r r

in E± ‚ Tn Q± , every relatively compact neighborhood K of K and every couple

(j0± q, z) ∈ E± — Z± there is an element g ∈ G∞ such that g — (j0± q, z) ∈ V — K .

r r r

±

Then every natural operator in question has ¬nite order on all objects of type

±.

Proof. For every relatively compact neighborhood K of K, there is an r ∈ N

∞

and a neighborhood of j0± s in the C r -topology such that χ± ¤ r on V — K .

But the assumptions of the corollary ensure that the orbit of V — K coincides

∞

with the whole space E± — Z± .

Next we deduce several simple applications of this procedure.

23.5. Proposition. Let F : Mfm ’ FM be a bundle functor of order r such

that its standard ¬ber Q together with the induced action of G1 ‚ Gr can be

m m

identi¬ed with a linear subspace in a ¬nite direct sum

ai bi

i

m

Rm—

R—

and bi > ai for all i. Let G1 : Mfm ’ FM be a bundle functor such that either

its standard ¬ber Z together with the induced G1 -action can be identi¬ed with

m

a linear subspace in a ¬nite direct sum

aj bj

j

m

Rm—

R—

and bj > aj for all j, or Z is compact.

Then every natural operator D : F (G1 , G2 ) de¬ned on all sections of the

bundles F M has ¬nite order.

Proof. Write •t : Rm ’ Rm , t ∈ R, for the homotheties x ’ tx. Let us consider

the canonical identi¬cation F Rm = Rm — Q and the zero section s = (idRm , 0)

in C ∞ (F Rm ). Further, consider an arbitrary section q : Rm ’ F Rm and let us

denote qt = F •t —¦ q —¦ •’1 and qt (x) = (x, qt (x)). Under our identi¬cation, s is

i

t

translation invariant and we can use formula 14.18.(2) to study the derivatives

of the maps qt at the origin. For all partial derivatives ‚ ± qt we get

i i

‚ ± qt (0) = tai ’bi ’|±| ‚ ± q i (0).

i

(1)

If the standard ¬ber Z is compact, then we can use lemma 23.4 with K = Z

and the zero section s. Indeed, if we choose an order r and a neighborhood V of

r r r

j0 s in Tm Q, then taking t large enough we obtain j0 qt ∈ V , so that the bound

r is valid everywhere. But if Z is not compact, then an analogous equality to

(1) holds for the sections of G1 Rm with ai ’ bi replaced by aj ’ bj and these are

also negative. Hence we can apply the same procedure taking K = {0}, where

0 is the zero element in the tensor space Z.

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

208 Chapter V. Finite order theorems

23.6. Examples. The assumptions of the proposition are satis¬ed by all tensor

bundles with more covariant then contravariant components. But clearly, these

are also satis¬ed for all a¬ne natural bundles with associated natural vector

bundles formed by the above tensor bundles. So in particular, F can equal to

QP 1 : Mfm ’ FM, the bundle functor of elements of linear connections, cf.

17.7, or to the bundle functors of elements of exterior forms. If G1 = IdC then Z

is a one-point-manifold, i.e. a compact. Hence we have proved that all natural

operators on connections or on exterior forms that do not extend the bases have

¬nite order.

23.7. Let us apply corollary 23.4 to the natural operators on the bundle func-

tor J 1 : FMm,n ’ FM, i.e. we want to derive the ¬niteness of the order for

geometric operations with general connections. For this purpose, consider the

maps •a,b : Rm+n ’ Rm+n , •(x, y) = (ax, by). In words, we will use the in-

clusion Gr — Gr ’ Gr r

m,n and the jets of homotheties in the jet groups Gm

m n

p

and Gr . In canonical coordinates (xi , y p , yi ) on J 1 (Rm+n ’ Rm ), i = 1, . . . , m,

n

p

j = 1, . . . , n, we get for every section s = yi (xi , y p ) and every local ¬bered

isomorphism • = (•i , •p )

‚•’1 ’1

‚•p ‚•p ’1 ‚•0

q

’1

—¦ •’1 ’1

1 0

J •—¦s—¦• —¦• (yj —¦ • )

= + .

‚xj ‚xi ‚y q ‚xi

In particular, for • = •a,b we obtain

p

•a,b — s(xi , y p ) = ba’1 yi —¦ •’1 .

a,b

Hence for every multi index ± = ±1 + ±2 , where ±1 includes all the derivatives

with respect to the indices i while ±2 those with respect to p™s, it holds

‚ ±1 +±2 (•a,b — s)(0) = a’1’|±1 | b1’|±2 | ‚ ±1 +±2 s(0).

(1)

Proposition. Let H : FMm,n ’ FM be an arbitrary bundle functor while

G : FMm,n ’ FM is either the identity functor or the functor J 1 or the vertical

tangent bundle V . Then every natural operator D : J 1 (G, H) de¬ned on all

sections of the ¬rst jet prolongations has ¬nite order.

