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

q

˜ij kl ˜i ˜j kq ˜j ˜i k ˜ij

q

On the other hand, the standard ¬ber of V — —2 T — B is Rn — —2 Rm— with

p

canonical coordinates zij and the following action

¯p q

zij = ap zkl ak al

˜i ˜j

q

We have to determine all G2 - equivariant maps S1 ’ Rn — —2 Rm— . Let

m,n

p p q r t

zij = fij (yk , y s , ymn ) be the coordinate expression of such a map. Consider the

canonical injection of GL(m)—GL(n) into G2 de¬ned by 2-jets of the products

m,n

m n

of linear transformations of R and R . The equivariance with respect to the

homotheties in GL(m) gives a homogeneity condition

p q p q

k 2 fij (yk , y rs , ymn ) = fij (kyk , ky rs , k 2 ymn ).

t t

When applying the homogeneous function theorem, we have to discuss the equa-

tion 2 = d1 + d2 + 2d3 . Hence fij is a sum gij + hp where gij is a linear map

p p p

ij

of Rn — Rm— — Rm— into itself and hp is a polynomial map Rn — Rm— — Rn —

ij

Rn— — Rm— ’ Rn — Rm— — Rm— . Then we see directly that both gij and hp are

p

ij

p

GL(m) — GL(n)-equivariant. For hij we have deduced in example 27.2

hp = ayi yjq + byi yjq + cyj yiq + dyj yiq

pq qp pq qp

ij

p

while for gij a direct use of theorem 27.1 yields

p p p

gij = eyij + f yji .

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

232 Chapter VI. Methods for ¬nding natural operators

Moreover, the equivariance with respect to the subgroup K ‚ G2 character-

m,n

ized by ai = δj , ap = δq leads to the relations a = 0 = c, e = ’f = ’b = d.

i p

q

j

p p p qp qp

Hence fij = e(yij ’ yji ’ yi yqj + yj yqi ), which is the coordinate expression of

eC, e ∈ R.

V ——2 T — B.

Step II. Assume we have an r-th order natural operator A : J 1

It corresponds to a Gr+1 -equivariant map from the standard ¬ber Sr of J r J 1

m,n

p p

n m— m—

into R —R —R . Denote by yi±β the partial derivative of yi with respect to a

multi index ± in xi and β in y p . Any map f : Sr ’ Rn —Rm— —Rm— is of the form

p

f (yi±β ), ± + β ¤ r. Similarly to the ¬rst part of the proof, GL(m) — GL(n) can

be considered as a subgroup of Gr+1 . One veri¬es easily that the transformation

m,n

p

law of yi±β with respect to GL(m) — GL(n) is tensorial. Using the homotheties

p p

in GL(m), we obtain a homogeneity condition k 2 f (yi±β ) = f (k |±|+1 yi±β ). This

implies that f is a polynomial linear in the coordinates with |±| = 1 and bilinear

in the coordinates with |±| = 0. Using the homotheties in GL(n), we ¬nd

p p

kf (yi±β ) = f (k 1’|β| yi±β ). This yields that f is independent of all coordinates

with |±| + |β| > 1. Hence A is a ¬rst order operator.

V ——2 T — B

Step III. Using 23.7 we conclude that every natural operator J 1

has ¬nite order. This completes the proof.

27.4. Curvature-like operators on pairs of connections. The Fr¨licher- o

Nijenhuis bracket [“, ∆] =: κ(“, ∆) of two general connections “ and ∆ on Y is

a section Y ’ V Y — Λ2 T — BY , which may be called the mixed curvature of “

and ∆. Since the pair “, ∆ can be interpreted as a section Y ’ J 1 Y —Y J 1 Y ,

V — Λ2 T — B between two bundle functors

κ is a natural operator κ : J 1 • J 1

V — Λ2 T — B or C2 : J 1 • J 1 V — Λ2 T — B

on FMm,n . Let C1 : J 1 • J 1

denote the curvature operator of the ¬rst or the second connection, respectively.

The following assertion can be deduced in the same way as proposition 27.3, see

[Kol´ˇ, 87a].

ar

V — —2 T — B form the following

Proposition. All natural operators J 1 • J 1

3-parameter family

k1 , k2 , k3 ∈ R.

k1 C1 + k2 C2 + k3 κ,

From a general point of view, this result enlightens us on the fact that the

mixed curvature of two general connections can be de¬ned in an ˜essentially

unique™ way, i.e. the possibility of de¬ning the mixed curvature is limited by the

above 3-parameter family with trivial terms C1 and C2 .

27.5. Remark. [Kurek, 91] deduced that the only natural operator J 1

V — Λ3 T — B is the zero operator. This result presents an interesting point of

view to the Bianchi identity for general connections.

