alternation. To specify f2 , consider the diagram

w Λ Ru

˜

u z— Λ R

f2

y

m m— p m— p m—

R —R

u u

id — id — Altp Alt

(5) p

w— R

Rm — —p+1 Rm— p m—

˜

where f2 is the linearization of f2 . Taking into account the maps in the bot-

tom row determined by the invariant tensor theorem, we conclude similarly as

j

above that f2 is a linear combination of the inner contraction Xj multiplied

j

by bi1 ...ip and of the contraction Xi1 bi2 ...ip j followed by the alternation. Thus,

the equivariance of f with respect to GL(m) leads to the following 4-parameter

family

j j

fi1 ...ip = aX j bi1 ...ip ,j + bX j bj[i2 ...ip ,i1 ] + cXj bi1 ...ip + eX[i1 bi2 ...ip ]j

(6)

a, b, c, e ∈ R.

The equivariance of f on the kernel ai = δj is expressed by the relation

i

j

0 = ’ aX j (bki2 ...ip ak1 j + · · · + bi1 ...ip’1 k akp j )+

i i

(7)

bX j bk[i2 ...ip ak1 ]j + cak X j bi1 ...ip + eX j ak 1 bi2 ...ip ]k .

kj

i j[i

This implies

(8) c = 0 and a=b+e

which gives the coordinate form of (1).

30.2. The Lie derivative. Proposition 30.1 gives a new look at the well known

formula expressing the Lie derivative LX ω of a p-form as the sum of diX ω and

iX dω. Clearly, the Lie derivative operator on p-forms (X, ω) ’ LX ω is a bilinear

natural operator T • Λp T — Λp T — . By proposition 30.1, there exist certain

real numbers a1 and a2 such that

LX ω = a1 diX ω + a2 iX dω

for every vector ¬eld X and every p-form ω on m-manifolds. If we evaluate

a1 = 1 = a2 in two suitable special cases, we obtain an interesting proof of the

classical formula.

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

252 Chapter VII. Further applications

30.3. Bilinear natural operators T •Λp T — Λq T — . These operators can be

determined in the same way as in 30.1, see [Kol´ˇ, 90b]. That is why we restrict

ar

ourselves to the result. The only natural bilinear operators T • Λp T — Λp’1 T —

or Λp+1 T — are the constant multiples of iX ω or d(iX dω), respectively. In the

case q = p ’ 1, p, p + 1, we have the zero operator only.

30.4. Bilinear natural operators of the Fr¨licher-Nijenhuis type. The

o

wedge product of a di¬erential q-form and a vector valued p-form is a bilin-

ear map „¦q (M ) — „¦p (M, T M ) ’ „¦p+q (M, T M ) characterized by ω § (• —

X) = (ω § •) — X for all ω ∈ „¦q (M ), • ∈ „¦p (M ), X ∈ X(M ). Further let

C : „¦p (M, T M ) ’ „¦p’1 (M ) be the contraction operator de¬ned by C(ω — X) =

i(X)ω for all ω ∈ „¦p (M ), X ∈ X(M ). In particular, for P ∈ „¦0 (M, T M ) we have

C(P ) = 0. Clearly C(i(P )Q) is a linear combination of C(i(Q)P ), i(P )(C(Q)),

i(Q)(C(P )), P ∈ „¦p (M, T M ), Q ∈ „¦q (M, T M ). By I we denote IdT M , viewed

as an element of „¦1 (M, T M ).

Theorem. For dimM ≥ p + q, all bilinear natural operators A : „¦p (M, T M ) —

„¦q (M, T M ) ’ „¦p+q (M, T M ) form a vector space linearly generated by the

following 10 operators

dC(P ) § Q, dC(Q) § P, dC(P ) § C(Q) § I,

[P, Q],

dC(Q) § C(P ) § I, dC(i(P )Q) § I, i(P )dC(Q) § I,

(1)

i(Q)dC(P ) § I, d(i(P )C(Q)) § I, d(i(Q)C(P )) § I.

These operators form a basis if p, q ≥ 2 and m ≥ p + q + 1.

30.5. Remark. If p or q is ¤ 1, then all bilinear natural operators in question

are generated by those terms from 30.4.(1) that make sense. For example, in the

extreme case p = q = 0 our result reads that the only bilinear natural operators

X(M ) — X(M ) ’ X(M ) are the constant multiples of the Lie bracket. This was

proved by [van Strien, 80], [Krupka, Mikol´ˇov´, 84], and in an ˜in¬nitesimal™

as a

sense by [de Wilde, Lecomte, 82]. For a detailed discussion of all special cases

we refer the reader to [Cap, 90]. Clearly, for m < p + q we have the zero operator

only.

30.6. To prove theorem 30.4, we start with the fact that the bilinear Peetre

theorem implies that every A has ¬nite order r. Denote by Pji1 ...jp or Qi 1 ...jq j

m p m— m q m—

the canonical coordinates on R — Λ R or R — Λ R , respectively. The

associated map A0 of A is bilinear in P ™s and Q™s and their partial derivatives up

to order r. Using equivariance with respect to homotheties in GL(m), we ¬nd

that A0 contains only the products Pji1 ...jp ,k Qm ...nq and Qi 1 ...jq ,k Pn1 ...np , where

m

n1 j

the ¬rst term in both expressions means the partial derivative with respect to

xk . In other words, A is a ¬rst order operator and A0 is a sum A1 + A2 where

A1 : Rm — Λp Rm— — Rm— — Rm — Λq Rm— ’ Rm — Λp+q Rm—

A2 : Rm — Λp Rm— — Rm — Λq Rm— — Rm— ’ Rm — Λp+q Rm—

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

30. The Fr¨licher-Nijenhuis bracket

o 253

are bilinear maps. One ¬nds easily that the transformation law of Pji1 ...jp ,k is

m

¯

Pji1 ...jp ,k = Pm1 ...mp ,n ai am1 . . . ajpp an

l

l ˜j1 ˜ ˜k

m m

+ Pm1 ...mp (ai am1 . . . ajpp an + ai am1 am2 . . . ajpp + . . .

l

ln ˜j1 ˜ ˜k l ˜j1 k ˜j2 ˜

(1)

m

+ ai am1 . . . ajpp ).

