Rk , Rk Rm = Rm —Wk , W0 = Rm —Rm— —Λ2 Rm— , Wk+1 = Wk —Rm— . Similarly,

the covariant di¬erentiation of sections of E forms natural operators dk : Q„ P 1 —

¯

Ek , where E0 = E, E0 Rm =: Rm —V0 , d0 is the inclusion, Ek Rm = Rm —Vk ,

E

Vk+1 = Vk — Rm— . Let us write Dk = (ρ0 , . . . , ρk’2 , d0 , . . . , dk ) : Q„ P 1 — E

Rk’2 — E k , where Rl = R0 — . . . — Rl , E l = E0 — . . . — El . All Dk are natural

operators. In 28.8 we de¬ned the Ricci sub bundles Z k ‚ Rk’2 — E k and we

know Dk : Q„ P 1 — E Zk.

Let us further de¬ne the functor Z ∞ as the inverse limit of Z k , k ∈ N, with

respect to the obvious natural transformations (projections) pk : Z k ’ Z , k > ,

and similarly D∞ : Q„ P 1 — E Z ∞ . As a corollary of 28.11 and the non linear

Peetre theorem we get

Proposition. For every natural operator D : Q„ P 1 — E F there is a unique

˜ ˜

∞ ∞

natural transformation D : Z ’ F such that D = D—¦D . Furthermore, for ev-

ery m-dimensional compact manifold M and every section s ∈ C ∞ (Q„ P 1 M —M

EM ), there is a ¬nite order k and a neighborhood V of s in the C k -topology

˜ ˜ ˜

such that DM |(D∞ )M (V ) = (πk )— (Dk )M , for some (Dk )M : (Dk )M (V ) ’

∞

˜

C ∞ (Z k M ), and DM |V = (Dk )M —¦ (Dk )M |V .

In words, a natural operator D : Q„ — E F is determined in all coordinate

charts of an arbitrary m-dimensional manifold M by a universal smooth mapping

de¬ned on the curvatures and all their covariant derivatives and on the sections

of EM and all their covariant derivatives, which depends ˜locally™ only on ¬nite

number of these arguments.

33.10. The Riemannian case. In section 28, we also applied the reduction

procedure to operators depending on Riemannian metrics and general vector

¬elds. In fact we have viewed the operators D : S+ T — — E

2

F as operators

¯ 2—

1

D : Q„ P — (S+ T — E) F independent of the ¬rst argument and we have

used the Levi-Civit` connection “ : S+ T —

2

Q„ P 1 to write D as a composition

a

¯

D = D —¦ (“, id). Since the covariant derivatives of the metric with respect to

the metric connection are zero, we can restrict ourselves to sub bundles in the

Ricci subspaces corresponding to the bundle S+ T — — E, which are of the form

2

S+ T — — Z k with Z k ‚ Rk’2 — E k , cf. 28.14. Let us notice that the bundles

2

Z k M involve the curvature of the Riemannian connection on M , its covariant

derivatives, and the covariant derivatives of the sections of EM . Similarly as

above, we de¬ne the inverse limits Z ∞ and D∞ and as a corollary of the non

linear Peetre theorem and 28.15 we get

Corollary. For every natural operator D : S+ T — — E

2

F there is a nat-

˜ ˜

ural transformation D : S+ T — Z ’ F such that D = D —¦ D∞ —¦ (“, id).

2— ∞

Furthermore, for every m-dimensional compact manifold M and every section

s ∈ C ∞ (S+ T — M —M EM ), there is a ¬nite order k and a neighborhood V of s

2

˜ ˜

in the C k -topology such that DM |(D∞ —¦ (“, id))M (V ) = (πk )— (Dk )M , where

∞

˜ ˜

(Dk )M : (Dk —¦ (“, id))M (V ) ’ C ∞ (Z k M ), and DM |V = (Dk )M —¦ (Dk )M —¦

(“, id)M |V .

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

33. Topics from Riemannian geometry 273

33.11. Polynomiality. Since the standard ¬ber V0 of E0 is embedded identi-

cally into Z0 Rm by the associated map to the operator Dk , we can use 28.16 and

k

add the following proposition to the statements of 33.9, or 33.10, respectively.

˜

Corollary. The operator D is polynomial if and only if the operators Dk are

polynomial. Further D is polynomial with smooth real functions on the values of

˜

E0 , or S+ T — , as coe¬cients if and only if the operators Dk are polynomial with

2

smooth real functions on the values of E0 , or S+ T — , as coe¬cients, respectively.

2

33.12. Natural operators D : S+ T — T (p,q) . According to 33.9 we ¬nd G∞ -

2

m

invariant open subsets Uk in J0 (S+ T — Rm ) forming a ¬ltration of the whole jet

∞ 2

space, such that on these subsets D factorizes through smooth Gk+1 -equivariant

m

mappings

i1 ...ip i1 ...ip

fj1 ...jq = fj1 ...jq (gij , . . . , gij 1 ... k )

de¬ned on πk Uk . For large k™s, the action of the homotheties c’1 δj on g™s is

∞ i

well de¬ned and we get

i ...i i ...i

cq’p fj1 ...jq (gij , . . . , gij ) = fj1 ...jq (c2 gij , . . . , c2+k gij

1 p 1 p

(1) ).

1 ... k 1 ... k

Now, let us add the assumption that D is homogeneous with weight », choose

the change g ’ c’2 g of the scale of the metric and insert this new metric into

(1). We get

i ...i i ...i

cq’p’» fj1 ...jq (gij , . . . , gij ) = fj1 ...jq (gij , c1 gij, 1 , . . . , ck gij

1 p 1 p

).

1 ... k 1 ... k

i ...i

1 p

This formula shows that the mappings fj1 ...jq are polynomials in all variables

except gij with functions in gij as coe¬cients.

According to 33.11 and 28.16, the map D is on Uk determined by a polynomial

mapping

i ...i i i

ω = (ωj1 ...jp (gij , Wjkl , . . . , Wjklm1 ...mk’2 ))

1 q

which is G1 -equivariant on the values of the covariant derivatives of the curva-

m

tures and the sections. If we apply once more the equivariance with respect to

the homothety x ’ c’1 x and at the same time the change of the scale of the

metric g ’ c’2 g, we get

i ...i

cq’p’» ωj1 ...jp (gij , Rjkl , . . . , Rjklm1 ...mk’2 ) =

i i

1 q

i ...i

= ωj1 ...jp (gij , c2 Rjkl , . . . , ck Rjklm1 ...mk’2 ).

i i

1 q

i ...i

This homogeneity shows that the polynomial functions ωj1 ...jp must be sums of

1 q

i

homogeneous polynomials with degrees a in the variables Rjklm1 ...m satisfying

2a0 + · · · + kak’2 = q ’ p ’ »

(2)

and their coe¬cients are functions depending on gij ™s.

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

274 Chapter VII. Further applications

Now, we shall ¬x gij = δij and use the O(m)-equivariance of the homogeneous

components of the polynomial mapping ω. For this reason, we shall switch to

a

the variables Rijklm1 ...ms = gia Rjklm1 ...ms . Using the standard polarization tech-

nique and H. Weyl™s theorem, we get that each multi homogeneous component

in question results from multiplication of variables Rijklm1 ,... ,ms , s = 0, 1, . . . , r,

and application of some O(m)-equivariant tensor operations on the target space.

Hence our operators result from a ¬nite number of the following steps.

(a) take tensor product of arbitrary covariant derivatives of the curvature

tensor

(b) tensorize by the metric or by its inverse

(c) apply arbitrary GL(m)-equivariant operation

(d) take linear combinations.

33.13. Remark. If q ’ p = » + 1, then there is no non negative integer solution

of 33.12.(2) and so all natural tensors in question are zero. The case q = 2,

p = 1, » = 0 implies that the Levi-Civit` connection is the only conformal

a

natural connection on Riemannian manifolds.

Indeed, the di¬erence of two such connections is a natural tensor twice co-

variant and once contravariant, and so zero.

33.14. Consider now Λp Rm— as the target tensor space. So in the above proce-

dure, all indices which were not contracted must be alternated at the end. Since

the metric is a symmetric tensor, we get zero whenever using the above step

(b) and alternating over both indices. But contracting over any of them has no

proper e¬ect, for δij Rjklnm1 ,... ,ms = Riklnm1 ,... ,ms . So we can omit the step (b)

at all.

The ¬rst Bianchi identity and 33.7.(1) imply Rijkl = Rklij . Then the lemma

in 33.5 and 33.7.(1) yield

Lemma. The alternation of Rijklm1 ...ms , 0 ¤ s, over arbitrary 3 indices among

the ¬rst four or ¬ve ones is zero.

