horizontal form of a connection on T M .

Proof. Let xi be local coordinates on M and y i = dxi be the induced coordinates

on T M . The coordinate expression of the horizontal form of a connection on

T M is

‚ ‚

+ Fij (x, y)dxi — j .

dxi —

(2)

‚xi ‚y

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

258 Chapter VII. Further applications

By (1), the coordinate expression of VM is

‚

dxi —

(3) .

‚y i

Having a second order di¬erential equation ξ of the form

‚ ‚

yi + ξ i (x, y) i

(4)

‚xi ‚y

we evaluate directly for Lξ VM

‚ξ i j

‚ ‚ ‚

’dx — i ’ j dx — i + dy i — i .

i

(5)

‚x ‚y ‚y ‚y

Hence 1 (1T M ’ Lξ VM ) has the required form

2

1 ‚ξ i j

‚ ‚

i

dx — i + dx — i .

(6)

2 ‚y j

‚x ‚y

31.6. Denote by A the operator from lemma 31.5. By the general theory, the

di¬erence of two general connections on T M ’ M is a section T M ’ V T M —

T — M = (T M • T M ) — T — M . The identity tensor of T M — T — M determines a

natural section IM : T M ’ V T M — T — M . Hence A + kI is a natural operator

for every k ∈ R.

Proposition. All ¬rst order natural operators transforming second order dif-

ferential equations on a manifold into connections on the tangent bundle form

the one-parameter family

k ∈ R.

A + kI,

Proof. We have the case of a morphism operator from 18.17 with C = Mfm ,

F1 = G1 = T , q = id, F2 = T1 , G2 = J 1 T and the additional conditions

2

that we consider the sections of T1 ’ T and J 1 T ’ T . Let S be the ¬ber of

2

J 1 (T1 Rm ’ T Rm ) over 0 ∈ Rm and Z be the ¬ber of J 1 T Rm over 0 ∈ Rm . By

2

18.19 we have to determine all G3 -equivariant maps f : S ’ Z over the identity

m

of T0 Rm .

i 2i

Denote by X i = dx , Y i = ddtx the induced coordinates on T1 Rm and by

2

2

dt

Xj = ‚Y i /‚xj , Yji = ‚Y i /‚X j the induced coordinates on S. By direct evalu-

i

ation, one ¬nds the following action of G3m

¯ ¯

X i = ai X j , Y i = ai X j X k + ai Y j

(1) j jk j

¯i

Xj = ai am X k X l + ai al Y k + 2ai ak am X l X n + ai Xlk al +

(2) klm ˜j kl ˜j kl ˜mj n ˜j

k

ai al am X n Ylk

k ˜mj n

¯

Yji = 2ai al X k + ai Ylk al

(3) kl ˜j ˜j

k

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

32. Jet functors 259

The standard coordinates Z i , Zj on Z have the transformation law

i

¯ ¯i

Z i = ai Z j , Zj = ai ak Z l + ai Zlk al .

(4) kl ˜j ˜j

j k

Let Z i = X i and Zj = fj (X p , Y q , Xn , Ylk ) be the coordinate expression of

i i m

f . The equivariance with respect to the homotheties in GL(m) ‚ G3 yields m

fj (X p , Y q , Xn , Ylk ) = fj (kX p , kY q , Xn , Ylk ).

i m i m

(5)

Hence fj do not depend on X p and Y q . Let ai = δj , ai = 0. Then the

i i

j jk

equivariance condition reads

fj (Xn , Ylk ) = fj (Xn + am X p X q , Ylk ).

i m i m

(6) npq

This implies fj are independent of Xn . Putting ai = δj , we obtain

i m i

j

fj (Ylk ) + ai X m = fj (Ylk + 2ak X m )

i i

(7) mj lm

with arbitrary ai . Di¬erentiating with respect to Ylk , we ¬nd ‚fj /‚Ylk = const.

i

jk

i

Hence fj are a¬ne functions. By the Invariant tensor theorem, we deduce

fj = k1 Yji + k2 δj Yk + k3 δj .

i ik i

(8)

1

Using (7) once again, we obtain k1 = 2 , k2 = 0. This gives the coordinate form

of our assertion.

31.7. Remark. If X is a spray, then the operator A from lemma 31.5 deter-

mines the classical linear symmetric connection induced by X. Indeed, 31.5.(4)

satis¬es the spray condition if and only if

‚ξ i j

y = 2ξ i .

‚y j

This kind of homogeneity implies ξ i = bi (x)y j y k . Then the coordinate form of

jk

A(X) is

dy i = bi (x)y j dxk .

jk

32. Jet functors

32.1. By 12.4, the construction of r-jets of smooth maps can be viewed as

a bundle functor J r on the product category Mfm — Mf . We are going to

determine all natural transformations of J r into itself. Denote by y : M ’ N

ˆ

r

the constant map of M into y ∈ N . Obviously, the assignment X ’ j±X βX

is a natural transformation of J r into itself called the contraction. For r = 1,

J 1 (M, N ) coincides with Hom(T M, T N ), which is a vector bundle over M — N .

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

260 Chapter VII. Further applications

Proposition. For r ≥ 2 the only natural transformations J r ’ J r are the

identity and the contraction. For r = 1, all natural transformations J 1 ’ J 1

form the one-parametric family of homotheties X ’ cX, c ∈ R.

Proof. Consider ¬rst the subcategory Mfm —Mfn ‚ Mfm —Mf . The standard

¬ber S = J0 (Rm , Rn )0 is a Gr — Gr -space and the action of (A, B) ∈ Gr — Gr

r

m n m n

on X ∈ S is given by the jet composition

¯

X = B —¦ X —¦ A’1 .

(1)

According to the general theory, the natural transformations J r ’ J r are in

bijection with the Gr — Gr -equivariant maps f : S ’ S.

m n

p p

Write A’1 = (˜i , . . . , ai 1 ...jr ), B = (bp , . . . , bp1 ...qr ), X = (Xi , . . . , Xi1 ...ir ) =

aj ˜j q q

(X1 , . . . , Xr ). Consider the equivariance of f = (f1 , . . . , fr ) with respect to the

homotheties in GL(m) ‚ Gr . This gives the homogeneity conditions

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 )

(2)

.

.

.

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

Taking into account the homotheties in GL(n), we further ¬nd

kf1 (X1 , . . . , Xr ) = f1 (kX1 , . . . , kXr )

.

.

(3) .

kfr (X1 , . . . , Xr ) = fr (kX1 , . . . , kXr ).

Applying the homogeneous function theorem to both (2) and (3), we deduce that

fs is linear in Xs and independent of the remaining coordinates, s = 1, . . . , r.

Consider furthemore the equivariance with respect to the subgroup GL(m) —