l ˜j1 ˜k

30.7. Taking into account the canonical inclusion GL(m) ‚ G2 , we see that

m

the linear maps associated with the bilinear maps A1 and A2 , which will be

denoted by the same symbol, are GL(m)-equivariant. Consider ¬rst the following

diagram

w

uz yu

A1

Rm — Λp Rm— — Rm— — Rm — Λq Rm— Rm — Λp+q Rm—

u u

id — Altp — id — Altq id — Altp+q

(1)

w

m p m— m— m q m—

R — —p+q Rm— m

R —— R —R —R —— R

where Alt denotes the alternator of the indicated degree. It su¬ces to determine

all equivariant maps in the bottom row, to restrict them and to take the alterna-

tor of the result. By the invariant tensor theorem, all GL(m)-equivariant maps

—2 Rm — —p+q+1 Rm— ’ Rm — —p+q Rm— are given by all kinds of permutations

of the indices, all contractions and tensorizing with the identity. Since we apply

this to alternating forms and use the alternator on the result, permutations do

not play a role.

In what follows we discuss the case p ≥ 2, q ≥ 2 only and we leave the other

cases to the reader. (A direct discussion shows that in the remaining cases the

list (2) below should be reduced by those terms that do not make sense, but

the next procedure leads to theorem 30.4 as well.) Constructing A1 , we may

contract the vector ¬eld part of P into a non-derivation entry of P or into the

derivation entry of P or into Q, and we may contract the vector ¬eld part of

Q into Q or into a non-derivation entry of P or into the derivation entry of

P , and then tensorize with the identity of Rm . This gives 8 possibilities. If

we perform only one contraction, we get 6 further possibilities, so that we have

a 14-parameter family denoted by the lower case letters in the list (2) below.

Constructing A2 , we obtain analogously another 14-parameter family denoted

by upper case letters in the list (2) below. Hence GL(m)-equivariance yields the

following expression for A0 (we do not indicate alternation in the subscripts and

we write ±, β for any kind of free form-index on the right hand side)

aPm±,k Qn δl + bP±,m Qn δl + cP±,k Qn δl + dPmn±,k Qn δl +

m i m i m i m i

nβ nβ nmβ β

ePn±,m Qn δl + f Pn±,k Qn δl + gPm±,n Qn δl + hP±,n Qn δl +

m i m i m i m i

β mβ β mβ

iPm±,k Qi + jP±,m Qi + kP±,k Qi + lP±,k Qn + mPn±,k Qn +

m m m i i

β β mβ nβ β

nP±,n Qn + APm± Qn δl + BPm± Qn δl + CPmn± Qn δl +

i m i m i m i

(2) β nβ,k β,n β,k

DP± Qn

m i mn i mn i mn i

nmβ,k δl + EP± Qmβ,n δl + F Pn± Qmβ,k δl + GP± Qnβ,m δl +

HPn± Qn δl + IP± Qn + JP± Qn + KPn± Qn +

m i i i i

β,m nβ,k β,n β,k

LPm± Qi + M P± Qi

m m mi

mβ,k + N P± Qβ,m .

β,k

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

254 Chapter VII. Further applications

30.8. Then we consider the kernel K of the jet projection G2 ’ G1 . Using

m m

30.5.(1) with ai = δj , we evaluate that A0 is K-equivariant if and only if the

i

j

following coordinate expression

(1)

BPm± Qt an + (’1)q qBPm± Qn at + bP± Qn am ’

m m t

β tn tβ nk nβ tm

(’1)p+q pbPt± Qn at + ((’1)q c ’ D ’ (’1)q (q ’ 1)G)P± Qn at +

m m

nβ mk ntβ mk

(C ’ (’1)q d ’ (’1)p+q (p ’ 1)g)Pmt± Qn at + ePn± Qn at +

m m

β nk β mt

(H ’ e)Pt± Qn at + EP± Qt an + (h ’ E)P± Qn at +

m m m

β mn mβ tn tβ mn

((’1)q qH ’ (’1)q f ’ F )Pn± Qn at +

m

tβ mk

(F + (’1)q f ’ (’1)p+q ph)Pn± Qt an ’ (’1)p+q (p ’ 1)ePnt± Qn at ’

m m

mβ tk β mk

(’1)q (q ’ 1)EP± Qn at δl + jP± Qi am + (’1)p+q pjPt± Qi at +

m i t m

mtβ nk β tm β mk

JP± Qt am + (’1)q qJP± Qm at ’ ((’1)q k ’ (’1)q qN + M )P± Qi at +

i i m

β tm tβ mk tβ mk

(K + (’1)p+q pn ’ (’1)q m)Pm± Qt am ’ l(’1)q P± Qm ai +

i t

β tk mβ tk

LPm± Qt ai ’ (’1)q mPm± Qm ai + M P± Qt ai + (n + N )P± Qt ai

m t m m

β tk β tk mβ tk β mt

represents the zero map Rm — Λp Rm— — Rm — Λq Rm— — Rm — S 2 Rm— ’ Rm —

Λp+q Rm— .

For dimM ≥ p + q + 1, the individual terms in (1) are linearly independent.

Hence (1) is the zero map if and only if all the coe¬cients vanish. This leads to

the following equations

b=B=e=E=h=H=j=J =l=L=m=M =0

c = (q ’ 1)G + (’1)q D, C = (’1)q d + (’1)p+q (p ’ 1)g

(2)

F = (’1)q’1 f, K = (’1)p+q’1 pn,

k = ’qn, N = ’n

while a, A, d, D, f , g, G, n, i, I are independent parameters. This yields the

coordinate form of 30.4.(1).

In the case m = p + q, p, q ≥ 2, there are certain linear relations between the

individual terms of (1). They are described explicitly in [Cap, 90]. But even in

this case we obtain the ¬nal result in the form indicated in theorem 30.4.

30.9. Linear and bilinear natural operators on vector valued forms.

Roughly speaking, we can characterize theorem 30.4 by saying that the Fr¨licher-

o

Nijenhuis bracket is the only non-trivial operator in the list 30.4.(1), since the

remaining terms can easily be constructed by means of tensor algebra and ex-

terior di¬erentiation. We remark that the natural operators on vector valued

forms were systematically studied by A. Cap. He deduced the complete list of

all linear natural operators „¦p (M, T M ) ’ „¦q (M, T M ) and all bilinear natu-

ral operators „¦p (M, T M ) — „¦q (M, T M ) ’ „¦r (M, T M ), which can be found

in [Cap, 90]. From a general point of view, the situation is analogous to 30.4:

except the Fr¨licher-Nijenhuis bracket, all other operators in question can easily

o

be constructed by means of tensor algebra and exterior di¬erentiation.