Consider a monomial P in the variables Rijkl± with degrees as in Rijklm1 ...ms .

In view of the above lemma, if P remains non zero after all alternations, then we

must contract over at least two indices in each Rijkl± and so we can alternate

over at most 2a0 + · · · + kak’2 indices. This means p ¤ 2a0 + · · · + kak’2 = p ’ ».

Consequently » ¤ 0 if there is a non zero natural form with weight ». This proves

the ¬rst assertion of theorem 33.8.

Let » = 0. Since the weight of g ij is ’2, any contraction on two indices

in the monomial decreases the weight of the operator by 2. Every covariant

derivative Rijklm1 ...ms of the curvature has weight 2. So we must contract on

exactly two indices in each Rijklm1 ...ms which implies there are s + 2 of them

under alternation. But then there must appear three alternated indices among

the ¬rst ¬ve if s = 0. This proves a1 = · · · = ak’2 = 0, so that p = 2a0 . Hence

all the natural forms have even degrees and they are generated by the forms

ωq , cf. 33.5. As we deduced in 33.7, these forms are zero if their degree is not

divisible by four.

This completes the proof of the theorem 33.8.

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

33. Topics from Riemannian geometry 275

33.15. Remark. The original proof of the Gilkey theorem assumes a poly-

nomial dependence of the natural forms on a ¬nite number of the derivatives

gij,± of the metric and on the entries of the inverse matrix g ij , but also the

homogeneity in the weight, [Gilkey, 73]. Under such polynomiality assumption,

our methods apply to all natural tensors. In particular, it follows easily that

the Levi-Civit` connection is the only second order polynomial connection on

a

Riemannian manifolds. Of course, the latter is not true in higher orders, for we

can contract appropriate covariant derivatives of the curvature and so we get

natural tensors in T — T — — T — of orders higher than two.

33.16. Operations on exterior forms. The approach from 33.4“33.5 can be

easily extended to the study of all natural operators D : Q„ P 1 — T (s,r) T (q,p)

with s < r or s = r = 0. This was done in [Slov´k, 92a], we shall present only

a

the ¬nal results. If we omit the assumption on s and r, we have to assume the

polynomiality.

Theorem. All natural operators D : Q„ P 1 —T (s,r) T (q,p) , s < r, are obtained

by a ¬nite iteration of the following steps: take tensor product of arbitrary

covariant derivatives of the curvature tensor or the covariant derivatives of the

tensor ¬elds from the domain, apply arbitrary GL(m)-equivariant operation,

take linear combinations. In the case s = r = 0 we have to add one more

step, the compositions of the functions from the domain with arbitrary smooth

functions of one real variable.

ΛT — , r > 0, is

The algebra of all natural operators D : Q„ P 1 — T (0,r)

generated by the alternation, the exterior derivative d and the Chern forms cq .

ΛT — is generated

The algebra of all natural operators D : Q„ P 1 — T (0,0)

by the compositions with arbitrary smooth functions of one real variable, the

exterior derivative d and the Chern forms cq .

The proof of these results follows the lines of 33.4“33.5 using two more lemmas:

First, the alternation on all indices of the second covariant derivative 2 t of an

arbitrary tensor t ∈ C ∞ (—s Rm— ) is zero (which is proved easily using the Bianchi

and Ricci identities) and , second, the alternation of the ¬rst covariant derivative

of an arbitrary tensor t ∈ C ∞ (—s Rm— ) coincides with the exterior di¬erential of

the alternation of t (this well known fact is proved easily in normal coordinates).

33.17. Operations on exterior forms on Riemannian manifolds. A mod-

i¬cation of our proof of the Gilkey theorem for operations on exterior forms on

Riemannian manifolds, which is also based on the two lemmas mentioned above,

appeared in [Slov´k, 92a]. The equality 33.7.(2) on the Riemannian curvatures

a

can be expressed as Rijkl = Rjikl , and this holds for curvatures of metrics

with arbitrary signatures. This observation extends our considerations to pseu-

doriemannian manifolds, see [Slov´k, 92b]. In particular, our proof of the Gilkey

a

theorem extends to the classi¬cation of natural forms on pseudoriemannian man-

ifolds. Let us write Sreg T — for the bundle functor of all non degenerate symmetric

2

two-forms. The de¬nition of the weight of the operators depending on metrics

and the de¬nition of the Pontryagin forms extend obviously to the pseudorie-

mannian case. All the considerations go also through for metrics with any ¬xed

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

276 Chapter VII. Further applications

signature.

Theorem. All natural operators D : Sreg T — — T (s,r)

2

T (q,p) , s < r, homoge-

neous in weight are obtained by a ¬nite iteration of the following steps: take

tensor product of arbitrary covariant derivatives of the curvature tensor or the

covariant derivatives of the tensor ¬elds from the domain, tensorize by the metric

or its inverse, apply arbitrary GL(m)-equivariant operation, take linear combi-

nations. In the case s = r = 0 we have to add one more step, the compositions

of the functions from the domain with arbitrary smooth functions of one real

variable.

There are no non zero operators D : Sreg T — — T (0,r) ΛT — , r ≥ 0, with a

2

positive weight. The algebra of all conformal natural operators Sreg T — —T (0,r)

2

ΛT — , r > 0, is generated by the alternation, the exterior derivative d and the

Pontryagin forms pq .

The algebra of all conformal natural operators D : Sreg T — — T (0,0) ΛT —

2

is generated by the compositions with arbitrary smooth functions of one real

variable, the exterior derivative d and the Pontryagin forms pq .

The discussion from the proof of these results can be continued for every ¬xed

negative weight. In particular, the situation is interesting for » = ’2 and linear

operators D : Λp T — Λp T — depending on the metric. Beside the compositions

d —¦ δ and δ —¦ d of the exterior di¬erential d and the well known codi¬erential

δ : Λp Λp’1 (the Laplace-Beltrami operator is ∆ = δ —¦ d + d —¦ δ), there are

only three other generators: the multiplication by the scalar curvature, the con-

traction with the Ricci curvature and the contraction with the full Riemmanian

curvature. This classi¬cation was derived under some additional assumptions in

[Stredder, 75], see also [Slov´k, 92b].

a

33.18. Oriented pseudoriemannian manifolds. It is also quite important

in Riemannian geometry to know what are the operators natural with respect to

the orientation preserving local isometries. We shall not go into details here since

this would require to extend the description from 33.2 to all SO(m)-invariant

linear maps and then to repeat some steps of the proof of the Gilkey theo-

rem. This was done in [Stredder, 75] (for the polynomial forms and Riemannian

manifolds), and in [Slov´k, 92b]. Let us only remark that on oriented pseu-

a

doriemannian manifolds we have a natural volume form ω : Sreg T — Λm T — and

2

natural transformations — : Λp T — ’ Λm’p T — . All natural operators on oriented

pseudoriemannian manifolds homogeneous in the weight are generated by those

described above, the volume form ω, and the natural transformations —.

As an example, let us draw a diagram which involves all linear natural con-

formal operators on exterior forms on oriented pseudoriemannian manifolds of

even dimensions which do not vanish on ¬‚at pseudoriemannian manifolds, up to

the possible omitting of the d™s on the sides in the operators indicated by the

long arrows. (More explicitely, we do not consider any contribution from the

curvatures.) The symbols „¦p refer, as usual, to the p-forms, the plus and minus

subscripts indicate the splitting into the selfdual and anti-selfdual forms in the

1

degree 2 m.

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

33. Topics from Riemannian geometry 277

In the even dimensional case, there are no natural conformal operators be-

tween exterior forms beside the exterior derivatives. For the proofs see [Slov´k,

a

92b].

We should also remark that the name ˜conformal™ is rather misleading in the

context of the natural operators on conformal (pseudo-) Riemannian manifolds

since we require the invariance only with respect to constant rescaling of the

metric (cf. the end of the next section). On the other hand, each natural operator

on the conformal manifolds must be conformal in our sense.

9 9 hh

Aj „¦p

d+ + d

w„¦ w ··· w„¦ w ··· w„¦ u w „¦u

e e „¦ u

eg ¢

d d d d d d

„¦0 1 p’1 p+1 m’1 m

d’ d

„¦p

’

Dp’1 =d—d=d—¦d+ ’d—¦d’

D1 =d—¦(—d)m’3

D0 =d—¦(—d)m’1

33.19. First order operators. The whole situation becomes much easier if

we look for ¬rst order natural operators D : S+ T —

2

(F, G), where F and G are

arbitrary natural bundles, say of order r. Namely, every metric g on a manifold

‚g

M satis¬es gij = δij and ‚xij = 0 at the center of any normal coordinate chart.

k

¯ are two such operators and if their values DRm (g), DRm (g)

¯

Therefore, if D, D

on the canonical Euclidean metric g on Rm coincide on the ¬ber over the origin,

¯

then D = D. Hence the whole classi¬cation problem reduces to ¬nding maps

between the standard ¬bers which are equivariant with respect to the action of

the subgroup O(m) B1 ‚ G1r

B1 = Gr . In fact we used this procedure in

r

m m

section 29.