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

28. The orbit reduction 233

28. The orbit reduction

We are going to explain another general procedure used in the theory of nat-

ural operators. From the computational point of view, the orbit reduction is an

almost self-evident assertion about independence of the maps in question on some

variables. This was already used e.g. for the simpli¬cation of (4) in 26.12. But

the explicit formulation of such a procedure presented below is useful in several

problems. First we discuss a concrete example, in which we obtain a Utiyama-

like theorem for general connections. Then we present a complete treatment of

the ˜classical™ reduction theorems from the theory of linear connections and from

Riemannian geometry.

28.1. Let p : G ’ H be a Lie group homomorphism with kernel K, M be a G-

space, Q be an H-space and π : M ’ Q be a p-equivariant surjective submersion,

i.e. π(gx) = p(g)π(x) for all x ∈ M , g ∈ G. Having p, we can consider every

H-space N as a G-space by gy = p(g)y, g ∈ G, y ∈ N .

Proposition. If each π ’1 (q), q ∈ Q is a K-orbit in M , then there is a bijection

between the G-maps f : M ’ N and the H-maps • : Q ’ N given by f = • —¦ π.

Proof. Clearly, • —¦ π is a G-map M ’ N for every H-map • : Q ’ N . Con-

versely, let f : M ’ N be a G-map. Then we de¬ne • : Q ’ N by •(π(x)) =

f (x). This is a correct de¬nition, since π(¯) = π(x) implies x = kx with k ∈ K

x ¯

by the orbit condition, so that •(π(¯)) = f (kx) = p(k)f (x) = ef (x). We have

x

f = • —¦ π by de¬nition and • is smooth, since π is a surjective submersion.

28.2. Example. We continue in our study of the standard ¬ber

S1 = J0 (J 1 (Rm+n ’ Rm ) ’ Rm+n )

1

corresponding to the ¬rst order operators on general connections from 27.3. If

p

we replace the coordinates yij by

p p pq

(1) Yij = yij + yiq yj ,

we ¬nd easily that the action of G2 on S1 is given by 27.3.(1), 27.3.(2) and

m,n

Yij = ap Ykl ak al + ap yk yl ak al + ap yk al ak + ap yk ak al

¯p q q q

rs

˜i ˜j ˜i ˜j ˜i ˜j ˜i ˜j

q rs ql ql

(2)

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

q

˜ij kl ˜i ˜j k ˜ij

q

De¬ne further

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

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

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

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

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 .

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

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

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

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

jection Gm ’ Gm . The product projection of Sr+1 — Tm Q — Wr’2 —

r+1

Wr’1 omitting the ¬rst factor satis¬es the orbit condition for Br . This

yields a Gr -map gr’1 : Tm Q — Wr’2 — Wr’1 ’ F such that gr = gr’1 —¦

r’2

m

r’1 r

(πr’2 , Cr’2 ) — idWr’1 ,i.e. f = gr’1 —¦ (πr’2 , Cr’2 , Cr’1 ).

In the last but one step we construct a G2 -map g1 : Q — W — . . . — Wr’1 ’ F

m

r

such that f = g1 —¦ (π0 , C, . . . , Cr’1 ). The product projection p1 of Q — W — . . . —

2

Wr’1 omitting the ¬rst factor satis¬es the orbit condition for the kernel B1 of the

jet projection G2 ’ G1 . By 28.1 there is a G1 -map g0 : W — . . . — Wr’1 ’ F

m m m

satisfying g1 = g0 —¦ p1 . Hence f = g0 —¦ C r’1 . Since K r’1 = C r’1 (Tm Q), the

r

restriction g = g0 |K r’1 is uniquely determined.

&(

&& & &

r

Tm Q

& & &&

&&

ψr

& &&

r

πr’1 —Cr’1

u

f

w T Q—W wF

pr gr

1 r’1 r’1

eege

—T Q—W

S r’1 r’1

r+2 m m

ee

e

ee

ψr’1 —idWr’1 r’1

πr’2 —Cr’2 —idWr’1

e

ee u

w T Q—W —W w Fu

pr’1 gr’1

’“

’

1 r’2 r’2

—T Q—W —W

Sr+1 r’2 r’1 r’2 r’1

u

m m

’’

’’

. g0

. g1

.

u’

wW p1

r’1 r’1

Q—W

T — — T — . By

28.7. Example. We determine all natural operators Q„ P 1

23.5, every such operator has a ¬nite order r. Let

u = f (“0 , “1 , . . . , “r )

“s ∈ Rm — S 2 Rm— — S s Rm— , be its associated map. The equivariance of f with

respect to the homotheties in G1 ‚ Gr+2 yields

m m

k 2 f (“0 , “1 , . . . , “r ) = f (k“0 , k 2 “1 , . . . , k r+1 “r ).