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

31. Two problems on general connections 255

30.10. Remark on the Schouten-Nijenhuis bracket. This is a bilinear

operator C ∞ Λp T M — C ∞ Λq T M ’ C ∞ Λp+q’1 T M introduced geometrically

by [Schouten, 40] and further studied by [Nijenhuis, 55]. In [Michor, 87b] the

natural operators of this type are studied. The problem is technically much

simpler than in the Fr¨licher-Nijenhuis case and the same holds for the result:

o

The only natural bilinear operators Λp T • Λq T Λp+q’1 T are the constant

multiples of the Schouten-Nijenhuis bracket.

31. Two problems on general connections

31.1. Vertical prolongation of connections. Consider a connection “ : Y ’

J 1 Y on a ¬bered manifold Y ’ M . If we apply the vertical tangent functor V ,

we obtain a map V “ : V Y ’ V J 1 Y . Let iY : V J 1 Y ’ J 1 V Y be the canonical

involution, see 39.8. Then the composition

VY “ := iY —¦ V “ : V Y ’ J 1 V Y

(1)

is a connection on V Y ’ M , which will be called the vertical prolongation of “.

Since this construction has geometrical character, V is an operator J 1 J 1V

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) —

GL(n). This yields that fs corresponds to an equivariant map of Rn — S s Rm—

into itself. By the generalized invariant tensor theorem, it holds fs = cs Xs with

any cs ∈ R.

For r = 1 we have deduced fip = c1 Xi . For r = 2 consider the equivariance

p

with respect to the kernel of the jet projection G2 — G2 ’ G1 — G1 . Taking

m n m n

into account the coordinate form of the jet composition, we ¬nd that the action

p p ¯p p

of an element ((δj , ai ), (δq , bp )) on (Xi , Xij ) is Xi = Xi and

i p

˜jk qr

¯p p qr p

Xij = Xij + bp Xi Xj + Xk ak .

(4) ˜ij

qr

p

Then the equivariance condition for fij reads

p qr p p qr p

c2 Xij + (c1 )2 bp Xi Xj + c1 Xk ak = c2 (Xij + bp Xi Xj + Xk ak ).

(5) ˜ij ˜ij

qr qr

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

32. Jet functors 261

This implies c1 = c2 = 0 or c1 = c2 = 1. Assume by induction that our assertion

holds in the order r ’ 1. Consider the equivariance with respect to the kernel

of the jet projection Gr — Gr ’ Gr’1 — Gr’1 . The action of an element

m n m n

((δj , 0, . . . , 0, ai 1 ...jr ), (δq , 0, . . . , 0, bp1 ...qr )) leaves X1 , . . . , Xr’1 unchanged and

i p

˜j q

it holds

Xi1 ...ir = Xi1 ...ir + bp1 ...qr Xi11 . . . Xirr + Xj aj1 ...ir .

¯p p q q p

(6) ˜i

q

Then the equivariance condition for fip ...ir requires

1

(7) cr Xi1 ...ir + (c1 )r bp1 ...qr Xi11 . . . Xirr + c1 Xj aj1 ...ir

p q q p

˜i

q

= cr (Xi1 ...ir + bp1 ...qr Xi11 . . . Xirr + Xj aj1 ...ir ).

p q q p

˜i

q

This implies cr = c1 = 0 or 1.

For r = 1 we have a homothety f n : X ’ kn X, kn ∈ R, on each subcat-

egory Mfm — Mfn ‚ Mfm — Mf . If we take the value of the transforma-

tion (f 1 , . . . , f n , . . . ) on the product of idRm with the injection ia,b : Ra ’ Rb ,

(x1 , . . . , xa ) ’ (x1 , . . . , xa , 0, . . . , 0), a < b, and apply it to 1-jet at 0 of the map

x1 = t1 , x2 = 0, . . . , xa = 0, (t1 , . . . , tm ) ∈ Rm , we ¬nd ka = kb . For r ≥ 2 we

have on each subcategory either the identity or the contraction. Applying the

latter idea once again, we deduce that the same alternative must take place in

all cases.

32.2. The construction of the r-th jet prolongation J r Y of a ¬bered manifold

Y ’ X can be considered as a bundle functor on the category FMm . This

functor is also denoted by J r . However, in order to distinguish from 32.1, we

shall use J¬b for J r in the ¬bered case here.

r

r r

Proposition. The only natural transformation J¬b ’ J¬b is the identity.

Proof. The construction of product ¬bered manifolds de¬nes an injection Mfm

— Mf ’ FMm and the restriction of J¬b to Mfm — Mf is J r . For r = 1,

r

proposition 31.1 gives a one-parameter family

p p

(yi ) ’ (cyi )

(1)

1 1

of possible candidates for the natural transformation J¬b ’ J¬b . But the trans-

p

formation law of yi with respect to the kernel of the standard homomorphism

¯p p p

G1 1 1

m,n ’ Gm — Gn is yi = yi + ai . The equivariance condition for (1) reads

ap = cap , which implies c = 1.

i i

For r ≥ 2, proposition 32.1 o¬ers the contraction and the identity. But the

contraction is clearly not natural on the whole category FMm , so that only the

identity remains.

32.3. Natural transformations J 1 J 1 ’ J 1 J 1 . It is well known that the

canonical involution of the second tangent bundle plays a signi¬cant role in ap-

plications. A remarkable feature of the canonical involution on T T M is that it

exchanges both the projections pT M : T T M ’ T M and T pM : T T M ’ T M .