Let us notice that the natural operators on oriented Riemannian manifolds

r r

are classi¬ed on replacing O(m) B1 by SO(m) B1 . If we modify 29.7 in such

a way, we obtain (cf. [Slov´k, 89])

a

Proposition. All ¬rst order natural connections on oriented Riemannian man-

ifolds are

(1) The Levi-Civit` connection “, if m > 3 or m = 2

a

(2) The one parametric family “ + kD1 where D1 means the scalar product

and k ∈ R, if m = 1

(3) The one parametric family “ + kD3 where D3 means the vector product

and k ∈ R, if m = 3.

33.20. Natural metrics on the tangent spaces of Riemannian mani-

folds. At the end of this section, we shall describe all ¬rst order natural opera-

tors transforming metrics into metrics on the tangent bundles. The results were

proved in [Kowalski, Sekizava, 88] by the method of di¬erential equations. Let

us start with some notation.

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

278 Chapter VII. Further applications

We write πM : T M ’ M for the natural projection and F for the natural

bundle with F M = πM (T — —T — )M ’ M , F f (Xx , gx ) = (T f.Xx , (T — —T — )f.gx )

—

for all manifolds M , local di¬eomorphisms f , Xx ∈ Tx M , gx ∈ (T — — T — )x M .

The sections of the canonical projection F M ’ T M are called the F-metrics

in literature. So the F-metrics are mappings T M • T M • T M ’ R which are

linear in the second and the third summand. We ¬rst show that it su¬ces to

describe all natural F-metrics, i.e. natural operators S+ T —

2

(T, F ).

There is the natural Levi-Civit` connection “ : T M • T M ’ T T M and the

a

natural equivalence ν : T M •T M ’ V T M . There are three F-metrics, naturally

derived from sections G : T M ’ (S 2 T — )T M . Given such G on T M , we de¬ne

γ1 (G)(u, X, Y ) = G(“(u, X), “(u, Y ))

γ2 (G)(u, X, Y ) = G(“(u, X), ν(u, Y ))

(1)

γ3 (G)(u, X, Y ) = G(ν(u, X), ν(u, Y )).

Since G is symmetric, we know also G(ν(u, X), “(u, Y )) = γ2 (G)(u, Y, X). No-

tice also that γ1 and γ3 are symmetric.

The connection “ de¬nes the splitting of the second tangent space into the

h v

vertical and horizontal subspaces. We shall write Xx,u = Xu + Xu for each

Xx,u ∈ Tu T M , π(u) = x. Since for every Xx,u there are unique vectors X h ∈

Tx M , X v ∈ Tx M such that “(u, X h ) = Xu and ν(u, X v ) = Xu , we can recover

h v

the values of G from the three F-metrics γi ,

G(Xx,u , Yx,u ) = γ1 (G)(u, X h , Y h ) + γ2 (G)(u, X h , Y v )

(2)

+ γ2 (G)(u, Y h , X v ) + γ3 (G)(u, X v , Y v ).

Lemma. The formulas (1) and (2) de¬ne a bijection between triples of natural

F-metrics where the ¬rst and the third ones are symmetric, and the natural

operators S+ T — (S 2 T — )T .

2

33.21. Let us call every section G : T M ’ (S 2 T — )T M a (possibly degenerated)

metric. If we ¬x an F-metric δ, then there are three distinguished constructions

of a metric G.

(1) If δ symmetric, we choose γ1 = γ3 = δ, γ2 = 0. So we require that G

coincides with δ on both vertical and horizontal vectors. This is called

the Sasaki lift and we write G = δ s . If δ is non degenerate and positive

de¬nite, the same holds for δ s .

(2) We require that G coincides with δ on the horizontal vectors, i.e. we put

γ1 = δ, γ2 = γ3 = 0. This is called the vertical lift and G is a degenerate

metric which does not depend on the underlying Riemannian metric. We

write G = δ v .

(3) The horizontal lift is de¬ned by γ2 = δ, γ1 = γ3 = 0 and is denoted by

G = δ h . If δ positive de¬nite, then the signature of G is (m, m).

We can reformulate the lemma 33.20 as

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

33. Topics from Riemannian geometry 279

Proposition. There is a bijective correspondence between the triples of nat-

ural F-metrics (±, β, γ), where ± and γ are symmetric, and natural (possibly

degenerated) metrics G on the tangent bundles given by

G = ±s + β h + γ v .

33.22. Proposition. All ¬rst order natural F-metrics ± in dimensions m > 1

form a family parameterized by two arbitrary smooth functions µ, ν : (0, ∞) ’ R

in the following way. For every Riemannian manifold (M, g) and tangent vectors

u, X, Y ∈ Tx M

(1) ±(M,g) (u)(X, Y ) = µ(g(u, u))g(X, Y ) + ν(g(u, u))g(u, X)g(u, Y ).

If m = 1, then the same assertion holds, but we can always choose ν = 0.

In particular, all ¬rst order natural F-metrics are symmetric.

Proof. We have to discuss all O(m)-equivariant maps ± : Rm ’ Rm— — Rm— .

Denote by g 0 = i i

i dx — dx the canonical Euclidean metric and by | | the

‚

induced norm. Each vector v ∈ Rm can be transformed into |v| ‚x1 0 . Hence ±

‚

is determined by its values on the one-dimensional subspace spanned by ‚x1 0 .

Moreover, we can also change the orientation on the ¬rst axis, i.e. we have to

‚

de¬ne ± only on t ‚x1 0 with positive reals t.

Let us consider the group G of all linear orthogonal transformations keeping

‚ ‚ m—

— Rm— is

‚x1 0 ¬xed. So for every t ∈ R the tensor β(t) = ±(t ‚x1 ) ∈ R

G-invariant. On the other hand, every such smooth map β determines a natural

F-metric.

So let us assume sij dxi — dxj is G-invariant. Since we can change the orien-

tation of any coordinate axis except the ¬rst one, all sij with di¬erent indices

must be zero. Further we can exchange any couple of coordinate axis di¬erent

from the ¬rst one and so all coe¬cients at dxi — dxi , i = 1, must coincide. Hence

all G-invariant tensors are of the form

νdx1 — dx1 + µg 0 .

(2)

The reals µ and ν are independent, if m > 1. In dimension one, G is the trivial

group and so the whole one dimensional tensor space consists of G-invariant

tensors.

Thus, our mapping β is de¬ned by (2) with two arbitrary smooth functions

‚

µ and ν (and they can be reduced to one if m = 1). Given v = t ‚x1 0 , we can

write

±(Rm ,g0 ) (v)(X, Y ) = β(|v|)(X, Y ) = µ(|v|)g 0 (X, Y ) + ν(|v|)|v|’2 g 0 (v, X)g 0 (v, Y )

In order to prove that all natural F-metrics are of the form (1), we only have

to express µ(t), ν(t) as ν (t2 ) = t’2 ν(t) and µ(t2 ) = µ(t) for all positive reals,

¯ ¯

see 33.19. Obviously, every such operator is natural and the proposition is

proved.

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

280 Chapter VII. Further applications

33.23. If we use the invariance with respect to SO(m) in the proof of the above

proposition, we get

Proposition. All ¬rst order natural F-metrics ± on oriented Riemannian mani-

folds of dimensions m form a family parameterized by arbitrary smooth functions

µ, ν, κ, » : (0, ∞) ’ R in the following way. For every Riemannian manifold

(M, g) of dimension m > 3 and tangent vectors u, X, Y ∈ Tx M

±(M,g) (u)(X, Y ) = µ(g(u, u))g(X, Y ) + ν(g(u, u))g(u, X)g(u, Y ).

If m = 3 then

±(M,g) (u)(X, Y ) = µ(g(u, u))g(X, Y ) + ν(g(u, u))g(u, X)g(u, Y )

+ κ(g(u, u))g(u, X — Y )

where — means the vector product. If m = 2, then

±(M,g) (u)(X, Y ) = µ(g(u, u))g(X, Y ) + ν(g(u, u))g(u, X)g(u, Y )

+ κ(g(u, u)) g(J g (u), X)g(u, Y ) + g(J g (u), Y )g(u, X)

+ »(g(u, u)) g(J g (u), X)g(u, Y ) ’ g(J g (u), Y )g(u, X)

where J g is the canonical almost complex structure on (M, g). In the dimension

m = 1 we get

±(M,g) (u)(X, Y ) = µ(g(u, u))g(X, Y ).

