1p 1p

p p p p

(Yij ’ Yji ).

(3) Sij = (Yij + Yji ), Rij =

2 2

Since the right-hand side of (2) except the ¬rst term is symmetric in i and j, we

¯p q q

obtain the action formula for Sij by replacing Ykl by Skl on the right-hand side

of (2). On the other hand,

¯p q

Rij = ap Rkl ak al .

˜i ˜j

q

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234 Chapter VI. Methods for ¬nding natural operators

q p

The map γ : S1 ’ Rn — Λ2 Rm— , γ(yk , y rs , ymn ) = Rij will be called the formal

t

curvature map.

Let Z be any (G2 — G2 )-space. The canonical projection G2 ’ G2 and

m n m,n m

2 2

the group homomorphism Gm,n ’ Gn determined by the restriction of local

isomorphisms of Rm+n ’ Rm to {0} — Rn ‚ Rm+n de¬ne a map p : G2 ’ m,n

2 2 i i i p p

Gm — Gn . The kernel K of p is characterized by aj = δj , ajk = 0, aq = δq ,

ap = 0. The group G2 acts on Rn —Λ2 Rm— by means of the jet homomorphism

qr m,n

2 1 1

π1 into Gm — Gn . One sees directly, that the curvature map γ satis¬es the orbit

condition with respect to K. Indeed, on K we have

yi = yi + ap ,

¯p p

yiq = yiq + ap ,

¯p p

Sij = Sij + ap yj + ap yi + ap .

¯p p q q

(5) i iq qi qj ij

Using ap , aq , as , we can transform every (yi , yjr , Sk ) into (0, 0, 0). In this

pq s

i jr k

situation, proposition 28.1 yields directly the following assertion.

Proposition. Every G2 -map S1 ’ Z factorizes through the formal curvature

m,n

map γ : S1 ’ Rn — Λ2 Rm— .

28.3. The Utiyama theorem and general connections. In general, an r-th

order Lagrangian on a ¬bered manifold Y ’ M is de¬ned as a base-preserving

morphism J r Y ’ Λm T — M , m = dim M . Roughly speaking, the Utiyama theo-

rem reads that every invariant ¬rst order Lagrangian on the connection bundle

QP ’ M of an arbitrary principal ¬ber bundle P ’ M factorizes through the

curvature map. This assertion will be formulated in a precise way in the frame-

work of the theory of gauge natural operators in chapter XII. At this moment we

shall apply proposition 28.2 to deduce similar results for the general connections

on an arbitrary ¬bered manifold Y ’ M .

Since the action 28.2.(5) is simply transitive, proposition 28.2 re¬‚ects exactly

the possibilities for formulating Utiyama-like theorems for general connections.

But the general interpretation of proposition 28.2 in terms of natural operators

is beyond the scope of this example and we restrict ourselves to one special case

only.

If we let the group G2 —G2 act on a manifold S by means of the ¬rst product

m n

2

projection, we obtain a Gm -space, which corresponds to a second order bundle

functor F on Mfm . (In the classical Utiyama theorem we have the ¬rst order

bundle functor Λm T — , which is allowed to be viewed as a second order functor

as well.) Obviously, F can be interpreted as a bundle functor on FMm,n , if

we compose it with the base functor B : FM ’ Mf and apply the pullback

construction. If we interpret proposition 28.2 in terms of natural operators

between bundle functors on FMm,n , we obtain immediately

Proposition. There is a bijection between the ¬rst order natural operators

F and the zero order natural operators A0 : V — Λ2 T — B

A: J1 F given by

2—

1

A = A0 —¦ C, where C : J V — Λ T B is the curvature operator.

28.4. The general Ricci identity. Before treating the classical tensor ¬elds

on manifolds, we deduce a general result for arbitrary vector bundles. Consider a

linear connection “ on a vector bundle E ’ M and a classical linear connection

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28. The orbit reduction 235

Λ on M , i.e. a linear connection on T M ’ M . The absolute di¬erential s of

a section s : M ’ E is a section M ’ E — T — M . Hence we can use the tensor

product “ — Λ— of connection “ and the dual connection Λ— of Λ, see 47.14, to

construct the absolute di¬erential of s. This is a section 2 s : E —T — M —T — M

Λ

called the second absolute di¬erential of s with respect to “ and Λ. We describe

the alternation Alt( 2 s) : M ’ E — Λ2 T — M . Let R : M ’ E — E — — Λ2 T — M

Λ

be the curvature of “ and S : M ’ T M — Λ2 T — M be the torsion of Λ. Then

the contractions R, s and S, s are sections of E — Λ2 T — M .

