The next section is devoted to some problems from Riemannian geometry.

Here we complete our study of natural connections on Riemannian manifolds,

we prove the Gilkey theorem on natural di¬erential forms and we ¬nd all natural

lifts of Riemannian metrics to the tangent bundles. We also deduce that all

natural operators transforming linear symmetric connections into exterior forms

are generated by the Chern forms. Since there are no natural forms of odd

degree, all of them are closed.

In the last section, we present a survey of some results concerning the multi-

linear natural operators which are based heavily on the (linear) representation

theory of Lie algebras. First we treat the naturality over the whole category

Mfm , where the main tools come from the representation theory of in¬nite di-

mensional algebras of vector ¬elds. At the very end we comment brie¬‚y on the

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

250 Chapter VII. Further applications

category of conformal (Riemannian) manifolds, which leads to ¬nite dimensional

representation theory of some parabolic subalgebras of the Lie algebras of the

pseudo orthogonal groups.

30. The Fr¨licher-Nijenhuis bracket

o

The main goal of this section is to determine all bilinear natural operators of

the type of the Fr¨licher-Nijenhuis bracket. But we ¬nd it useful to start with a

o

technically simpler problem, which can serve as an introduction.

30.1. Bilinear natural operators T • Λp T — Λp T — . We are going to study

the natural operators transforming a vector ¬eld and an exterior p-form into an

exterior p-form. In order to get results of geometric interest, it is reasonable to

restrict ourselves to the bilinear operators. The two simplest examples of such

operators are (X, ω) ’ diX ω and (X, ω) ’ iX dω.

Proposition. All bilinear natural operators T • Λp T — Λp T — form the 2-

parameter family

k1 , k2 ∈ R.

(1) k1 diX ω + k2 iX dω,

Proof. First of all, every such operator has ¬nite order r by the bilinear Pee-

tre theorem. The canonical coordinates on the standard ¬ber S = J0 T Rm — r

J0 Λp T — Rm are X± , bi1 ...ip ,β , |±| ¤ r, |β| ¤ r, while the canonical coordinates

r i

on the standard ¬ber Z = Λp Rm— are ci1 ...ip . Since we consider the bilinear

i

operators, even the associated maps f : S ’ Z are bilinear in X± and bi1 ...ip ,β .

Using the homotheties in GL(m) ‚ Gr+1 , we obtain

m

k p f (X± , bi1 ...ip ,β ) = f (k |±|’1 X± , k p+|β| bi1 ...ip ,β ).

i i

(2)

This implies that only the products X i bi1 ...ip ,j and Xj bi1 ...ip can appear in f .

i

(In particular, every natural bilinear operator T • Λp T — Λp T — is a ¬rst order

operator.) Denote by f = f1 + f2 the corresponding decomposition of f .

The transformation laws of bi1 ...ip , bi1 ...ip ,j can be found in 25.4 and one

deduces easily

¯ ¯i

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

(3) kl ˜j ˜j

j k

In particular, the transformation laws with respect to the subgroup GL(m) ‚

G2 are tensorial in all cases. Hence we ¬rst have to determine the GL(m)-

m

equivariant bilinear maps Rm — Λp Rm— — Rm— ’ Λp Rm— . Consider the following

diagram

w Λ Ru

uz

f1

y

Rm — Λp Rm— — Rm— p m—

u u

id — Altp — id Alt

(4) p

w— R

Rm — —p+1 Rm— p m—

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30. The Fr¨licher-Nijenhuis bracket

o 251

where Alt denotes the alternator of the indicated degree. The vertical maps are

also GL(m)-equivariant and the GL(m)-equivariant map in the bottom row can

be determined by the invariant tensor theorem. This implies that f1 is a linear

combination of the contraction of X i with the derivation entry in bi1 ...ip ,j and

of the contraction of X i with a non-derivation entry in bi1 ...ip ,j followed by the

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—

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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 “.