33.24. If we combine the results from 33.20“33.23 we deduce that all natural

metrics on tangent bundles of Riemannian manifolds depend on six arbitrary

smooth functions on positive real numbers if m > 1, and on three functions in

dimension one.

The same result remains true for oriented Riemannian manifolds if m > 3

or m = 1, but the metrics depend on 7 real functions if m = 3 and on 10 real

functions in dimension two.

34. Multilinear natural operators

We have already discussed several ways how to ¬nd natural operators and

all of them involve some results from representation theory. Our general proce-

dures work without any linearity assumption and we also used them in section

30 devoted to the bilinear operators of the type of Fr¨licher-Nijenhuis bracket.

o

However, there are very e¬ective methods involving much more linear represen-

tation theory of the jet groups in question which enable us to solve more general

classes of problems concerning linear geometric operations.

In fact, the representation theory of the Lie algebras of the in¬nite jet groups,

i.e. the formal vector ¬elds with vanishing absolute terms, plays an important

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

34. Multilinear natural operators 281

role. Thus, the methods di¬er essentially if these Lie algebras have ¬nite dimen-

sion in the geometric category in question. The best known example beside the

Riemannian manifolds is the category of manifolds with conformal Riemannian

structure.

Although we feel that this theory lies beyond the scope of our book, we would

like to give at least a survey and a sort of interface between the topics and the

terminology of this book and some related results and methods available in the

literature. For a detailed survey on the subject we recommend [Kirillov, 80,

pp. 3“29]. The linear natural operators in the category of conformal pseudo-

Riemannian manifolds are treated in the survey [Baston, Eastwood, 90].

Some basic concepts and results from representation theory were treated in

section 13.

34.1. Recall that every natural vector bundles E1 , . . . , Em , E : Mfn ’ FM

of order r correspond to Gr -modules V1 , . . . , Vm , V . Further, m-linear natural

n

operators D : C ∞ (E1 • · · · • Em ) = C ∞ (E1 ) — . . . — C ∞ (Em ) ’ C ∞ (E) are

of some ¬nite order k (depending on D), cf. 19.9, and so they correspond to

m-linear Gk+r -equivariant mappings D de¬ned on the product of the standard

n

¬bers Tn Vi of the k-th prolongations J k Ei , D : Tn V1 — . . . — Tn Vm ’ V , see

k k k

k+r

14.18 or 18.20. Equivalently, we can consider linear Gn -equivariant maps

k k

D : Tn V1 — · · · — Tn Vm ’ V . We can pose the problem at three levels.

First, we may ¬x all bundles E1 , . . . , Em , E and ask for all m-linear operators

D : E 1 • · · · • Em E. This is what we always have done.

Second, we ¬x only the source E1 •· · ·•Em , so that we search for all m-linear

geometric operations with the given source. The methods described below are

e¬cient especially in this case.

Third, both the source and the target are not prescribed.

We shall ¬rst proceed in the latter setting, but we derive concrete results only

in the special case of ¬rst order natural vector bundles and m = 1. Of course, the

results will appear in a somewhat implicit way, since we have to assume that the

bundles in question correspond to irreducible representations of G1 = GL(n).

n

We do not lose much generality, for all representations of GL(n) are completely

reducible, except the exceptional indecomposable ones (cf. [Boerner, 67, chapter

V]). But although all tensorial representations are decomposable, it might be a

serious problem to ¬nd the decompositions explicitly in concrete examples. This

also concerns our later discussion on bilinear operations. In particular, we do

not know how to deduce explicitly (in some short elementary way) the results

from section 30 from the more general results due P. Grozman, see below.

34.2. Given linear representations π, ρ of a connected Lie group G on vector

spaces V , W , we know that a linear mapping • : V ’ W is a G-module ho-

momorphism if and only if it is a g-module homomorphism with respect to the

induced representations T π, T ρ of the Lie algebra g on V , W , see 5.15. So if

we ¬nd all gk+r -module homomorphisms D : Tn V1 — · · · — Tn Vm ’ V , we de-

k k

n

scribe all (Gk+r )+ -equivariant maps and so all operators natural with respect to

n

orientation preserving di¬eomorphisms. Hence we shall be able to analyze the

problem on the Lie algebra level. But we ¬rst continue with some observations

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

282 Chapter VII. Further applications

concerning the Gr+k -modules.

n

Recall that for every Gr -module V with homogeneous degree d (as a G1 -

n n

module) the induced Gr+k -module Tn V decomposes as GL(n)-module into the

k

n

k

sum Tn V = V0 • · · · • Vk of GL(n)-modules Vi with homogeneous degrees d ’ i.

Hence given an irreducible G1 -module W and a Gk+r -module homomorphism

n n

k

• : Tn V ’ W such that ker• does not include Vk , W must have homogeneous

degree d ’ k and Tn V is a decomposable Gr+k -module by virtue of 13.14. Hence

k

n

in order to ¬nd all Gk+r -module homomorphisms with source Tn V we have to

k

n

k

discuss the decomposability of this module. Note Tn V is always reducible if

k > 0, cf. 13.14. A corollary in [Terng, 78, p. 812] reads

If V is an irreducible G1 -module, then Tn V is indecomposable except V =

k

n

Λp Rn— , k = 1.

So an explicit decomposition of Tn (Λp Rn— ) leads to

1

Theorem. All non zero linear natural operators D : E1 E between two nat-

1

ural vector bundles corresponding to irreducible Gn -modules are

(1) E1 = Λp T — , E = Λp+1 T — , D = k.d, where k ∈ R, n > p ≥ 0

(2) E1 = E, D = k.id, k ∈ R.

This theorem was originally formulated by J. A. Schouten, partially proved

by [Palais, 59] and proved independently by [Kirillov, 77] and [Terng, 78]. Terng

proved this result by direct (rather technical) considerations and she formulated

the indecomposability mentioned above as a consequence. Her methods are not

suitable for generalizations to m-linear operations or to more general categories

over manifolds.

34.3. If we pass to the Lie algebra level, we can include more information ex-

tending the action of gk+r to an action of the whole algebra g = g’1 • g0 • . . .

n

∞ ‚

of formal vector ¬elds X = |±|=0 aj x± ‚xj on Rn . In particular, the action of

±

the (abelian) subalgebra of constant vector ¬elds g’1 will exclude the general

k

reducibility of Tn V .

Lemma. The induced action of gk+r on Tn V = (J k E)0 Rn is given by the Lie

k

n

r+k k k

di¬erentiation j0 X.j0 s = j0 (L’X s) and this formula extends the action to

the Lie algebra g of formal vector ¬elds. Every gk+r -module homomorphism

n

k

• : Tn V ’ W is a g-module homomorphism.

Proof. We have

r+k k k

‚

j0 X.j0 s = ‚t 0 expt.j0 X (j0 s) = (by 13.2)

r+k

k

‚

= ‚t 0 j0 FlX (j0 s) = (by 14.18)

r+k

t

k X X

‚

‚t 0 j0 (E(Flt ) —¦ s —¦ Fl’t ) =

= (by 6.15)

k

j0 L’X s

=

Each gk+r -module homomorphism • : Tn V ’ W de¬nes an operator D natural

k

n

with respect to orientation preserving local di¬eomorphisms. It follows from 6.15

that every natural linear operator commutes with the Lie di¬erentiation (this

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34. Multilinear natural operators 283

can be seen easily also along the lines of the above computation and we shall

∞

discuss even the converse implication in chapter XI). Hence for all j0 X ∈ g,

k k

j0 s ∈ Tn V

∞ ∞

k k k

j0 X.•(j0 s) = L’X Ds(0) = D(L’X s)(0) = •(j0 (L’X s)) = •(j0 X.j0 s)

and so • is a g-module homomorphism.

34.4. Consider a Gr -module V , a g-module homomorphism • : Tn V ’ W and

k

n

its dual •— : W — ’ (Tn V )— . If W is a Gq -module, then the subalgebra bq =

k

n

gq • gq+1 • . . . in g acts trivially on the image Im•— ‚ (Tn V )— .

k

We say that a g-module V is of height p if gq .V = 0 for all q > p and gp .V = 0.

k

De¬nition. The vectors v ∈ Tn V with trivial action of all homogeneous com-

ponents of degrees greater then the height of V are called singular vectors.