By the homogeneous function theorem, f is a ¬rst order operator. According to

are in bijection with G1 -maps K ’ Rm— — Rm— .

28.6, the ¬rst order operators m

Let u = g(W ) be such a map. The equivariance with respect to the homotheties

yields k 2 g(W ) = g(k 2 W ), so that g is linear. Consider the injection i : K ’

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

240 Chapter VI. Methods for ¬nding natural operators

Rm — —3 Rm— . Since Rm — —3 Rm— is a completely reducible GL(m)-module,

there is an equivariant projection p : Rm — —3 Rm— ’ K satisfying p —¦ i = idK .

Hence we can proceed analogously to 24.8. By the invariant tensor theorem, all

linear G1 -maps Rm — —3 Rm— ’ Rm— — Rm— form a 6-parameter family. Its

m

restriction to K gives the following 2-parameter family

k k

k1 Wkij + k2 Wikj .

Let R1 and R2 be the corresponding contractions of the curvature tensor. By

T — —T — form a two parameter family

theorem 28.6, all natural operators Q„ P 1

linearly generated by two contractions R1 and R2 of the curvature tensor.

˜

28.8. Ricci subspaces. Let V = Rn be a GL(m)-module and V denote

the corresponding ¬rst order natural vector bundle over m-manifolds. Write

Vr = V — —r Rm— , V r = V — V1 — . . . — Vr . The formal r-th order absolute

di¬erentiation de¬nes a map Dr = r : Tm Q — Tm V ’ Vr , D0 = idV . If v p ,

V r’1 r V

p p r

vi , . . . , vi1 ...ir are the jet coordinates on Tm V (symmetric in all subscripts) and

Vip ...ir are the canonical coordinates on Vr , then Dr is of the form

V

1

Vip ...ir = vi1 ...ir + polyn(Tm Q — Tm V ).

p r’1 r’1

(1) 1

Set DV = (D0 , D1 , . . . , Dr ) : Tm Q — Tm V ’ V r .

r V V V r’1 r

V

We de¬ne the r-th order Ricci equations Er , r ≥ 2, as follows. For r = 2,

˜

E2 are the formal Ricci identities of V (Rm ). By 28.4, they are of the form

V

p

V[ij] ’ bilin(W, V ) = 0.

(2)

V V

For r > 2, Er are the absolute derivatives of Er’1 and the formal Ricci identities

˜

of V (Rm ) — —r’2 T — Rm . These equations are of the form

Vip ···[is’1 is ]···ir ’ bilin(W r’2 , V r’2 ) = 0.

(3) 1

De¬nition. The r-th order Ricci subspace ZV ‚ K r’2 — V r is de¬ned by E2 =

r V

0, . . . , Er = 0, r ≥ 2. For r = 0, 1 we set ZV = V and ZV = V 1 .

V 0 1

Lemma. ZV is a submanifold of K r’2 —V r , it holds ZV = (C r’2 , DV )(Tm Q—

r r r r’1

Tm V ) and the restricted map (C r’2 , DV ) : Tm Q—Tm V ’ ZV is a submersion.

r r r’1 r r

0 0 1 1

Proof. For r = 0 we have ZV = V and DV = idV . For r = 1, DV : Q — Tm V ’

V 1 = ZV is of the form

1

Vip = vi + bilin(Q, V )

p

V p = vp ,

r’1

so that our claim is trivial. Assume by induction ZV is a submanifold and the

r’1

restriction of the ¬rst product projection of K r’3 — V r’1 to ZV is a surjective

r’1

submersion. Consider the ¬ber product K r’2 —K r’3 ZV and the product vector

r’1

bundle (K r’2 —K r’3 ZV ) — Vr . By (3) ZV is characterized by a¬ne equations

r

of constant rank. This proves ZV is a subbundle and ZV ’ K r’2 is a surjective

r r

submersion.

r’1 r’1

Assume by induction (C r’3 , DV ) : Tm Q — Tm V ’ ZV

r’2 r’1

is a surjec-

r r’1 r m—

tive submersion. We have Tm V = Tm V — V — S R . By (1) and (3),

r’1

(C r’2 , DV ) : (Tm Q — Tm V ) — V — S r Rm— ’ (ZV ’ K r’2 —K r’3 ZV ) is

r r’1 r’1 r

bijective on each ¬ber. This proves our lemma.

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

28. The orbit reduction 241

28.9. The following result is of technical character, but it covers the core of

several applications. Let F be a G1 -space.

m

Proposition. For every Gr+1 -map f : Tm Q — Tm V ’ F there exists a unique

r’1 r

m

G1 -map g : ZV ’ F satisfying f = g —¦ (C r’2 , DV ).

r r

m

Proof. First we deduce a lemma.