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

262 Chapter VII. Further applications

Nowadays, in several problems of the ¬eld theory the role of the tangent bun-

dle of a smooth manifold is replaced by the ¬rst jet prolongation J 1 Y of a

¬bered manifold p : Y ’ M . On the second iterated jet prolongation J 1 J 1 Y =

J 1 (J 1 Y ’ M ) there are two analogous projections to J 1 Y , namely the target

jet projection β1 : J 1 J 1 Y ’ J 1 Y and the prolongation J 1 β : J 1 J 1 Y ’ J 1 Y of

the target jet projection β : J 1 Y ’ Y . Hence one can ask whether there exists

a natural transformation of J 1 J 1 Y into itself exchanging the projections β1 and

J 1 β, provided J 1 J 1 is considered as a functor on FMm,n . But the answer is

negative.

Proposition. The only natural transformation J 1 J 1 ’ J 1 J 1 is the identity.

This assertion follows directly from proposition 32.6 below, so that we shall

not prove it separately. It is remarkable that we have a di¬erent situation on

¯

the subspace J 2 Y = {X ∈ J 1 J 1 Y, β1 X = J 1 β(X)}, which is called the second

semiholonomic prolongation of Y . There is a one-parametric family of natural

¯ ¯

transformations J 2 ’ J 2 , see 32.5.

32.4. An exchange map. However, one can construct a suitable exchange

map eΛ : J 1 J 1 Y ’ J 1 J 1 Y by means of a linear connection Λ on the base man-

ifold M as follows. Interpreting Λ as a principal connection on the ¬rst order

frame bundle P 1 M of M , we ¬rst explain how Λ induces a map hΛ : J 1 J 1 Y •

QP 1 M ’ Tm (Tm Y ). Every X ∈ J 1 J 1 Y is of the form X = jx ρ(z), where ρ is

1 1 1

a local section of J 1 Y ’ M , and for every u ∈ Px M we have Λ(u) = jx σ(z),

1 1

where σ is a local section of P 1 M ‚ J0 (Rm , M ). Taking into account the canon-

1

ical inclusion J 1 Y ‚ J 1 (M, Y ), the jet composition ρ(z) —¦ σ(z) de¬nes a local

map M ’ J0 (Rm , Y ) = Tm Y , the 1-jet of which jx (ρ(z) —¦ σ(z)) ∈ Jx (M, Tm Y )

1 1 1 1 1

depends on X and Λ(u) only. Since u ∈ J0 (Rm , M ), we have hΛ (X, u) =

1

1 11

jx (ρ(z) —¦ σ(z)) —¦ u ∈ Tm Tm Y . Furthermore, there is a canonical exchange map

11 11

κ : Tm Tm Y ’ Tm Tm Y , the de¬nition of which will be presented in the frame-

work of the theory of Weil bundles in 35.18. Using κ and hΛ , we construct a

map eΛ : J 1 J 1 Y ’ J 1 J 1 Y .

Lemma. For every X ∈ (J 1 J 1 Y )y there exists a unique element eΛ (X) ∈

J 1 J 1 Y satisfying

(1) κ(hΛ (X, u)) = hΛ (eΛ (X), u)

˜

˜

1

for any frame u ∈ Px M , x = p(y), provided Λ means the conjugate connection

of Λ.

Proof consists in direct evaluation, for which the reader is referred to [Kol´ˇ,

ar

Modugno, 91]. The coordinate form of eΛ is

yi = Yip ,

p

Yip = yi ,

p p p p p

yij = yji + (yk ’ Yk )Λk

(2) ji

where Yip = ‚y p /‚xi , yij = ‚yi /‚xj are the additional coordinates on J 1 (J 1 Y

p p

’ M ).

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

32. Jet functors 263

32.5. Remark. The subbundle J 2 Y ‚ J 1 J 1 Y is characterized by yi = Yip .

p

¯

¯

Formula 32.4.(2) shows that the restriction of eΛ to J 2 Y does not depend on Λ,

¯ ¯ ¯

so that we have a natural map e : J 2 Y ’ J 2 Y . Since J 2 Y ’ J 1 Y is an a¬ne

¯ ¯

bundle, e generates a one-parameter family of natural transformations J 2 ’ J 2

X ’ kX + (1 ’ k)e(X), k ∈ R.

¯

One proves easily that this family represents all natural transformations J 2 ’

¯2

J , see [Kol´ˇ, Modugno, 91].

ar

32.6. The map eΛ was introduced by M. Modugno by another construction, in

which the naturality ideas were partially used. Hence it is interesting to study

the whole problem purely from the naturality point of view.

Our goal is to ¬nd all natural transformations J 1 J 1 Y • QP 1 M ’ J 1 J 1 Y .

Since J 1 Y ’ Y is an a¬ne bundle with associated vector bundle V Y — T — M ,

we can de¬ne a map

δ : J 1 J 1 Y ’ V Y — T — M, A ’ β1 (A) ’ J 1 β(A).

(1)

On the other hand, proposition 25.2 implies directly that all natural operators

T M — T — M — T — M form the 3-parameter family

N : QP 1 M

ˆ ˆ

N : Λ ’ k1 S + k2 I — S + k3 S — I

(2)

ˆ

where S is the torsion tensor of Λ, S is the contracted torsion tensor and I is

the identity of T M . Using the contraction with respect to T M , we construct a

3-parameter family of maps

δ, N (Λ) : J 1 J 1 Y ’ V Y — T — M — T — M.

(3)

The well known exact sequence of vector bundles over J 1 Y

Vβ

0 ’ V Y — T —M ’ V J 1Y ’’ V Y ’ 0

’

(4)

shows that V Y —T — M —T — M can be considered as a subbundle in V J 1 Y —T — M ,

which is the vector bundle associated with the a¬ne bundle β1 : J 1 J 1 Y ’ J 1 Y .

Proposition. All natural transformations f : J 1 J 1 Y ’ J 1 J 1 Y depending on

a linear connection Λ on the base manifold form the two 3-parameter families

(5) I. f = id + δ, N (Λ) , II. f = eΛ + δ, N (Λ) .