An analogous de¬nition applies to subalgebras a ‚ g with grading and a-

modules.

So the linear natural operations between irreducible ¬rst order natural vec-

tor bundles are described by gk+1 -submodules of singular vectors in (Tn V )— .

k

n

Similarly we can treat m-linear operators on replacing (Tn V )— by (Tn V1 )— —

k k

· · · — (Tn Vm )— . Since all modules in question are ¬nite dimensional, it su¬ces

k

to discuss the highest weight vectors (see 34.8) in these submodules which can

also lead to the possible weights of irreducible modules V . For this purpose, one

can use the methods developed (for another aim) by Rudakov. Remark that the

Kirillov™s proof of theorem 34.2 also analyzes the possible weights of the modules

V , but by discussing the possible eigen values of the Laplace-Casimir operator.

First we have to derive some suitable formula for the action of g on (Tn V )— .

k

‚

In what follows, V and W will be G1 -modules and we shall write ‚i = ‚xi ∈ g’1 .

n

k

34.5. Lemma. (Tn V )— = S i (g’1 ) — V — .

k

i=0

Proof. Every multi index ± = i1 . . . i|±| , i1 ¤ · · · ¤ i|±| , yields the linear map

k k

Tn V ’ V, = (L’‚i1 —¦ . . . —¦ L’‚i|±| s)(0).

±: ± (j0 s)

Since the elements in g’1 commute, we can view the elements in S |±| (g’1 ) as

linear combinations of maps ± . Now the contraction with V — yields a linear

k

map i=0 S i (g’1 ) — V — ’ (Tn V )— . This map is bijective, since (Tn V )— has a

k k

basis induced by the iterated partial derivatives which correspond to the maps

±.

This identi¬cation is important for our computations. Let us denote i =

L’‚i ∈ g— = S 1 (g’1 ), so the elements ± can be viewed as ± = i1 —¦ . . . —¦ i|±| ∈

’1

|±|

S (g’1 ) and we have ± = 0 if |±| > k. Further, for every ∈ g we shall denote

ad ± . = (’1)|±| [‚i1 , [. . . [‚i|±| , ] . . . ]].

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

284 Chapter VII. Further applications

— v ∈ S p — V — is

∈ gq on

34.6. Lemma. The action of ±

—v = — (ad γ . ).v + —¦ (ad γ . ) — v.

. ± β β

β+γ=± β+γ=±

|γ|=q |γ|=q+1

k

= j0 X ∈ gq

Proof. We compute with

k k k

— v)(j0 s) = ’( — v)( .j0 s) = ( — v)(j0 (LX s)) = ( —¦ LX s)(0), v .

.( ± ± ± ±

Since j —¦ LY = LY —¦ + L[’‚j ,Y ] for all Y ∈ g, 1 ¤ j ¤ n, and [‚j , gl ] ‚ gl’1 ,

j

we get

k

— v)(j0 s) = ip’1 LX ip s(0), v ip’1 L[’‚ip ,X] s(0), v

.( ... + ...

± i1 i1

and the same procedure can be applied p times in order to get the Lie derivative

terms just at the left hand sides of the corresponding expressions. Each choice

of indices among i1 , . . . , ip determines just one summand of the outcome. Hence

we obtain (the sum is taken also over repeating indices)

k

— v)(j0 s) =

.( (ad γ . ). β s(0), v .

±

β+γ=±

Further ad γ . = 0 whenever |γ| > q +1 and for all vector ¬elds Y ∈ g0 •g1 •. . .

we have

(LY —¦ β s)(0), v = ’ ( β s)(0), LY v

so that only the terms with |γ| = q or |γ| = q + 1 can survive in the sum (notice

k

Y ∈ gp , p ≥ 1, implies LY v = 0). Since = j0 Y ∈ g0 acts on (the jet of constant

section) v by .v = L’Y v(0), we get the result.

34.7. Example. In order to demonstrate the computations with this formula,

let us now discuss the linear operations in dimension one.

We say that V is a gk -module homogeneous in the order if there is k0 such

n

that gk0 .v = 0 implies v = 0 and gl .v = 0 for all v and l > k0 . Each gk -module

n

includes a submodule homogeneous in order. Indeed, the isotropy algebra of

each vector v contains some kernel bl , l ¤ k, denote lv the minimal one. Let p

be the minimum of these l™s. Then the set of vectors with lv = p is a submodule

homogeneous in order. In particular, every irreducible module is homogeneous

in order.

Consider a g1 module V homogeneous in order. For every non zero vector

1

a = p — v ∈ S p (g’1 ) — V — ‚ (T1 V )— and ∈ g1 we get

k

1

p’1 p’2

p

.a = ’p — [‚1 , ]v + —¦ [‚1 , [‚1 , ]] — v.

1 1

2

d d

= x2 dx so that [‚1 , ] = 2x dx =: 2h and [‚1 , [‚1 , ]] = 2‚1 .

Take

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

34. Multilinear natural operators 285

Assume now b1 .a = 0. Then

p’1

— (’2ph.v + p(p ’ 1)v)

0 = .a = 1

so that 2h.v = (p ’ 1)v or p = 0.

d

Further, set = x3 dx . We get

p p’2

— [‚1 , [‚1 , ]].v ’ p p’3 .[‚1 , [‚1 , [‚1 , ]]] — v

0 = .a = 21 31

p’2

— (3p(p ’ 1)h.v ’ p(p ’ 1)(p ’ 2)v).

= 1

Hence either p = 0 or p = 1 or 3h.v = 3 (p ’ 1)v = (p ’ 2)v. The latter is not

2

possible, for it says p = ’1. The case p = 0 is not interesting since the action

of b1 on all vectors in V — = S 0 (g’1 ) — V — is trivial. But if p = 1 we get h.v = 0

and so the homogeneity in order implies the action of g1 on V is trivial. Hence

1

V = R if irreducible. Moreover, the submodule generated by a in (T1 R)— is 1

1

1 — R with the action h.t 1 = 0 + t 1 . Hence if • : T1 V ’ W is a g-module

homomorphism and if both V and W are irreducible, then either • factorizes

through • : V ’ W which means V = W , • = k.idV , or V = R, W = R— with

the minus identical action of g1 . In this way we have proved theorem 34.2 in the

1

dimension one.

34.8. The situation in higher dimensions is much more di¬cult. Let us mention

some concepts and results from representation theory. Our source is [Zhelobenko,

Shtern, 83] and [Naymark, 76].

Consider a Lie algebra g and its representation ρ in a vector space V . An

element » ∈ g— is called a weight if there is a non zero vector v ∈ V such that

ρ(x)v = »(x)v for all x ∈ g. Then v is called a weight vector (corresponding

to »). If h ‚ g is a subalgebra, then the weights of the adjoint representation

of h in g are called roots of the algebra g with respect to h. The corresponding

weight vectors are called the root vectors (with respect to h).

A maximal solvable subalgebra b in a Lie algebra g is called a Borel subalgebra.

A maximal commutative subalgebra h ‚ g with the property that all operators

adx, x ∈ h, are diagonal in g is called a Cartan subalgebra.

In our case g = gl(n), the upper triangular matrices form the Borel subalgebra

b+ while the diagonal matrices form the Cartan subalgebra h. Let us denote

n+ the derived algebra [b+ , b+ ], i.e. the subalgebra of triangular matrices with

zeros on the diagonals. Consider a gl(n)-module V . A vector v ∈ V is called

the highest weight vector (with respect to b+ ) if there is a root » ∈ h— such that

x.v ’ »(x)v = 0 for all x ∈ h and x.v = 0 for all x ∈ n+ . The root » is called

the highest weight. In our case we identify h— with Rn .

The highest weight vectors always exist for complex representations of com-

plex algebras and are uniquely determined for the irreducible ones. The pro-

cedure of complexi¬cation allows to use this for the real case as well. So each

¬nite dimensional irreducible representation of gl(n) is determined by a high-

est weight (»1 , . . . , »n ) ∈ C such that all »i ’ »i+1 are non negative integers,

i = 1, . . . , n ’ 1.

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

286 Chapter VII. Further applications

34.9. Examples. Let us start with the weight of the canonical representation

on Rn corresponding to the tangent bundle T . The action of a = (ak ), ak = δj δl ki

l l

‚

for some j < i, (corresponding to the action of X = xi ‚xj given by the negative

of the Lie derivative) on a highest weight vector v must be zero, so that only its

¬rst coordinate can be nonzero. Hence the weight is (1, 0, . . . , 0).