Lemma. If y, y ∈ Tm Q satisfy C r’2 (y) = C r’2 (¯), then there is an element

r’1

¯ y

r+1 r+1

h ∈ B1 of the kernel B1 of the jet projection Gm ’ G1 such that h(¯) = y.

r+1

y

m

r+1

r’1

Indeed, consider the orbit set Tm Q/B1 . (We shall not need a manifold

structure on it, as one checks easily that 28.1 and 28.6 work at the set-theoretical

r+1 r+1

level as well.) This is a G1 -set under the action a(B1 y) = aB1 (y), y ∈

m

Tm Q, a ∈ G1 ‚ Gr+1 . Clearly, the factor projection

r’1

m m

r+1

r’1 r’1

p : Tm Q ’ Tm Q/B1

r+1

is a Gr+1 -map. By 28.6 there is a map g : K r’2 ’ Tm Q/B1 r’1

satisfying

m

r’2 r’2 r’2

p=g—¦C . If C (y) = C (¯) = x, then p(y) = p(¯) = g(x). This proves

y y

our lemma.

Consider the map (idTm Q , DV ) : Tm Q — Tm V ’ Tm Q — V r and denote

r r’1 r r’1

r’1

˜

by V r ‚ Tm Q — V r its image. By 28.8.(1), the restricted map DV : Tm Q —

r’1 r r’1

˜

Tm V ’ V r is bijective for every y ∈ Tm Q, so that DV is an equivariant di¬eo-

r r’1 r

˜ ˜ ˜

morphism. De¬ne C r’2 : V r ’ ZV , C r’2 (y, z) = (C r’2 (y), z), y ∈ Tm Q,

r r’1

˜

z ∈ V r . By lemma 28.5, C r’2 is a surjective submersion. By de¬nition,

˜ ˜

C r’2 (y, z) = C r’2 (¯, z ) means C r’2 (y) = C r’2 (¯) and z = z . Thus, the above

y¯ y ¯

˜ r’2 satis¬es the orbit condition for B r+1 . By 28.1 there is a

lemma implies C 1

˜ r’2 . Composing both sides

r ’1

1 r

Gm -map g : ZV ’ F satisfying f —¦ (DV ) = g —¦ C

with DV , we ¬nd f = g —¦ (C r’2 , DV ).

r r

28.10. Remark. The idea of the proof of proposition 28.9 can be applied

to suitable invariant subspaces of V as well. We shall need the case P =

RegS 2 Rm— ‚ S 2 Rm— of the standard ¬ber of the bundle of pseudoriemannian

metrics over m-manifolds. In this case we only have to modify the de¬nition

of Pr to Pr = S 2 Rm— — —r Rm— , but the rest of 28.8 and 28.9 remains to be

unchanged. Thus, for every Gr+1 -map f : Tm Q — Tm P ’ F there exists a

r’1 r

m

unique G1 -map g : ZP ’ F satisfying f = g —¦ (C r’2 , DP ).

r r

m

˜

28.11. Linear symmetric connection and a general vector ¬eld. Let F

denote the ¬rst order natural bundle over m-manifolds determined by G1 -space

m

˜ ˜ with associated

1

F . Consider an r-th order natural operator Q„ P • V F

¯ r ‚ K r’1 — V r be the pre-image of

r+2 r r

Gm -map f : Tm Q — Tm V ’ F . Let ZV

ZV ‚ K r’2 — V r with respect to the canonical projection K r’1 ’ K r’2 .

r

1 r’1 r

Take the map ψr : Sr+2 — Tm Q — Wr’1 ’ Tm Q from 28.6 and construct

1 r’1 r r r

ψr — idTm V : Sr+2 — Tm Q — Wr’1 — Tm V ’ Tm Q — Tm V . If we apply the

r

orbit reduction to f —¦ (ψr — idTm V ) in the previous way, we obtain a Gr+1 -

r

m

r’1 r r

map h : Tm Q — Wr’1 — Tm V ’ F such that f = h —¦ (πr’1 , Cr’1 ) — idTm V .r

Applying proposition 28.9 (with ˜parameters™ from Wr’1 ) to h, we obtain

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

242 Chapter VI. Methods for ¬nding natural operators

Proposition. For every Gr+2 -map f : Tm Q — Tm V ’ F there exists a unique

r r

m

¯r

G1 -map g : ZV ’ F satisfying f = g —¦ (C r’1 , DV ).

r

m

˜ ˜

Roughly speaking, every r-th order natural operator Q„ P 1 • V F factorizes

through the curvature operator and its absolute derivatives up to order r ’ 1

˜

and through the absolute derivatives on vector bundle V up to order r.

