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

¯