Proof. The standard ¬bers V = (yi , Yip , yij ) and Z = (Λi ) are G2 -spaces

p p

m,n

jk

2

and we have to ¬nd all Gm,n -equivariant maps f : V — Z ’ V . The action of

G2 on V is

m,n

yi = ap yj aj + ap aj ,

¯p q

Yip = ap Yjq aj + ap aj

¯

˜i j ˜i ˜i j ˜i

q q

yij = ap ykl ak al + ap yk Ylr ak al + ap Ylq ak al +

¯p q q

(6) ˜i ˜j ˜i ˜j ˜i ˜j

q qr qk

+ ap yk ak al + ap yk ak + ap ak al + ap ak

q q

˜i ˜j ˜ij kl ˜i ˜j k ˜ij

q

ql

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

264 Chapter VII. Further applications

while the action of G2 on Z is given by 25.2.(3).

m,n

The coordinate form of an arbitrary map f : V — Z ’ V is

y = F (y, Y, y2 , Λ)

(7) Y = G(y, Y, y2 , Λ)

y2 = H(y, Y, y2 , Λ)

where y = (yi ), Y = (Yip ), y2 = (yij ), Λ = (Λi ). Considering equivariance of

p p

jk

(7) with respect to the base homotheties we ¬nd

kF (y, Y, y2 , Λ) = F (ky, kY, k 2 y2 , kΛ)

kG(y, Y, y2 , Λ) = G(ky, kY, k 2 y2 , kΛ)

(8)

k 2 H(y, Y, y2 , Λ) = H(ky, kY, k 2 y2 , kΛ).

By the homogeneous function theorem, F and G are linear in y, Y , Λ and

independent of y2 , while H is linear in y2 and bilinear in y, Y , Λ. The ¬ber

homotheties then yield

kF (y, Y, Λ) = F (ky, kY, Λ)

(9) kG(y, Y, Λ) = G(ky, kY, Λ)

kH(y, Y, Λ) = H(ky, kY, ky2 , Λ).

Comparing (9) with (8) we ¬nd that F and G are independent of Λ and H is

linear in y2 and bilinear in (y, Λ) and in (Y, Λ).

Since f is GL(m)—GL(n)-equivariant, we can apply the generalized invariant

tensor theorem. This yields

Fip = ayi + bYip

p

Gp = cyi + dYip

p

i

p p p

Hij = eyij + f yji +

(10)

p p p p p p

gyi Λk + hyi Λk + iyj Λk + jyj Λk + kyk Λk + lyk Λk +

jk kj ik ki ij ji

mYip Λk + nYip Λk + pYjp Λk + qYjp Λk + rYk Λk + sYk Λk .

p p

jk kj ik ki ij ji

The last step consists in expressing the equivariance of (10) with respect to the

subgroup of G2 characterized by ai = δj , ap = δq . This leads to certain simple

i p

m,n q

j

algebraic identities, which are equivalent to (5).

32.7. Remark. The only map in 32.6.(5) independent of Λ is the identity.

This proves proposition 32.3.

If we consider a linear symmetric connection Λ, then the whole family N (Λ)

vanishes identically. This implies

Corollary. The only two natural transformations J 1 J 1 Y ’ J 1 J 1 Y depending

on a linear symmetric connection Λ on the base manifold are the identity and

eΛ .

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

33. Topics from Riemannian geometry 265

¯

32.8. Remark. The functors J 2 and J 1 J 1 restricted to the category Mfm —

Mf ‚ FMm de¬ne the so called semiholonomic and non-holonomic 2-jets in the

sense of [Ehresmann, 54]. We remark that all natural transformations of each of

those restricted functors into itself are determined in [Kol´ˇ, Vosmansk´, 87].

ar a

Further we remark that [Kurek, to appear b] described all natural transfor-

mations T r— ’ T s— between any two one-dimensional covelocities functors from

12.8. He also determined all natural tensors of type 1 on T r— M , [Kurek, to

1

appear c].

33. Topics from Riemannian geometry

33.1. Our aim is to outline the application of our general procedures to the

study of geometric operations on Riemannian manifolds. Since the Riemannian

metrics are sections of a natural bundle (a subbundle in S 2 T — ), we can always

add the metrics to the arguments of the operation in question instead of spe-

cializing our general approach to categories over manifolds for the category of

Riemannian manifolds and local isometries. In this way, we reduce the problem

to the study of some equivariant maps between the standard ¬bers, in spite of

the fact that the Riemannian manifolds are not locally homogeneous in the sense

of 18.4. However, at some stage we mostly have to ¬x the values of the metric

entry by restricting ourselves to the invariance with respect to the isometries and

so we need description of all tensors invariant under the action of the orthogonal

group.

Let us write S+ T — for the natural bundle of elements of Riemannian metrics.

2

33.2. O(m)-invariant tensors. An O(m)-invariant tensor is a tensor B ∈

—p Rm — —q Rm— satisfying aB = B for all a ∈ O(m). The canonical scalar

product on Rm de¬nes an O(m)-equivariant isomorphism Rm ∼ Rm— . This

=

p+q m—

identi¬es B with an element from — R , i.e. with an O(m)-invariant linear

map —p+q Rm ’ R. Let us de¬ne a linear map •σ : —2s Rm ’ R, by

•σ (v1 — · · · — v2s ) = (vσ(1) , vσ(2) ).(vσ(3) , vσ(4) ) · · · (vσ(2k’1) , vσ(2s) ),

where ( , ) means the canonical scalar product de¬ned on Rm and σ ∈ Σ2s

is a permutation. The maps •σ are called the elementary invariants. The

fundamental result due to [Weyl, 46] is

Theorem. The linear space of all O(m)-invariant linear maps —k Rm ’ R is

spanned by the elementary invariants for k = 2s and is the zero space if k is

odd.

Proof. We present a proof based on the Invariant tensor theorem (see 24.4),

following the lines of [Atiyah, Bott, Patodi, 73]. The idea is to involve explicitly

all metrics gij ∈ S+ Rm— and then to look for GL(m)-invariant maps. So together

2

with an O(m)-invariant map • : —k Rm ’ R we consider the map • : S+ Rm— —

¯2

—k Rm ’ R, de¬ned by •(Im , x) = •(x) and •(G, x) = •((A’1 )T GA’1 , Ax)

¯ ¯ ¯

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

266 Chapter VII. Further applications

for all A ∈ GL(m), G ∈ S+ Rm— . By de¬nition, • is GL(m)-invariant. With

2

¯

the help of the next lemma, we shall be able to extend the map • to the whole

¯

2 m— km

S R —— R .