28.12. Linear non-symmetric connections. An arbitrary linear connection

on T M can be uniquely decomposed into its symmetrization and its torsion

tensor. In other words, QP 1 M = Q„ P 1 M • T M — Λ2 T — M . Hence we have

the situation of 28.11, in which the role of standard ¬ber V is played by Rm —

Λ2 Rm— =: H . This proves

Corollary. For every Gr+2 -map f : J0 (QP 1 Rm ) ’ F there exists a unique

r

m

¯r

G1 -map g : ZH ’ F satisfying f = g —¦ (C r’1 , DH ).

r

m

T — — T — . In

28.13. Example. We determine all natural operators QP 1

the same way as in 28.7 we deduce that such operators are of the ¬rst order.

By 28.12 we have to ¬nd all G1 -maps f : K — H — H1 ’ Rm— — Rm— . The

m

equivariance with respect to the homotheties yields the homogeneity condition

k 2 f (W, H, H1 ) = f (k 2 W, kH, k 2 H1 ).

Hence f is linear in W and H1 and quadratic in H. The term linear in W was

determined in 28.7. By the invariant tensor theorem, the term quadratic in H

is generated by the permutations of m, n, p, q in

mnpq k l

δi δj δk δl Hmn Hpq .

kl kl kl

This yields the 3 di¬erent double contractions Sik Sjl , Sij Skl , Sil Sjk of the tensor

product S — S of the torsion tensor with itself. Finally, the term linear in H1

corresponds to the permutations of l, m, n in

lmn k

δi δj δk Hlmn .

This gives 3 generators

k k k

(1) Hijk , Hikj , Hjki .

Thus, all natural operators QP 1 ’ T — — T — form an 8-parameter family linearly

generated by 2 di¬erent contractions of the curvature tensor of the symmetrized

connection, by 3 di¬erent double contractions of S — S and by 3 operators

constructed from the covariant derivatives of the torsion tensor with respect

to the symmetrized connection according to (1).

We remark that the ¬rst author determined all natural operators QP 1 ’ T — —

T — by direct evaluation in [Kol´ˇ, 87b]. Some of his generators are geometrically

ar

di¬erent of our present result, but both 8-parameter families are, of course,

linearly equivalent.

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

28. The orbit reduction 243

28.14. Pseudoriemannian metrics. Using the notation of 28.10, we deduce

a reduction theorem for natural operators on pseudoriemannian metrics. Let

¯

P r = ZP © (K r’2 — P — {0} — . . . — {0}) be the subspace determined by 0 ∈

r

P1 , . . . , 0 ∈ Pr .

¯

Lemma. P r is a submanifold of ZP .

r

Proof. By [Lichnerowicz, 76, p. 69], the Ricci identity in the case of the bundle

of pseudoriemannian metrics has the form

m m

(1) Pij[kl] + Wikl Pmj + Wjkl Pim = 0.

¯

Thus, for r = 2, P 2 ‚ W — P 2 is characterized by the curvature equations E2 ,

by Pijk = 0, Pijkl = 0 and by

m m

(2) Wikl Pmj + Wjkl Pim = 0.

Equations (2) are G1 -equivariant. We know that P is divided into m+1 compo-

m

nents Pσ according to the signature σ of the metric in question. Every element

in each component can be transformed by a linear isomorphism into a canonical

¯

form ±δij . This implies that P 2 is characterized by linear equations of constant

¯

rank over each component Pσ . Assume by induction P r’1 ‚ W r’3 — P r’1 is a

¯ ¯

r r’1

submanifold. Then P ‚ P — {0} — Wr’2 is characterized by the curvature

equations Er and by

n n

(3) Wiklm1 ...mr’2 Pnj + Wjklm1 ...mr’2 Pin = 0.

By the above argument we deduce that this is a system of a¬ne equations of

constant rank over each Pσ .

Consider a Gr+1 -map f : Tm P ’ F . Applying 28.10 to f —¦ p2 = Tm Q —

r r’1

m

Tm P ’ F , where p2 is the second product projection, we obtain a G1 -map

r

m

r

h : ZP ’ F satisfying

f —¦ p2 = h —¦ (C r’2 , DP ).

r

(4)

r r’1

Let »r : Tm P ’ Tm Q be the map determined by constructing the r-jets of the

r r’1 r

Levi-Civit` connection. Composing (4) with (»r , id) : Tm P ’ Tm Q — Tm P ,

a

we ¬nd

f = h —¦ (C r’2 , DP ) —¦ (»r , id).

r

(5)

¯

Let g be the restriction of h to P r . Since the Levi-Civit` connection is char-

a

acterized by the fact that the absolute di¬erential of the metric tensor van-

¯

ishes, the values of (C r’2 , DP ) —¦ (»r , id) lie in P r . Write Lr’2 = (C r’2 , DP ) —¦

r r

¯

(»r , id) : Tm P ’ P r . Then we can summarize by

r

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

244 Chapter VI. Methods for ¬nding natural operators

Proposition. For every Gr+1 -map f : Tm P ’ F there exists a G1 -map

r

m m

¯

g : P r ’ F satisfying f = g —¦ Lr’2 .