Now we compute the weights of the irreducible modules Λp Rn— . The action

‚

of X = xi ‚xj on a (constant) form ω is L’X ω. Since LX dxl = δj dxi we l

get (cf. 7.6) that if X.ω = 0 for all j < i then ω is a constant multiple of

dxn’p+1 § · · · § dxn . Further, the action of L’xi /‚xi on dxi1 § · · · § dxip is

minus identity if i appears among the indices ij and zero if not. Hence the

highest weight is (0, . . . , 0, ’1, . . . , ’1) with n ’ p zeros. Similarly we compute

the highest weight of the dual Λp Rn , (1, . . . , 1, 0, . . . , 0) with n ’ p zeros.

Analogously one ¬nds that the highest weight vector of S p Rm— is the sym-

metric tensor product of p copies of dxn and the weight is (0, . . . , 0, ’p).

34.10. Let us come back to our discussion on singular vectors in (Tn V )— for an

k

irreducible gl(n)-module V . In our preceding considerations we can take suitable

subalgebras with grading instead of the whole algebra g of formal vector ¬elds.

It turns out that one can describe in detail the singular vectors in dimension two

and for the subalgebra of divergence free formal vector ¬elds. We shall denote

this algebra by s(2) and we shall write sr for the Lie algebras of the corresponding

2

jet groups. We shall not go into details here, they can be found in [Rudakov, 74,

pp. 853“859]. But let us indicate why this description is useful. A subalgebra

a ‚ g is called a testing subalgebra if there is an isomorphism s(2) ’ a ‚ g

of algebras with gradings and a distinguished subspace w(a) ‚ g’1 such that

g’1 = a’1 • w(a), [a, w(a)] = 0.

k

Lemma. Let V be a g1 -module, (Tn V )— = i=0 S i (g’1 ) — V — and a ‚ g be a

k

n

k

¯

testing subalgebra. Then V = i=0 S i (w(a)) — V — ‚ (Tn V )— is an a0 -module

k

k ¯

and there is an a-module isomorphism (Tn V )— ’ i=0 S i (a’1 ) — V onto the

k

image.

∞

¯ S i (w(a)) — V — is an a0 -module, for [a, w(a)] = 0. We have

Proof. V = i=0

∞ ∞ ∞

¯ S j (w(a)) — V —

S i (a’1 ) — V = S i (a’1 ) —

i=0 i=0 j=0

∞

S i (a’1 • w(a)) — V — .

= i=0

34.11. It turns out that there are enough testing subalgebras in the algebra of

formal vector ¬elds. Using the results on s(2), Rudakov proves that for every g1 - n

— 1

module V the homogeneous singular vectors can appear only in V • S (g’1 ) —

V — ‚ (Tn V )— . This is equivalent to the assertion that all linear natural operators

k

are of order one.

Let us remark that this was also proved by [Terng, 78] in a very interesting

way. She proved that every tensor ¬eld is locally a sum of ¬elds with polynomial

coe¬cients of degree one in suitable coordinates (di¬erent for each summand)

and so the naturality and linearity imply that the orders must be one.

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

34. Multilinear natural operators 287

34.12. Now, we know that if there is a homogeneous singular vector x which

does not lie in V — ‚ (Tn V )— then there must be a highest weight singular vector

1

x ∈ g— — V — , for all linear representations in question are ¬nite dimensional.

’1

k

i=1 li — ui , where k ¤ n and all i are assumed linearly

Let us write x =

independent.

k

Proposition. Let x = i=1 li — ui be a singular vector of highest weight with

respect to the Borel algebra b+ ‚ g1 . If ui = 0, i = 1, . . . , p, and up+1 = 0,

n

then ui = 0, i = p + 1, . . . , k, and up is a highest weight vector with weight

» = (1, . . . , 1, 0, . . . , 0) with n ’ p + 1 zeros. Then the weight of x is µ =

(1, . . . , 1, 0, . . . , 0) with n ’ p zeros.

Proof. Since x is singular, we have for all k, j, l (we do not use summation

conventions now)

(1)

0 = ’xk xl ‚xj . p p — up = p 1 — [ ‚xp , xk xl ‚xj ].up = xl ‚xj .uk + xk ‚xj .ul .

‚ ‚ ‚ ‚ ‚

In particular, for all k, j

xk ‚xj .uk = 0

‚

(2)

xj ‚xj .uk = ’xk ‚xj .uj .

‚ ‚

(3)

Further, x is a highest weight vector with weight µ = (µ1 , . . . , µn ) and for all i,

j we have

xi ‚xj .x =

‚ ‚ ‚ i‚

— xi ‚xj .up + — up

(4) p [’ ‚xp , x ‚xj ]

pp

‚

— xi ‚xj .up + — ui .

= pp j

‚

If i > j, we have xi ‚xj .x = 0 and so

xi ‚xj .up = 0

‚

(5) p=j

xi ‚xj .uj = ’ui .

‚

(6)

‚

Further, xi ‚xi .x = µi x and so (4) implies for all p, i

p

xi ‚xi .up = (µi ’ δi )up .

‚

(7)

The latter formula shows that the vectors up are either zero or root vectors

of V — with respect to the Cartan algebra h with weights »(p) = (»1 , . . . , »n ),

p

»i = µi ’ δi . Formula (2) implies that either up = 0 or µp = 1. If up = 0,

then all ul = 0, l ≥ p, by (6). Assume up = 0 and up+1 = 0, i.e. µi = 1, i ¤ p.

‚

Then (5) and (6) show that up is a highest weight vector. By (3), xj ‚xj .uk =

‚

»(k)j uk = ’xk ‚xj .uj , so that for k = p, j > p, (7) implies »(p)j up = µj .up =

‚

’xp ‚xj .uj = 0. Hence µi = 1, i = 1, . . . , p, and µi = 0, i = p + 1, . . . , n.

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

288 Chapter VII. Further applications

34.13. Now it is easy to prove theorem 34.2. If D : E1 E is a linear natural

1

operator between bundles corresponding to irreducible Gn -modules V , W , then

either V — = Λp Rn , p = 0, . . . , n ’ 1, and W — = Λp+1 Rn , or D is a zero order

operator. The dual of the inclusion W — ’ (Tn V )— corresponds to the exterior

1

di¬erential up to a constant multiple.

Let us remark, that the only part of the proof we have not presented in detail

is the estimate of the order, but we mentioned a purely geometric way how to

prove this, cf. 34.11. It might be useful in concrete situations to combine some

general methods with ¬nal computations in the above style.

34.14. The method of testing subalgebras is heavily used in [Rudakov, 75] deal-

ing with subalgebras of divergence free formal vector ¬elds and Hamiltonian vec-

tor ¬elds. The aim of all the mentioned papers by Rudakov is the study of in¬nite

dimensional representations of in¬nite dimensional Lie algebras of formal vector

∞

¬elds. His considerations are based on the study of the space i=0 S i (g’1 ) — V —

and so the results are relevant for our purposes as well. We should remark that in

‚

the cited papers the action slightly di¬ers in notation and the vector ¬elds xi ‚xj

are identi¬ed with the transposed matrix (ai ) to our (aj ) and so the weights cor-

j i

respond to the Borel subalgebra of lower triangular matrices. Due to Rudakov™s

results, a description of all linear operations natural with respect to unimodular

or symplectic di¬eomorphisms is also available. In the unimodular case we get

the following result. We write S n for the category of n-dimensional manifolds

with ¬xed volume forms and local di¬eomorphisms preserving the distinguished

forms.

Theorem. All non zero linear natural operators D : E1 E between two ¬rst

order natural bundles on category S n corresponding to irreducible representa-

tions of the ¬rst order jet group are

(1) E1 = E, D = k.id, k ∈ R

(2) E1 = Λp T — , E = Λp+1 T — , D = k.d, k ∈ R, n > p ≥ 0

i

(3) E1 = Λn’1 T — , E = Λ1 T — , D = k.(d —¦ i —¦ d) : Λn’1 T — ’ Λn T — ’ Λ0 T — ’

’

∼ =

Λ1 T — , k ∈ R.

Let us point out that this theorem describes all linear natural operations not

only up to decompositions into irreducible components but also up to natural

equivalences. For example, to ¬nd linear natural operations with vector ¬elds

we have to notice Rn ∼ Λn’1 Rn— , ‚xp ’ i( ‚xp )(dx1 § · · · § dxn ). Hence the

‚ ‚

=

Lie di¬erentiation of the distinguished volume forms corresponds to the exte-

rior di¬erential on (n ’ 1)-forms, the identi¬cation of n-forms with functions

yields the divergence of vector ¬elds and the exterior di¬erential of the diver-

gence represents the ˜composition™ of exterior derivatives from point (3). Beside

the constant multiples of identity, there are no other linear operations (with

irreducible target).