Let us write V = —k Rm . The map • induces a map GL(m) — V ’ R,

¯

T

(A, x) ’ •(A A, x) = •(Ax) and this map is extended by the same formula to

¯

a polynomial map f : gl(m) — V ’ R, linear in V . So fx (A) = f (A, x) = •(Ax)

is polynomial and O(m)-invariant for all x ∈ V , and f (A, x) = •(AT A, x) if A

¯

invertible.

Lemma. Let h : gl(m) ’ R be a polynomial map such that h(BA) = h(A)

for all B ∈ O(m). Then there is a polynomial F on the space of all symmetric

matrices such that h(A) = F (AT A).

Proof. In dimension one, we deal with the well known assertion that each even

polynomial, i.e. h(x) = h(’x), is a polynomial in x2 . However in higher dimen-

sions, the proof is quite non trivial. We present only the main ideas and refer

the reader to our source, [Atiyah, Bott, Patodi, 73, p. 323], for more details.

First notice that it su¬ces to prove the lemma for non singular matrices, for

then the assertion follows by continuity. Next, if AT A = P with P non singular

and if there is a symmetric Q, Q2 = P , then A lies in the O(m)-orbit of Q.

Indeed, Q is also non singular and B = AQ’1 satis¬es B T B = Q’1 AT AQ’1 =

Im . So it su¬ces to restrict ourselves to symmetric matrices.

Hence we want to ¬nd a polynomial map g satisfying h(Q) = g(Q2 ) for all

symmetric matrices. For every symmetric matrix P , there is the square root

√

P = Q if we extend the ¬eld of scalars to its algebraic closure. This can be

computed easily if we express √ = B T DB with an√

P orthogonal matrix B and

√

T

diagonal matrix D, since then P = B DB and D is the diagonal matrix

with the square roots of the eigen values of P on its diagonal. But we should

express Q as a universal polynomial in the elements pij of the matrix P . Let us

assume that all eigenvalues »i of P are di¬erent. Then we can write

m

P ’ »j

Q= »i .

»i ’ » j

i=1 j=i

Notice that the eigen values »i are given by rational functions of the elements

pij of P . Thus, in order to make this to a polynomial expression, we have ¬rst to

extend the ¬eld of complex numbers to the ¬eld K of rational functions (i.e. the

elements are ratios of polynomials in pij ™s). So for matrices with entries from K,

all eigen values depend polynomially on pij ™s. We also need their square roots

√

to express Q, but next we shall prove that after inserting Q = P into h(Q) all

square roots will factor out. For any ¬xed P , let us consider the splitting ¬eld

√

L over K with respect to the roots of the equation det(P ’ »2 ) = 0. So P is

polynomial over L. As a polynomial map, h extends to gl(m, L) and the next

sublemma shows that it is in fact O(m, L)-invariant.

Sublemma. Let L be any algebraic extension of R and let f : O(m, L) ’ L be

a rational function. If f vanishes on O(m, R) then f is zero.

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

33. Topics from Riemannian geometry 267

Proof. The Cayley map C : o(m, R) ’ O(m, R) is a birational isomorphism of

the orthogonal group with an a¬ne space. Hence there are ˜enough real points™

to make zero all coe¬cients of the rational map. For more details see [Atiyah,

Bott, Patodi, 73]

Now the basic fact is, that for any automorphism σ : L ’ L of the Galois

group of L over K we have (σQ)2 = σP = P and since both Q and σQ are sym-

metric, B = σQQ’1 is orthogonal. Hence we get σh(Q) = h(σQ) = h(BQ) =

h(Q). Since this holds for all σ, h(Q) lies in K and so h(Q) = g(Q2 ) for a

rational function g.

The latter equality remains true if P is a real symmetric matrix such that all

its eigen values are distinct and the denominator of g(P ) is non zero. If g = F/G

for two polynomials F and G, we get F (AT A) = h(A)G(AT A). If we choose A

so that G(AT A) = 0, we get F (AT A) = 0. Hence g is a globally de¬ned rational

function without poles and so a polynomial.

Thus, we have found a polynomial F on the space of symmetric matrices

such that h(A) = F (AT A) holds for a Zariski open set in gl(m). This proves

our lemma.

Let us continue in the proof of the Weyl™s theorem. By the lemma, every

fx satis¬es fx (A) = gx (AT A) for certain polynomial gx and so we get a poly-

nomial mapping g : S 2 Rm— — V ’ R linear in V . For all B, A ∈ GL(m, C) we

have g((B ’1 )T AT AB ’1 , Bx) = f (AB ’1 , Bx) = f (A, x) = g(AT A, x) and so

g : S 2 Rm— — V ’ R is GL(m)-invariant. Then the composition of g with the

symmetrization yields a polynomial GL(m)-invariant map —2 Rm— ——k Rm ’ R,

linear in the second entry. Each multi homogeneous component of degree s+1 in

the sense of 24.11 is also GL(m)-invariant and so its total polarization is a linear

GL(m)-invariant map H : —2s Rm— ——k Rm ’ R. Hence, by the Invariant tensor

theorem, k = 2s and H is a sum of complete contractions over possible permu-

tations of indices. Since the original mapping • is given by •(x) = g(Im , x),

Weyl™s theorem follows.

33.3. To explain the coordinate form of 33.2, it is useful to consider an ar-

bitrary metric G = (gij ) ∈ S+ Rm— . Let O(G) ‚ GL(m) be the subgroup

2

of all linear isomorphisms preserving G, so that O(m) = O(Im ). Clearly,

theorem 33.2 holds for O(G)-invariant tensors as well. Every O(G)-invariant

i1 ...i

tensor B = (Bj1 ...jp ) ∈ —p Rm — —q Rm— induces an O(G)-invariant tensor

p

k ...k

gi1 k1 . . . gip kp Bj11...jqp ∈ —p+q Rm— . Hence theorem 33.2 implies that all O(G)-

invariant tensors in —p Rm — —q Rm— with p + q even are linearly generated by

g i1 k1 . . . g ip kp gσ(k1 )σ(k2 ) . . . gσ(jq’1 )σ(jq )

where g ik gjk = δj , g ij = g ji , for all permutations σ of p + q letters.

i

Consequently, all O(G)-equivariant tensor operations are generated by: ten-

˜

sorizing by the metric tensor G : Rm ’ Rm— or by its inverse G : Rm— ’ Rm ,

applying contractions and permutations of indices, and taking linear combina-

tions.