Proposition. It holds

2

= ’ R, s + S,

(1) Alt( Λ s) s.

2

Proof. This follows directly from the coordinate formula for Λs

‚ ‚sp r

p ‚s

pq

’ “r sq + Λk p

’ “qi s ’ “rj ks .

qi ij

j ‚xi i

‚x ‚x

The coordinate form of (1) will be called the general Ricci identity of E. If

E is a vector bundle associated to P 1 M and “ is induced from a principal con-

nection on P 1 M , we take for Λ the connection induced from the same principal

connection. In this case we write 2 s only. For the classical tensor ¬elds on M

our proposition gives the classical Ricci identity, see e.g. [Lichnerowicz, 76, p.

69].

28.5. Curvature subspaces. We are going to describe some properties of

the absolute derivatives of curvature tensors of linear symmetric connections on

m-manifolds. Let Q = (Q„ P 1 Rm )0 denote the standard ¬ber of the connection

bundle in question, see 25.3, let W = Rm — Rm— — Λ2 Rm— , Wr = W — —r Rm— ,

W r = W — W1 — . . . — Wr . The formal curvature is a map C : Tm Q ’ W , 1

its formal r-th absolute di¬erential is Cr = r C : Tm Q ’ Wr . We write

r+1

C r = (C, C1 , . . . , Cr ) : Tm Q ’ W r , where the jet projections Tm Q ’ Tm Q,

r+1 r+1 s

s < r + 1, are not indicated explicitly. (Such a slight simpli¬cation of notation

will be used even later in this section.)

We de¬ne the r-th order curvature equations Er on W r as follows.

i) E0 are the ¬rst Bianchi identity

i i i

(1) Wjkl + Wklj + Wljk = 0

ii) E1 are the absolute derivatives of (1)

i i i

(2) Wjklm + Wkljm + Wljkm = 0

and the second Bianchi identity

i i i

(3) Wjklm + Wjlmk + Wjmkl = 0

iii) Es , s > 1, are the absolute derivatives of Es’1 and the formal Ricci

identity of the product vector bundle Ws’2 — Rm . By 28.4, the latter equations

are of the form

i

(4) Wjklm1 ···[ms’1 ms ] = bilin(W, Ws’2 )

where the right-hand sides are some bilinear functions on W — Ws’2 .

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236 Chapter VI. Methods for ¬nding natural operators

De¬nition. The r-th order curvature subspace K r ‚ W r is de¬ned by

E0 = 0, . . . , Er = 0.

We write K = K 0 ‚ W . For r = 1 we denote by K1 ‚ W1 the subspace

de¬ned by E1 = 0. Hence K 1 = K — K1 .

Lemma. K r is a submanifold of W r , it holds K r = C r (Tm Q) and the re-

r+1

stricted map C r : Tm Q ’ K r is a submersion.

r+1

Proof. To prove K r is a submanifold we proceed by induction. For r = 0 we

have a linear subspace. Assume K r’1 ‚ W r’1 is a submanifold. Consider the

product bundle K r’1 — Wr . Equations Er consist of the following 3 systems

i

(5) W{jkl}m1 ...mr = 0

i

(6) Wj{klm1 }m2 ...mr = 0

Wjklm1 ···[ms’1 ms ]···mr + polyn(W r’2 ) = 0

i

(7)

where {. . . } denotes the cyclic permutation and polyn(W r’2 ) are some poly-

nomials on W r’2 . The map de¬ned by the left-hand sides of (5)“(7) repre-

sents an a¬ne bundle morphism K r’1 — Wr ’ K r’1 — RN of constant rank,

N = the number of equations (5)“(7). Analogously to 6.6 we ¬nd that its kernel

K r is a subbundle of K r’1 — Wr .

To prove K r = C r (Tm Q) we also proceed by induction.

r+1

1 2

Sublemma. It holds K = C(Tm Q) and K1 = C1 (Tm Q).

