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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

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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

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


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