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=±