This is the classical assertion that every r-th order natural operator on pseu-

doriemannian metrics with values in an arbitrary ¬rst order natural bundle fac-

torizes through the metric itself and through the absolute derivatives of the

curvature tensor of the Levi-Civit` connection up to order r ’ 2.

a

We remark that each component Pσ of P can be treated separately in course

of the proof of the above proposition. Hence the result holds for any kind of

pseudoriemannian metrics (in particular for the proper Riemannian metrics).

28.15. Pseudoriemannian metric and a general vector ¬eld. A simple

modi¬cation of 28.11 and 28.14 leads to a reduction theorem for the r-th order

natural operators transforming a pseudoriemannian metric and a general vector

¬eld into a section of a ¬rst order natural bundle. In the notation from 28.11 and

28.14, let f : Tm P — Tm V ’ F be a Gr+1 -map. Consider the product projection

r r

m

r’1 r r r r

p : Tm Q — Tm P — Tm V ’ Tm P — Tm V . Then we can apply 28.9 and 28.10 to

the product P — V . Hence there exists a G1 -map h : ZP —V ’ F satisfying

r

m

f —¦ p = h —¦ (C r’2 , DP —V ).

r

(1)

¯r

Denote by PV ‚ ZP P — V ‚ K r’2 — P r — V r the subspace determined by

r

¯r

0 ∈ P1 , . . . , 0 ∈ Pr . Analogously to 28.14 we deduce that PV is a submani-

¯r

fold. Write Lr’2 = (»r , idTm P ) — idTm V : Tm P — Tm V ’ PV , i.e. Lr’2 (u, v) =

r r

r r

V V

(C r’2 (»r (u)), u0 , 0, . . . , 0, DV (»r (u), v)), u ∈ Tm P , v ∈ Tm V , u0 = π0 (u). Then

r r r r

¯r

(1) implies f = h —¦ Lr’2 . If we denote by g the restriction of h to PV , we obtain

V

the following assertion.

Proposition. For every Gr+1 -map f : Tm P — Tm V ’ F there exists a G1 -map

r r

m m

¯r

g : PV ’ F satisfying f = g —¦ Lr’2 .

V

Hence every r-th order natural operator transforming a pseudoriemannian

metric and a general vector ¬eld into a section of a ¬rst order natural bundle

factorizes through the metric itself, through the absolute derivatives of the cur-

vature tensor of the Levi-Civit` connection up to the order r ’ 2 and through

a

the absolute derivatives with respect to the Levi-Civit` connection of the general

a

vector ¬eld up to the order r.

28.16. Remark. Since Q„ P 1 M ’ M is an a¬ne bundle, the standard ¬ber

r

Tm Q of its r-th jet prolongation is an a¬ne space by 12.17. In other words,

Gr+2 acts on Tm Q by a¬ne isomorphisms. Consider an a¬ne action of G1

r

m m

of F (with the linear action as a special case). Then we can introduce the

r

concept of a polynomial map Tm Q ’ F analogously to 24.10. Analyzing the

proof of theorem 28.6, we observe that all the maps ψr and •r are polynomial.

This implies that for every polynomial Gr+2 equivariant map f : Tm Q ’ F ,

r

m

the unique G1 -equivariant map g : K r’1 ’ F from the theorem 28.6 is the

m

restriction of a polynomial map g : W r’1 ’ F .

¯

1

Consider further a Gm -module V as in 28.8 or an invariant open subset of such

a module as in 28.10. Then we also have de¬ned the concept of a polynomial

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

29. The method of di¬erential equations 245

map of Tm Q — Tm V into an a¬ne G1 -space F . Quite similarly to the ¬rst

r’1 r

m

part of this remark we deduce that for every polynomial Gr+1 -equivariant map

m

f : Tm Q — Tm V ’ F the unique G1 -equivariant map g : ZV ’ F from the

r’1 r r

m

proposition 28.9 is the restriction of a polynomial map g : W r’2 — V r ’ F .

¯

29. The method of di¬erential equations

29.1. In chapter IV we have clari¬ed that the ¬nite order natural operators

between any two bundle functors are in a canonical bijection with the equivariant

maps between certain G-spaces. We recall that in 5.15 we deduced the following

in¬nitesimal characterization of G-equivariance. Given a connected Lie group G

and two G-spaces S and Z we construct the induced fundamental vector ¬eld

S Z

ζA and ζA on S and Z for every element A ∈ g of the Lie algebra of G. Then

Q

S

f : S ’ Z is a G-equivariant map if and only if vector ¬elds ζA and ζA are

f -related for every A ∈ g, i.e.

S Z

T f —¦ ζA = ζA —¦ f for all A ∈ g.