Proof. If G = IdF Mm,n , then we can take b = 1, a > 0 and corollary 23.4

together with (1) imply the assertion. The same choice of a and b leads also to

p p

the case G = J 1 , for J 1 •a,b (yi ) = a’1 yi on the standard ¬ber over 0 ∈ Rm+n .

In the third case we have to be more careful. On the standard ¬ber Rn of

V Rm+n we have V •a,b (ξ p ) = (bξ p ). Let us ¬x some r ∈ N and choose a = b’r ,

0 < b < 1 arbitrary. Then

|‚ ±1 +±2 (•a,b — s)(0)| = br(1+|±1 |)+1’|±2 | |‚ ±1 +±2 s(0)|

and so for all |±| ¤ r we get

|‚ ± (•a,b — s)(0)| ¤ b|‚ ± s(0)|.

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

23. The order of natural operators 209

Hence also in this case corollary 23.4 implies our assertion.

At the end of this section, we illustrate on two examples how bad things

may be. First we construct a natural operator which essentially depends on

in¬nite jets and the next example presents a non-regular natural operator. This

contrasts the results on bundle functors where the regularity follows from the

other axioms.

23.8. Example. Consider the bundle functor F = T • T — : Mfm ’ Mf and

let G be the bundle functor de¬ned by GM = M — R, Gf = f — idR , for all

m-dimensional manifolds M and local di¬eomorphisms f , i.e. ˜the bundle of real

functions™. The contraction de¬nes a natural function, i.e. a natural operator

G, of order zero. The composition with any ¬xed real function R ’ R

F

+

is a natural transformation G ’ G and also the addition G • G ’ G and

’

.

multiplication G • G ’ G are natural transformations. Moreover, there is the

’

T — , a natural operator of order 1.

exterior di¬erential d : G

By induction, let us de¬ne operators Dk : T • T — ’ G. We set

(D0 )M (X, ω) = iX ω and (Dk+1 )M (X, ω) = iX d (Dk )M (X, ω)

for k = 0, 1, . . . . Further, consider a smooth function a : R2 ’ R satisfying

a(t, x) = 0 if and only if |x| > t > 0. We de¬ne

∞

a(k, ’) —¦ (iX ω) . (Dk )M (X, ω) .

DM (X, ω) =

k=0

Since the sum is locally ¬nite for every (X, ω) ∈ C ∞ (F M ), this is a natural

operator of in¬nite order.

23.9. Example. Consider once more the bundle functors F , G and operators

Dk from example 23.8. Let a and g : R2 ’ R be the functions used in 19.15.

We shall modify operator D from example 19.15 to get a non-regular natural

operator. Let us de¬ne operators DM : C ∞ (F M ) C ∞ (GM ) by

∞

a(k, ’) —¦ g —¦ ((iX ω) — (iX d(iX ω))) . (Dk )M (X, ω)

DM (X, ω) =

k=0

for all (X, ω) ∈ C ∞ (T • T — M ). We have used only natural operators in our

construction, but, unfortunately, the values DM (X, ω) need not be smooth (or

even de¬ned) if dimension m is greater then one. This is caused by the in¬nite

value of lim supx’(0,1) g(x). But if m = 1, then all values are smooth and the

system DM satis¬es all axioms of natural operators except the regularity. Indeed,

it su¬ces to verify the smoothness of the values of DR . But if (iX ω)(t0 ) = 0

d d

and (iX d(iX ω))(t0 ) = 1, i.e. X(t0 ) dx (Xω)(t0 ) = 1, then dx (Xω)(t0 ) = 0 and

therefore the curve

d

t ’ (Xω)(t), X (Xω)(t)

dx

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

210 Chapter V. Finite order theorems

lies on a neighborhood of t0 inside the unit circles centered in (’1, 1) and (1, 1).

Hence DR (X, ω) = 0 on some neighborhood of t0 .

Let us note that our operator D is not only non-regular, but also of in¬nite

order and it shows that the assertion of lemma 23.3 does not hold for all maps

s ∈ DC± , in general. A non-regular natural operator of order 4 on Riemannian

metrics for dimension m = 2 can be found in [Epstein, 75].

23.10. If we consider natural operators D : F (G1 , G2 ) with domains formed

by all sections of the bundles F M ’ M , then we can use the regularity of

D and apply the stronger version of nonlinear Peetre theorem 19.10 instead of

19.7 in the proof of 23.3. Hence we do not need the invariance of the section s.

Consequently, the assertion of lemma 23.3 holds for all sections s ∈ EC± . That

is why, under the assumptions of corollary 23.4 we can strengthen its assertion.