Proof. The coordinate form of C is

Wjkl = “i ’ “i + “i “m ’ “i “m .

i

(8) jk,l jl,k ml jk mk jl

1

This is an a¬ne bundle morphism of a¬ne bundle Tm Q ’ Q into W of constant

rank. We know that the values of C lie in K, so that it su¬ces to prove that the

¯

image is the whole K at one point 0 ∈ Q. The restricted map C : Rm — S 2 Rm— —

Rm— ’ W is of the form

Wjkl = “i ’ “i .

i

(9) jk,l jl,k

Denote by dimE0 the number of independent equations in E0 , so that dimK =

¯

dimW ’ dimE0 . From linear algebra we know that K is the image of C if

¯

dimW ’ dimE0 = dimRm — S 2 Rm— — Rm— ’ dim KerC.

(10)

Clearly, dimW = m3 (m ’ 1)/2 and dimRm — S 2 Rm— — Rm— = m3 (m + 1)/2. By

¯ ¯

(9) we have KerC = Rm — S 3 Rm— , so that dim KerC = m2 (m + 1)(m + 2)/6.

One ¬nds easily that (1) represents one equation on W for any i and mutually

di¬erent j, k, l, while (1) holds identically if at least two subscripts coincide.

Hence dimE0 = m2 (m ’ 1)(m ’ 2)/6. Now (10) is veri¬ed by simple evaluation.

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

28. The orbit reduction 237

The absolute di¬erentiation of (8) yields that C1 is an a¬ne morphism of

2 1

a¬ne bundle Tm Q ’ Tm Q into W1 of constant rank. We know that the values

of C1 lie in K1 so that it su¬ces to prove that the image is the whole K1 at one

¯

point 0 ∈ Tm Q. The restricted map C1 : Rm — S 2 Rm— — S 2 Rm— ’ W1 is

1

Wjklm = “i

i i

jk,lm ’ “jl,km .

(11)

Analogously to (10) we shall verify the dimension condition

¯

dimW1 ’ dimE1 = dimRm — S 2 Rm— — S 2 Rm— ’ dim KerC1 .

(12)

Clearly, dimW1 = m4 (m ’ 1)/2, dimRm — —2 S 2 Rm— = m3 (m + 1)2 /4. We have

¯ ¯

KerC1 = Rm — S 4 Rm— , so that dim KerC1 = m2 (m + 1)(m + 2)(m + 3)/24.

For any i and mutually di¬erent j, k, l, m, (2) and (3) represent 8 equations,

but one ¬nds easily that only 7 of them are linearly independent. This yields

7m2 (m ’ 1)(m ’ 2)(m ’ 3)/24 independent equations. If exactly two subscripts

coincide, (2) and (3) represent 2 independent equations. This yields another

m2 (m ’ 1)(m ’ 2) equations. In the remaining cases (2) and (3) hold identically.

Now a direct evaluation proves our sublemma.

Assume by induction C r’1 : Tm Q ’ K r’1 is a surjective submersion. The

r

iterated absolute di¬erentiation of (8) yields the following coordinate form of Cr

Wjklm1 ...mr = “i

i r

(13) j[k,l]m1 ...mr + polyn(Tm Q)

where polyn(Tm Q) are some polynomials on Tm Q. This implies C r is an a¬ne

r r

bundle morphism

w

Cr

r+1

Kr

Tm Q

u u

wK

C r’1

r r’1

Tm Q

r

of constant rank. Hence it su¬ces to prove at one point 0 ∈ Tm Q that the

¯

image is the whole ¬ber of K r ’ K r’1 . The restricted map Cr : Rm — S 2 Rm— —

S r+1 Rm— ’ Wr is of the form

Wjklm1 ...mr = “i

i i

jk,lm1 ...mr ’ “jl,km1 ...mr .

(14)

¯

By (7) the values of Cr lie in W — S r Rm— . Then (5) and (6) characterize

(K — S r Rm— ) © (K1 — S r’1 Rm— ). Consider an element X = (Xjklm1 ...mr ) of the

i

¯

latter space. Since C1 (Rm —S 2 Rm— —S 2 Rm— ) = K1 by the sublemma, the tensor

¯

product C1 — idS r’1 Rm— : Rm — S 2 Rm— — S 2 Rm— — S r’1 Rm— ’ K1 — S r’1 Rm— is

a surjective map. Hence there is a Y ∈ Rm — S 2 Rm— — S 2 Rm— — S r’1 Rm— such

that

i i i

Xjklm1 ...mr = Yjklm1 ...mr ’ Yjlkm1 ...mr .