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268 Chapter VII. Further applications

33.4. Our next main goal is to prove the famous Gilkey theorem on natural

exterior forms on Riemannian metrics, i.e. to determine all natural operators

S+ T — Λp T — . This will be based on 33.2 and on the reduction theorems from

2

section 28. But since the resulting forms come from the Levi-Civit` connection

a

via the Chern-Weil construction, we ¬rst determine all natural operators trans-

forming linear symmetric connections into exterior forms. This will help us to

describe easily the metric operators later on.

Let us start with a description of natural tensors depending on symmetric

linear connections, i.e. natural operators Q„ P 1 T (p,q) , where T (p,q) Rm =

Rm — —p Rm — —q Rm— . Each covariant derivative of the curvature R(“) ∈

C ∞ (T M — T — M — Λ2 T — M ) of the connection “ on M is natural. Further every

tensor multiplication of two natural tensors and every contraction on one covari-

ant and one contravariant entry of a natural tensor give new natural tensors.

Finally we can tensorize any natural tensor with a GL(m)-invariant tensor, we

can permute any number of entries in the tensor products and we can repeat

each of these steps and take linear combinations.

Lemma. All natural operators Q„ P 1 T (p,q) are obtained by this procedure.

In particular, there are no non zero operators if q ’ p = 1 or q ’ p < 0.

Proof. By 23.5, every such operator has some ¬nite order r and so it is deter-

mined by a smooth Gr+2 -equivariant map f : Tm Q ’ V , where Q is the standard

r

m

¬ber of the connection bundle and V = —p Rm ——q Rm— . By the proof of the the-

orem 28.6, there is a G1 -equivariant map g : W r’1 ’ V such that f = g —¦ C r’1 .

m

Here W r’1 = W — . . . — Wr’1 , W = Rm — Rm— — Λ2 Rm— , Wi = W — —i Rm— ,

i = 1, . . . , r ’ 1. Therefore the coordinate expression of a natural tensor is given

by smooth maps

i ...i i i

ωj1 ...jp (Wjkl , . . . , Wjklm1 ...mr’1 ).

1 q

Hence we can apply the Homogeneous function theorem (see 24.1). The action

of the homotheties c’1 δj ∈ G1 gives

i

m

i ...i i ...i

cq’p ωj1 ...jp (Wjkl , . . . , Wjklm1 ...mr’1 ) = ωj1 ...jp (c2 Wjkl , . . . , cr+1 Wjklm1 ...mr’1 ).

i i i i

1 q 1 q

Hence the ω™s must be sums of homogeneous polynomials of degrees ds in the

i

variables Wjklm1 ...ms satisfying

2d0 + · · · + (r + 1)dr’1 = q ’ p.

(1)

Now we can consider the total polarization of each multi homogeneous compo-

nent and we obtain linear mappings

S d0 W — · · · — S dr’1 Wr’1 ’ V.

According to the Invariant tensor theorem, all the polynomials in question are

linearly generated by monomials obtained by multiplying an appropriate number

i

of variables Wjkl± and applying some of the GL(m)-equivariant operations.

If q = p, then the polynomials would be of degree zero, and so only the

GL(m)-invariant tensors can appear. If q ’ p = 1 or q ’ p < 0, there are no non

negative integers solving (1).

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

33. Topics from Riemannian geometry 269

33.5. Natural forms depending on linear connections. To determine

Λq T — , we have to consider the case p = 0 and

the natural operators Q„ P 1

apply the alternation to the subscripts. It is well known that the Chern-Weil

construction associates a natural form to every polynomial P which is de¬ned

on Rm — Rm— and invariant under the action of GL(m). This natural form is

obtained by substitution of the entries of the matrix valued curvature 2-form

R for the variables and taking the wedge product for multiplication. So if P

is homogeneous of degree j, then P (R) is a natural 2j-form. Let us denote by

ωq the form obtained from the tensor product of q copies of the curvature R

by taking its trace and alternating over the remaining entries. In coordinates,

kq kq’1

k1

ωq = (Rk1 ab Rk2 cd . . . Rkq ef ), where we alternate over all indices a, . . . , f . One

¬nds easily that the polynomials Pq depending on the entries of the matrix 2-

form R correspond to the homogeneous components of degree q in det(Im + R)

and so the forms ωq equal the Chern forms cq up to the constant factor (i/(2π))q .

The wedge product on the linear space of all natural forms depending on

connections de¬nes the structure of a graded algebra.

•m Λp T — is gener-

Theorem. The algebra of all natural operators Q„ P 1 p=0

ated by the Chern forms cq .

In particular, there are no natural forms with odd degrees and consequently

all natural forms are closed.

Proof. We have to continue our discussion from the proof of the lemma 33.4.

i

However, we need some relations on the absolute derivatives Rjklm1 ...ms of the

curvature tensor. First recall the antisymmetry, the ¬rst and the second Bianchi

identity, cf. 28.5

i i

Rjkl = ’Rjlk

(1)

i i i

(2) Rjkl + Rklj + Rljk = 0

i i i

(3) Rjklm + Rjlmk + Rjmkl = 0

i

Lemma. The alternation of Rjklm1 ...ms over any 3 indices among the ¬rst four

subscripts is zero.

Proof. Since the covariant derivative commutes with the tensor operations like

i i

the permutation of indices, it su¬ces to discuss the variables Rjkl and Rjklm .

i

By (2), the alternation over the subscripts in Rjkl is zero and (3) yields the same

i

for the alternation over k, l, m in Rjklm . In view of (1), it remains to discuss

i i i

the alternation of Rjklm over j, l, m. (1) implies Rjkml = ’Rjmkl and so we

can rewrite this alternation as follows

i i i i

Rjklm + Rjmkl + Rjlmk ’ Rjlmk

i i i i

+Rmkjl + Rmlkj + Rmjlk ’ Rmjlk

i i i i

+Rlkmj + Rljkm + Rlmjk ’ Rlmjk .

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

270 Chapter VII. Further applications

The ¬rst three entries on each row form a cyclic permutation and hence give

zero. The same applies to the last column.