(1)

The coordinate expression of (1) is a system of partial di¬erential equations

for the coordinate components of f . If we can ¬nd the general solution of this

system, we obtain all G-equivariant maps. This procedure is sometimes called

the method of di¬erential equations.

29.2. Remark. If G is not connected and G+ denotes its connected component

of unity, then the solutions of 29.1.(1) determine all G+ -equivariant maps S ’ Z.

Obviously, there is an algebraic procedure how to decide which of these maps

are G-equivariant. We select one element ga in each connected component of

G and we check which solutions of 29.1.(1) are invariant with respect to all

ga . However, one usually interprets the solutions of 29.1.(1) geometrically. In

practice, if we succeed in ¬nding the geometrical constructions of all solutions

of 29.1.(1), it is clear that all of them determine the G-equivariant maps and we

are not obliged to discuss the individual connected components of G.

29.3. From 5.12 we have that for each left G-space S the map of the fundamental

S S

vector ¬elds A ’ ζA , A ∈ g, is a Lie algebra antihomomorphism, i.e. ζ[A,B] =

SS

’[ζA , ζB ] for all A, B ∈ g, where on the left-hand side is the Lie bracket in g

and on the right-hand side we have the bracket of vector ¬elds. Hence if some

vectors A± , ± = 1, . . . , q ¤ dim G generate g as a Lie algebra, i.e. A± with all

their iterated brackets generate g as a vector space, then the equations

S Z

T f —¦ ζ A± = ζ A± —¦ f ± = 1, . . . , q

imply T f —¦ ζA = ζA —¦ f for all A ∈ g. In particular, for the group Gr the

S Z

m

generators of its Lie algebra are described in 13.9 and 13.10.

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

246 Chapter VI. Methods for ¬nding natural operators

29.4. The Levi-Civit` connection. We are going to determine all ¬rst order

a

natural operators transforming pseudoriemannian metrics into linear connec-

tions. We denote by RegS 2 T — M the bundle of all pseudoriemannian metrics

over an m-manifold M , so that the standard ¬ber of the corresponding natural

bundle over m-manifolds is the subset RegS 2 Rm— ‚ S 2 Rm— of all elements gij

satisfying det(gij ) = 0. Since the zero of S 2 Rm— does not lie in RegS 2 Rm— , the

homogeneous function theorem is of no use for our problem. (Of course, this

analytical fact is deeply re¬‚ected in the geometry of pseudoriemannian mani-

folds.) Hence we shall try to apply the method of di¬erential equations. In the

canonical coordinates gij = gji , gij,k on the standard ¬ber S = J0 RegS 2 T — Rm ,

1

the action of G2 has the following form

m

gij = gkl ak al

(1) ¯ ˜i ˜j

gij,k = glm,n al am an + glm (˜l am + al am ).

(2) ¯ ˜i ˜j ˜k aik ˜j ˜i ˜jk

Since we deal with a classical problem, we shall use the classical Christo¬el™s on

the standard ¬ber Z = (QP 1 Rm )0 . In this case we have the following action of

G2m

¯ jk

“i = ai “l am an + ai al

(3) l mn ˜j ˜k l ˜jk

see 17.15.

We shall not need all di¬erential equations of our problem, since we shall

proceed in another way in the ¬nal step. It is su¬cient to deduce the funda-

i i

mental vector ¬elds Sjk on S and Zjk on Z corresponding to the one-parameter

subgroups ai = δj , ai = t for j = k and ai = δj , ai = 2t. From (1)“(3) we

i i

˜jk

j j jj

deduce easily

‚ ‚

i

(4) Sjk = 2gil +

‚glj,k ‚glk,j

and

‚ ‚

i

(5) Zjk = +

i ‚“i

‚“jk kj

Hence the corresponding part of the di¬erential equations for a G2 -equivariant

m

i

map “ : S ’ Z with components “jk (glm , glm,n ) is

‚“i ‚“i

jk jk i mn nm

(6) 2glp + = δl δj δk + δj δk .

‚gpm,n ‚gpn,m

Multiplying by g lq and replacing q by l, we ¬nd

‚“i ‚“i 1

jk jk

= g il δj δk + δk δj .

mn mn

(7) +

‚glm,n ‚gln,m 2

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

29. The method of di¬erential equations 247

Let (7™) or (7”) be the equations derived from (7) by the permutation (l, m, n) ’

(m, n, l) or (l, m, n) ’ (n, l, m), respectively. Then the sum (7)+(7 )’(7 ) yields

‚“i 1 il m n

jk

g (δj δk + δk δj ) + g im (δj δk + δk δj )

mn nl nl

2 =

‚glm,n 2

(8)

’ g in (δj δk + δk δj ) .

lm lm

The right-hand sides are independent on gij,k . Since we meet such a situation

frequently, it is useful to formulate a simple lemma of general character.