We shall not describe the Hamiltonian case. We remark only that then not

even the di¬erential forms correspond to irreducible representations and that

the interesting operations live on irreducible components of them.

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

34. Multilinear natural operators 289

34.15. Next we shall shortly comment some results concerning m-linear oper-

ations. We follow mainly [Kirillov, 80]. So ρ— will denote a representation dual

to a representation ρ of G1 + and we write ∆ for the one-dimensional represen-

n

tation given by a ’ deta’1 . Further “ρ (M ) is the space of all smooth sections

of the vector bundle Eρ over M corresponding to ρ. In particular “∆ (M ) co-

incides with „¦n M . To every representation ρ we associate the representation

ρ := ρ— — ∆. The pointwise pairing on “ρ (M ) — “ρ— (M ) gives rise to a bilinear

˜

mapping “ρ (M ) — “ρ (M ) ’ „¦n (M ), a natural bilinear operation of order zero.

˜

Given two sections s ∈ “ρ (M ), s ∈ “ρ (M ) with compact supports we can inte-

˜ ˜

grate the resulting n-form, let us write s, s for the outcome. We have got a

˜

bilinear functional invariant with respect to the di¬eomorphism group Di¬M .

For every m-linear natural operator D of type (ρ1 , . . . , ρm ; ρ) we de¬ne an

(m + 1)-linear functional

FD (s1 , . . . , sm , sm+1 ) = D(s1 , . . . , sm ), sm+1 ,

de¬ned on sections si ∈ “ρi (M ), i = 1, . . . , m, sm+1 ∈ “ρ (M ) with compact

˜

supports. The functional FD satis¬es

(1) FD is continuous with respect to the C ∞ -topology on “ρi and “ρ ˜

(2) FD is invariant with respect to Di¬M

(3) FD = 0 whenever ©m+1 suppsi = ….

i=1

We shall call the functionals with properties (1)“(3) the invariant local func-

tionals of the type (ρ1 , . . . , ρm ; ρ).

˜

Theorem. The correspondence D ’ FD is a bijection between the m-linear

natural operators of type (ρ1 , . . . , ρm ; ρ) and local linear functionals of type

(ρ1 , . . . , ρm ; ρ).

˜

The proof is sketched in [Kirillov, 80] and consists in showing that each such

functional is given by an integral operator the kernel of which recovers the natural

m-linear operator.

34.16. The above theorem simpli¬es essentially the discussion on m-linear nat-

ural operations. Namely, there is the action of the permutation group Σm+1

on these operations de¬ned by (σFD )(s1 , . . . , sm+1 ) = FD (sσ1 , . . . , sσ(m+1) ),

σ ∈ Σm+1 . Hence a functional of type (ρ1 , . . . , ρm ; ρ) is transformed into a

functional of type (ρσ’1 (1) , . . . , ρσ’1 (m+1) ) and so for every operation D of the

type (ρ1 , . . . , ρm ; ρ) there is another operation σD. If σ(m + 1) = m + 1, then

this new operation di¬ers only by a permutation of the entries but, for example,

if σ transposes only m and m + 1, then σD is of type (ρ1 , . . . , ρm’1 , ρ; ρm ).

˜˜

In the simplest case m = 1, the exterior derivative d : „¦ M ’ „¦p+1 M is

p

transformed by the only non trivial element in Σ2 into d : „¦n’p’1 M ’ „¦n’p M .

If m = 2, the action of Σ3 becomes signi¬cant. We shall now describe all

operations in this case. Those of order zero are determined by projections of

ρ1 — ρ2 onto irreducible components.

34.17. First order bilinear natural operators. We shall divide these op-

erations into ¬ve classes, each corresponding to some intrinsic construction and

the action of Σ3 .

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

290 Chapter VII. Further applications

1. Write „ for the canonical representation of G1 + on Rn , i.e. “„ (M ) are the

n

smooth vector ¬elds on M . For every representation ρ we have the Lie derivative

L : “„ (M ) — “ρ (M ) ’ “ρ (M ), a natural operation of type („, ρ; ρ). The action

of Σ3 yields an operation of type (ρ, ρ; „ ) allowing to construct invariantly a

˜˜

covector density from any two tensor ¬elds which admit a pointwise pairing into

a volume form. This operation appears often in the lagrangian formalism and

Nijenhuis called it the lagrangian Schouten concomitant.

2. This class contains the operations of the types (Λk „ —∆κ , Λl „ —∆» ; Λm „ —

∆µ ), where k, l, m are certain integers between zero and n while κ, », µ are

certain complex numbers.

Assume ¬rst k + l > n + 1. Then an operation exists if m = k + l ’ n ’ 1,

µ = κ + » ’ 1. Let us choose an auxiliary volume form v ∈ “∆ (M ) and use the

identi¬cation Λk „ —∆κ ∼ Λn’k „ — —∆κ’1 , i.e. we shall construct an operation of

=

the type (Λ „ —∆ , Λ „ —∆» ; Λm „ — —∆µ ) with k +l ¤ n’1, m = k +l +1

k— l—

κ

and µ = κ + » . Then we can write a ¬eld of type Λk „ — ∆κ in the form ω.v κ’1 ,

ω ∈ Λn’k T — M . We de¬ne

D(ω1 .v κ’1 , ω2 .v »’1 ) = (c1 dω1 § ω2 + c2 ω1 § dω2 ).v µ’1

(1)

where ω1 is a (n’k)-form, ω2 is a (n’l)-form, and c1 , c2 are constants. The right

hand side in (1) should not depend on the choice of v. So let us write v = •.˜ v

κ’1 κ’1 »’1 »’1

where • is a positive function. Then ω1 .v = ω1 .˜

˜v , ω2 .v = ω2 .˜

˜v ,

κ’1 »’1

with ω1 = •

˜ .ω1 , ω2 = •

˜ .ω2 . After the substitution into (1), there appears

the extra summand

(c1 d•κ’1 § ω1 § •»’1 ω2 + c2 •κ’1 ω1 § d•»’1 § ω2 )˜mu’1

v

= (κ ’ 1)c1 + (’1)k (» ’ 1)c2 .d(ln•) § ω1 § ω2 .v µ’1 .

Thus (1) is a correct de¬nition of an invariant operation if and only if

(κ ’ 1)c1 + (’1)k (» ’ 1)c2 = 0.

(2)

Now take k + l ¤ n + 1. We ¬nd an operation if and only if m = k + l ’ 1

and µ = κ + ». As before, we ¬x an auxiliary volume form v and we write

the ¬elds of type Λk „ — ∆κ as a.v κ where a is a k-vector ¬eld. The usual

divergence of vector ¬elds extends to a linear operation δv on k-vector ¬elds,

k

δv (X1 § · · · § Xk ) = i=1 (’1)i+1 divXi .X1 § · · · §i · · · § Xk , where §i means that

the entry is missing. Of course, this divergence depends on the choice of v. We

have

(3)

k

(’1)i+1 Xi (•).X1 § · · · §i · · · § Xk .

δ•v (X1 § · · · § Xk ) = •.δv (X1 § · · · § Xk ) +

i=1

Let us look for a natural operator D of the form

D(a.v κ , b.v » ) = (c1 δv (a) § b + c2 a § δv (b) + c3 δv (a § b)) .v µ .

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

34. Multilinear natural operators 291

Formula (3) implies that D is natural if and only if

(κ ’ 1)c1 + (κ + » ’ 1)c3 = 0, (» ’ 1)c2 + (κ + » ’ 1)c3 = 0.

(4)

The formulas (2) and (4) de¬ne the constants uniquely except the case κ =

» = 1 when we get two independent operations, see also the ¬fth class. Let

us point out that the second class involves also the Schouten-Nijenhuis bracket

Λp T • Λq T Λp+q’1 T (the case κ = » = 0, k + l ¤ n + 1), cf. 30.10, sometimes

also caled the antisymmetric Schouten concomitant, which de¬nes the structure

of a graded Lie algebra on the ¬elds in question. This bracket is given by

[X1 § · · · § Xk , Y1 § · · · § Yl ]

§ X1 § · · · §i · · · § Xk § Y1 § · · · §j · · · § Yl .

i+j

= i,j (’1) [Xi , Yj ]

The second class is invariant under the action of Σ3 .