(15)

¯

Consider the symmetrization Y = (Yjkl(m1 m2 )···mr ) ∈ Rm — S 2 Rm— — S r+1 Rm— .

i

The second condition X ∈ K — S r Rm— implies X is symmetric in m1 and m2 ,

¯¯

so that Cr (Y ) = X.

Finally, since C r’1 : Tm Q ’ K r’1 is a submersion and C r : Tm Q ’ K r is

r r+1

an a¬ne bundle morphism surjective on each ¬ber, C r is also a submersion.

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

238 Chapter VI. Methods for ¬nding natural operators

28.6. Linear symmetric connections. A fundamental result on the r-th

order natural operators on linear symmetric connections with values in a ¬rst

order natural bundle is that they factorize through the curvature operator and

its absolute derivatives up to order r ’ 1. We present a formal version of this

result, which involves a precise description of the factorization.

Let F be a G1 -space, which is considered as a Gr+2 -space by means of the

m m

jet homomorphism Gr+2 ’ G1 .

m m

Theorem. For every Gr+2 -map f : Tm Q ’ F there exists a unique G1 -map

r

m m

g : K r’1 ’ F satisfying f = g —¦ C r’1 .

Proof. We use a recurrence procedure, in the ¬rst step of which we apply the

r+2

orbit reduction with respect to the kernel Br+1 of the jet projection Gr+2 ’

m

Gr+1 . Let Sr : Tm Q ’ Rm — S r+2 Rm— =: Sr+2 be the symmetrization

r 1

m

Sj1 ...jr+2 = “i 1 j2 ,j3 ...jr+2 )

i

(1) (j

r r r’1

and πr’1 : Tm Q ’ Tm Q be the jet projection. De¬ne

r r 1 r’1

•r = (Sr , πr’1 , Cr’1 ) : Tm Q ’ Sr+2 — Tm Q — Wr’1 .

The map Cr’1 is of the form

Wjkl1 ...lr = “i 1 ...lr ’ “i 1 ,kl2 ...lr + polyn(Tm Q).

i r’1

(2) jk,l jl

One sees easily that in the formula

“i 1 ...lr = Sjkl1 ...lr + (“i 1 ...lr ’ “i 1 ...lr ) )

i

(3) jk,l jk,l (jk,l

the expression in brackets can be rewritten as a linear combination of terms of

the form “i i i

mn,p1 ...pr ’ “mp1 ,np2 ...pr . If we replace each of them by Wmnp1 ...pr ’

r’1

polyn(Tm Q) according to (2), we obtain a map (not uniquely determined)

1 r’1 r

ψr : Sr+2 — Tm Q — Wr’1 ’ Tm Q over idTm Q satisfying

r’1

ψr —¦ •r = idTm Q .

(4) r

r+2 1

Consider the canonical action of Abelian group Br+1 = Sr+2 on itself, which

is simply transitive. From the transformation laws of “i it follows that ψr is

jk

r+2 1 r’1

a Br+1 -map. Thus the composed map f —¦ ψr : Sr+2 — Tm Q — Wr’1 ’ F

r+2

satis¬es the orbit condition for Br+1 with respect to the product projection

pr : Sr+2 — Tm Q — Wr’1 ’ Tm Q — Wr’1 . By 28.1 there is a Gr+1 -map

1 r’1 r’1

m

r’1

gr : Tm Q — Wr’1 ’ F satisfying f —¦ ψr = gr —¦ pr . Composing both sides with

r

•r , we obtain f = gr —¦ (πr’1 , Cr’1 ).

In the second step we de¬ne analogously

r’1 r’1 1 r’2

•r’1 = (Sr’1 , πr’2 , Cr’2 ) : Tm Q ’ Sr+1 — Tm Q — Wr’2

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28. The orbit reduction 239

1 r’2 r’1

and construct ψr’1 : Sr+1 — Tm Q — Wr’2 ’ Tm Q satisfying ψr’1 —¦ •r’1 =

1 r’2

idTm Q . The composed map gr —¦ (ψr’1 — idWr’1 ) : Sr+1 — Tm Q — Wr’2 —

r’1

r+1

Wr’1 ’ F is equivariant with respect to the kernel Br of the jet pro-

r+1 r 1 r’2