29.5. Lemma. Let U be an open subset in Ra with coordinates z ± and let

f (z ± , w» ) be a smooth function on U —Rb , (w» ) ∈ Rb , satisfying ‚f (z,w) = g» (z).

‚w»

Then

b

g» (z)w» + h(z)

(1) f (z, w) =

»=1

where h(z) is a smooth function on U .

b »

Proof. Notice that the di¬erence F (z, w) = f (z, w) ’ »=1 g» (z)w satis¬es

‚F

= 0.

‚w»

Applying lemma 29.5 to 29.4.(8), we ¬nd

1 il

“i = i

g (glj,k + glk,j ’ gjk,l ) + γjk (glm ).

jk

2

i

For γjk = 0 we obtain the coordinate expression of the Levi-Civit` connection Λ,

a

which is natural by its standard geometric interpretation. Hence the di¬erence

“ ’ Λ is a GL(m)-equivariant map RegS 2 Rm— ’ Rm — Rm— — Rm— .

29.6. Lemma. The only GL(m)-equivariant map f : RegS 2 Rm— ’ Rm —Rm— —

Rm— is the zero map.

Proof. Let Is be the matrix gii = 1 for i ¤ s, gjj = ’1 for j > s and gij = 0 for

i = j. Since every g ∈ RegS 2 Rm— can be transformed into some Is , it su¬ces to

i

deduce fjk (Is ) = 0 for all i, j, k. If j = i = k or j = i = k, the equivariance with

i i

respect to the change of orientation on the i-th axis gives fjk (Is ) = ’fjk (Is ). If

j = i = k, we obtain the same result by changing the orientation on both the

i-th and k-th axes.

Lemma 29.6 implies “ ’ Λ = 0. This proves

29.7. Proposition. The only ¬rst order natural operator transforming pseu-

doriemannian metrics into linear connections is the Levi-Civit` operator.

a

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

248 Chapter VI. Methods for ¬nding natural operators

Remarks

The ¬rst version of our systematical approach to the problem of ¬nding nat-

ural operators was published in [Kol´ˇ, 87b]. In the same paper both geometric

ar

results from section 25 are deduced. The smooth version of the tensor evaluation

theorem is ¬rst presented in this book. Proposition 26.12 was proved by [Kol´ˇ, ar

Radziszewski, 88]. The generalized invariant tensor theorem was ¬rst used in

[Kol´ˇ, 87b]. We remark that the natural equivalence s : T T — ’ T — T from 26.11

ar

was ¬rst studied in [Tulczyjew, 74].

The reduction theorems for symmetric linear connections and pseudorieman-

nian metrics are classical, see e.g. [Schouten, 54]. Some extensions or reformula-

tions of them are presented in [Lubczonok, 72] and [Krupka, 82]. The method of

di¬erential equations is used systematically e.g. in the book [Krupka, Janyˇka,

s

90]. The complete version of proposition 29.7 was deduced in [Slov´k, 89].

a

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

249

CHAPTER VII.

FURTHER APPLICATIONS

In this chapter we discuss some further geometric problems about di¬erent

types of natural operators. First we deduce that all natural bilinear operators

transforming a vector ¬eld and a di¬erential k-form into a di¬erential k-form

form a 2-parameter family. This further clari¬es the well known relation be-

tween Lie derivatives and exterior derivatives of k-forms. From the technical

point of view this problem can be considered as a preparatory exercise to the

problem of ¬nding all bilinear natural operators of the type of the Fr¨licher-

o

Nijenhuis bracket. We deduce that in general case all such operators form a

10-parameter family. Then we prove that there is exactly one natural operator

transforming general connections on a ¬bered manifold Y ’ M into general con-

nections on its vertical tangent bundle V Y ’ M . Furthermore, starting from

some geometric problems in analytical mechanics, we deduce that all ¬rst-order

natural operators transforming second-order di¬erential equations on a manifold

M into general connections on its tangent bundle T M ’ M form a one param-

eter family. Further we study the natural transformations of the jet functors.

The construction of the bundle of all r-jets between any two manifolds can be

interpreted as a functor J r on the product category Mfm — Mf . We deduce

that for r ≥ 2 the only natural transformations of J r into itself are the identity

and the contraction, while for r = 1 we have a one-parameter family of homo-

theties. This implies easily that the only natural transformation of the functor

of the r-th jet prolongation of ¬bered manifolds into itself is the identity. For

the second iterated jet prolongation J 1 (J 1 Y ) of a ¬bered manifold Y we look

for an analogy of the canonical involution on the second iterated tangent bundle

T T M . We prove that such an exchange map depends on a linear connection on

the base manifold and we give a simple list of all natural transformations of this

type.

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—

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

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