Now it is easy to complete the proof of the theorem. Consider ¬rst a monomial

i

containing at least one quantity Rjklm1 ...ms with s > 0. Then there exists

one term of the product with three free subscripts among the ¬rst four ones

i

or one term Rjkl with all free subscripts, so that the monomial vanishes after

i i i

alternation. Further, (1) and (2) imply Rjkl ’ Rlkj = ’Rklj . Hence we can

restrict ourselves to contractions with the ¬rst subscripts and so all the possible

kq kq’1

k1

natural forms are generated by the expressions Rk1 ab Rk2 cd . . . Rkq ef where the

indices a, . . . , f remain free for alternation. But these are coordinate expressions

of the forms ωq .

33.6. Characteristic classes. The dimension of the homogeneous component

of the algebra of natural forms of degree 2s equals the number π(s) of the parti-

tions of s into sums of positive integers. Since all natural forms are closed, they

determine cohomology classes in the De Rham cohomologies of the underlying

manifolds. It is well known from the Chern-Weil theory that these classes do

not depend on the connection. This can be deduced as follows.

¯ ¯

Consider two linear connections “, “ expressed locally by “i , “i ∈ (T — M —

j j

i ¯i

— — 2—

T M )—T M , and their curvatures Rj , Rj ∈ (T M —T M )—Λ T M . Write “t =

¯

t“ + (1 ’ t)“ and analogously Rt for the curvatures. Let Pq be the polynomial

¯

de¬ning the form ωq and Q be its total polarization. We de¬ne „q (“, “) =

1 ¯ d d

q 0 Q(“ ’ “, Rt , . . . , Rt )dt. The structure equation yields dt Rt = dt (d“t ) ’

¯

d d

dt “t § “t ’ “t § dt “t = d(“ ’ “) and we calculate easily in normal coordinates

1 1

d d

¯

ωq (“) ’ ωq (“) = Q(Rt , . . . , Rt )dt = q Q( Rt , Rt , . . . , Rt )dt

dt dt

0 0

1

¯ ¯

dQ(“ ’ “, Rt , . . . , Rt )dt = d„q (“, “).

=q

0

Λ2q’1 T — and the

In fact, „q is one of many natural operators Q„ P 1 — Q„ P 1

integration helps us to ¬nd the proper linear combination of more elementary

operators which are obtained by a procedure similar to that from 33.4“33.5. The

¯

form „q (“, “) is called the transgression.

33.7. Natural forms on Riemannian manifolds. Since there is the natural

Levi-Civit` connection, we can evaluate the natural forms from 33.5 using the

a

curvature of this connection. In this case 28.14.(3) holds, i.e.

n n

gin Wjklm1 ...mr = ’gjn Wiklm1 ...mr .

(1)

For gij = δij , r = 0, this implies

j

i

Rjkl = ’Rikl

(2)

k k

k1

q q’1

and so the contractions in a monomial Rk1 ab Rk2 cd . . . Rkq ef yield zero if q is

odd. The natural forms pj = (2π)’2j ω2j are called the Pontryagin forms. The

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

33. Topics from Riemannian geometry 271

dimension of the homogeneous component of degree 4s of the algebra of forms

generated by the Pontryagin forms is π(s), cf. 33.6.

If we assume the dependence of the natural operators on the metric, then

every two indices of any tensor can be contracted. In particular, the complete

contractions of covariant derivatives of the curvature of the Levi-Civit` connec-

a

tion give rise to natural functions of all even orders grater then one. Composing

k natural functions with any ¬xed smooth function Rk ’ R, we get a new natu-

ral function. Since every natural form can be multiplied by any natural function

without loosing naturality, we see that there is no hope to describe all natural

forms in a way similar to 33.5. However, in Riemannian geometry we often meet

operations with a sort of homogeneity with respect to the change of the scale of

the metric and these can be described in more details.

Our operators will have several arguments as a rule and we shall use the

following brief notation in this section: Given several natural bundles Fa , . . . , Fb ,

we write Fa — . . . — Fb for the natural bundle associating to each m-manifold M

the ¬bered product Fa M —M . . .—M Fb M and similarly on morphisms. (Actually,

this is the product in the category of functors, cf. 14.11.) Hence D : F1 —F2 G

∞

means a natural operator transforming couples of sections from C (F1 M ) and

C ∞ (F2 M ) to sections from C ∞ (GM ) (which is also denoted by D : F1 •F2 G

in this book). Analogously, given natural operators D1 : F1 G1 and D2 : F2

G2 , we use the symbol D1 — D2 : F1 — F2 G1 — G2 .

De¬nition. Let E and F be natural bundles over m-manifolds. We say that a

natural operator D : S+ T — — E

2

F is conformal, if D(c2 g, s) = D(g, s) for all

metrics g, sections s, and all positive c ∈ R. If F is a natural vector bundle and

D satis¬es D(c2 g) = c» D(g), then » is called the weight of D.

Let us notice that the weight of the metric gij is 2 (we consider the inclusion

g : S+ T — S 2 T — ), that of its inverse g ij is ’2, while the curvature and all its

2

covariant derivatives are conformal.

33.8. Gilkey theorem. There are no non zero natural forms on Riemannian

manifolds with a positive weight. The algebra of all conformal natural forms on

Riemannian manifolds is generated by the Pontryagin forms.

33.9. Let us start the proof with a discussion on the reduction procedure de-

veloped in section 28. Even if we have no estimate on the order, we can get

an analogous result. Consider an arbitrary natural operator Q„ P 1 — E F.

By the non-linear Peetre theorem, D is of order in¬nity and so it is determined

by the restriction D of its associated mapping J ∞ ((Q„ P 1 — E)Rm ) ’ F Rm

to the ¬ber over the origin. Moreover, we obtain an open ¬ltration of the

whole ¬ber J0 ((Q„ P 1 — E)Rm ) consisting of maximal G∞ -invariant open sub-

∞

m

∞

sets Uk where the associated mapping D factorizes through Dk : πk (Uk ) ‚

J0 ((Q„ P 1 — E)Rm ) ’ F0 Rm . Now, we can apply the same procedure as in

k

∞

the section 28 to this invariant open submanifolds πk (Uk ).

Let F be a ¬rst order bundle functor on Mfm , E be an open natural sub

¯

bundle of a vector bundle functor E on Mfm . The curvature and its covariant

derivatives are natural operators ρk : Q„ P 1 Rk , with values in tensor bundles

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

272 Chapter VII. Further applications