3. The third class is represented by the so called symmetric Schouten con-

comitant. This is an operation of type (S k „, S l „ ; S k+l’1 „ ) with a nice geometric

de¬nition. The elements in S k T M can be identi¬ed with functions on T — M ¬ber-

wise polynomial of degree k. Since there is a canonical symplectic structure on

T — M , there is the Poisson bracket on C ∞ (T — M ). The bracket of two ¬berwise

polynomial functions is also ¬berwise polynomial and so the bracket gives rise

to our operation.

The action of Σ3 yields an operation of the type (S k „, S l „ — —∆; S l’k+1 „ — —∆).

If k = 1, this is the Lie derivative and if k = l, we get the lagrangian Schouten

concomitant.

4. This class involves the Fr¨licher-Nijenhuis bracket, an operation of the type

o

k— l— k+l —

(„ — Λ „ , „ — Λ „ ; „ — Λ „ ), k + l ¤ n. The tensor spaces in question are not

irreducible, „ — Λk „ — is a sum of Λk’1 „ — and an irreducible representation ρk

of highest weight (1, . . . , 1, 0, . . . , 0, ’1) where 1 appears k-times (the trace-free

vector valued forms). The Fr¨licher-Nijenhuis bracket is a sum of an operation

o

of type (ρk , ρl ; ρk+l ) and several other simpler operations.

If we apply the action of Σ3 to the Fr¨licher-Nijenhuis bracket, we get an

o

operation of the type („ — Λ „ , „ — Λ „ ; „ — Λk+m „ — ) which is expressed

m— — k— —

through contractions and the exterior derivative.

5. Finally, there are the natural operations which reduce to compositions of

wedge products and exterior di¬erentiation. Such operations are always de¬ned

if at least one of the representations ρ1 , ρ2 , or one of the irreducible components

of ρ1 — ρ2 coincides with Λk „ — . Since Λk „ — = Λn’k „ — , this class is also invariant

under the action of Σ3 .

In [Grozman, 80b] we ¬nd the next theorem. Unfortunately its proof based

on the Rudakov™s algebraic methods is not available in the literature. In an

earlier paper, [Grozman, 80a], he classi¬ed the bilinear operations in dimension

two, including the unimodular case.

34.18. Theorem. All natural bilinear operators between natural bundles cor-

responding to irreducible representations of GL(n) are exhausted by the zero

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

292 Chapter VII. Further applications

order operators, the ¬ve classes of ¬rst order operators described in 34.17, the

operators of second and third order obtained by the composition of the ¬rst and

zero order operators and one exceptional operation in dimension n = 1, see the

example below.

In particular, there are no bilinear natural operations of order greater then

three.

34.19. Example. A tensor density on the real line is determined by one com-

plex number », we write f (x)(dx)’» ∈ C ∞ (E» R) for the corresponding ¬elds of

geometric objects. There is a natural bilinear operator D : E2/3 • E2/3 E’5/3

f g df /dx dg/dx

D(f (dx)’2/3 , g(dx)’2/3 ) = 2 .(dx)5/3

+3

d3 f /dx3 d3 g/dx3 d2 f /dx2 d2 g/dx2

This is a third order operation which is not a composition of lower order ones.

34.20. The multilinear natural operators are also related to the cohomology

theory of Lie algebras of formal vector ¬elds. In fact these operators express

zero dimensional cohomologies with coe¬cients in tensor products of the spaces

of the ¬elds in question, see [Fuks, 84]. The situation is much further analyzed

in dimension n = 1 in [Feigin, Fuks, 82]. In particular, they have described all

skew symmetric operations E» • · · · • E» Eµ . They have deduced

Theorem. For every » ∈ C, m > 0, k ∈ Z, there is at most one skew symmetric

operation D : Λm C ∞ E» ’ C ∞ Eµ with µ = m»’ 2 m(m’1)’k, up to a constant

1

multiple. A necessary and su¬cient condition for its existence is the following:

either k=0, or 0 < k ¤ m and » satis¬es the quadratic equation

(» + 1 )(k2 + 1) ’ m = 1 (k2 ’ k1 )2

1

(» + 2 )(k1 + 1) ’ m 2 2

with arbitrary positive k1 ∈ Z, k2 ∈ Z, k1 .k2 = k.

The operator corresponding to the ¬rst possibility k = 0, D : Λm C ∞ (E» R) ’

C ∞ (Em»’ 1 m(m’1) R), admits a simple expression

2

(m’1)

f1 f1 ... f1

(m’1) 1

f1 (dx)’» § · · · § fm (dx)’» ’ (dx)’m»+ 2 m(m’1)

f2 f2 ... f2

.............

(m’1)

fm fm ... fm

Grozman™s operator from 34.19 corresponds to the choice m = 2, k = 2,

» = 2/3, k1 = 2, k2 = 1. The proof of this theorem is rather involved. It

is based on the structure of projective representations of the algebra of formal

vector ¬elds on the one-dimensional sphere.

34.21. The problem of ¬nding all natural m-linear operations has been also for-

mulated for super manifolds. As far as we know, only the linear operations were

classi¬ed, see [Bernstein, Leites, 77], [Leites, 80], [Shmelev, 83], but their results

include also the unimodular, and Hamiltonian cases. Some more information is

also available in [Kirillov, 80].

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

34. Multilinear natural operators 293

34.22. The linear natural operations on conformal manifolds. As we

have seen, the description of the linear natural operators is heavily based on the

structure of the subalgebra in the algebra of formal vector ¬elds which corre-

sponds to the jet groups in the category in question. If the category involves

very few morphisms, these algebras become small. In particular, they might

have ¬nite dimensions like in the case of Riemannian manifolds or conformal

Riemannian manifolds. The former example is not so interesting for the follow-

ing reasons: Since all irreducible representations of the orthogonal groups are

O(m, R)-invariant irreducible subspaces in tensor spaces, we can work in the

whole category of manifolds in the way demonstrated in section 33. On the

other hand, if we include the so called spinor representations of the orthogonal

group, we get serious problems with the whole setting. However, the second

example is of highest interest for many reasons coming both from mathematics

and physics and it is treated extensively nowadays. Let us conclude this section

with a very short overview of the known results, for more information see the

survey [Baston, Eastwood, 90] or the papers [Baston, 90], [Branson, 85].

Let us write C for the category of manifolds with a conformal Riemannian

structure, i.e. with a distinguished line bundle in S+ T — M , and the morphisms

2

keeping this structure. More explicitely, two metrics g, g on M are called confor-

ˆ

mal if there is a positive smooth function f on M such that g = f 2 g. A conformal

ˆ

structure is an equivalence class with respect to this equivalence relation. The

conformal structure on M can also be described as a reduction of the ¬rst order

frame bundle P 1 M to the conformal group CO(m, R) = R O(m, R), and the

conformal morphisms • are just those local di¬eomorphisms which preserve this

reduction under the P 1 •-action. Thus, each linear representation of CO(m, R)

on a vector space V de¬nes a bundle functor on C. The category C is not locally

homogeneous, but it is local.

The main di¬erence from the situations typical for this book is that there

are new natural bundles in the category C. In fact, we can take any linear

representation of O(m, R) and a representation of the center R ‚ GL(m, R)

and combine them together. The representations of the center are of the form

(t.id)(v) = t’w .v with an arbitrary real number w, which is called the confor-

mal weight of the representation or of the corresponding bundle functor. Each

tensorial representation of GL(m, R) induces a representation of CO(m, R) with

the conformal weight equal to the di¬erence of the number of covariant and

contravariant indices. In particular, the convention for the weight is chosen in

such a way that the bundle of metrics has conformal weight two. If we restrict

our considerations to the tensorial representations, we exclude nearly all natural

linear operators.

Each isometry of a conformal manifold with respect to an arbitrary metric

from the distinguished class is a conformal morphism. Thus, the Riemannian

natural operators described in section 33 can be taken for candidates in the

classi¬cation. But the remaining problems are still so di¬cult that a general

solution has not been found yet.

Let us mention at least two possibilities how to treat the problem. The

¬rst one is to restrict ourselves to locally conformally ¬‚at manifolds, i.e. we

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

294 Chapter VII. Further applications

consider only a subcategory in C which is admissible in our sense. Thus, the

classi¬cation problem for linear operators reduces to a (di¬cult) problem from

the representation theory. But what remains then is to distinguish those natural

operators on the conformally ¬‚at manifolds which are restrictions of natural

operators on the whole category, and to ¬nd explicite formulas for them. For

general reasons, there must be a universal formula in the terms of the covariant

derivatives, curvatures and their covariant derivatives. The best known example

is the conformal Laplace operator on functions in dimension 4

1

a

D= +R

a

6

where a a means the operator of the covariant di¬erentiation applied twice

and followed by taking trace, and R is the scalar curvature. The proper confor-