Corollary. Let s ∈ C ∞ (F C± ) be a section and K ‚ Z be a compact set.

r r r

Assume that for every order r ∈ N, every neighborhood V of j0± s in E± ‚ Tn Q± ,

r

every relatively compact neighborhood K of K and every couple (j0± q, z) ∈

E± — Z± there is an element g ∈ G∞ such that g — (j0± q, z) ∈ V — K . Then every

r r

±

natural operator D : F (G1 , G2 ) has a ¬nite order on all objects of type ±.

Remarks

The general setting for bundle functors and natural operators extends the

original categorical approach to geometric objects and operators due to [Nijen-

huis, 72] and we follow mainly [Kol´ˇ, 90] and partially [Slov´k, 91].

ar a

The multilinear version of Peetre theorem, proved in [Cahen, De Wilde, Gutt,

80], seems to be the ¬rst non-linear generalization of the famous Peetre theorem,

[Peetre, 60]. The study of general nonlinear operators started in [Chrastina, 87]

and [Slov´k, 87b]. The original aim of the nonlinear version 19.7, ¬rst proved in

a

[Slov´k, 87b], was the reduction of the problem of ¬nding natural operators to a

a

¬nite order. The pure analytical results were further generalized and completed

in a setting of H¨lder-continuous maps and metric spaces in [Slov´k, 88] and

o a

it became clear that they should help to unify the approach to the ¬niteness

of the orders of both natural operators and bundle functors and to avoid the

original manipulation with in¬nite dimensional Lie algebras, see [Palais, Terng,

77]. Let us remark that nearly all categories over manifolds used in di¬erential

geometry are admissible and locally ¬‚at, however the veri¬cation of the Whitney

extendibility might present a serious analytical problem in concrete examples.

In the most technical part of the description of bundle functors, i.e. in the proof

of the regularity, we mainly follow [Mikulski, 85] which generalizes the original

proof due to [Epstein, Thurston, 79] to natural bundles with in¬nite dimensional

values. Let us point out that our proof also applies to continuous regularity of

bundle functors on the categories in question with values in in¬nite dimensional

manifolds.

Our sharp estimate on the orders of jet groups acting on manifolds is a gener-

alization of [Zajtz, 87], where similar results are obtained for the full group Gr .

m

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

Remarks 211

The results on the order of bundle functors on FMm follow some ideas from

[Kol´ˇ, Slov´k, 89] and [Mikulski, 89 a, b]. The methods used in our discussion

ar a

on the order of natural operators never exploit the regularity of the natural op-

erators which we have incorporated into our de¬nition. So the results of section

23 can be applied to non-regular natural operators which can also be classi¬ed

in some concrete situations.

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

212

CHAPTER VI.

METHODS FOR FINDING

NATURAL OPERATORS

We present certain general procedures useful for ¬nding some equivariant

maps and we clarify their application by solving concrete geometric problems.

The equivariance with respect to the homotheties in GL(m) gives frequently a

homogeneity condition. The homogeneous function theorem reads that under

certain assumptions a globally de¬ned smooth homogeneous function must be

polynomial. In such a case the use of the invariant tensor theorem and the

polarization technique can specify the form of the polynomial equivariant map

up to such an extend, that all equivariant maps can then be determined by

direct evaluation of the equivariance condition with respect to the kernel of

the jet projection Gr ’ G1 . We ¬rst deduce in such a way that all natural

m m

operators transforming linear connections into linear connections form a simple

3-parameter family. Then we strengthen a classical result by Palais, who deduced

that all linear natural operators Λp T — ’ Λp+1 T — are the constant multiples of

the exterior derivative. We prove that for p > 0 even linearity follows from

naturality. We underline, as a typical feature of our procedures, that in both

cases we ¬rst have guaranteed by the results from chapter V that the natural

operators in question have ¬nite order. Then the homogeneous function theorem

implies that the natural operators have zero order in the ¬rst case and ¬rst

order in the second case. In section 26 we develop the smooth version of the

tensor evaluation theorem. As the ¬rst application we determine all natural

transformations T T — ’ T — T . The result implies that, unlike to the case of

cotangent bundle, there is no natural symplectic structure on the tangent bundle.

As an example of a natural operator related with ¬bered manifolds we discuss

the curvature of a general connection. An important tool here is the generalized

invariant tensor theorem, which describes all GL(m) — GL(n)-invariant tensors.

We deduce that all natural operators of the curvature type are the constant

multiples of the curvature and that all such operators on a pair of connections

are linear combinations of the curvatures of the individual connections and of

the so-called mixed curvature of both connections. The next section is devoted

to the orbit reduction. We develop a complete version of the classical reduction

theorem for linear symmetric connections and Riemannian metrics, in which

the factorization procedure is described in terms of the curvature spaces and

the Ricci spaces. The so-called method of di¬erential equations is based on the

simple fact that on the Lie algebra level the equivariance condition represents

a system of partial di¬erential equations. As an example we deduce that the

only ¬rst order natural operator transforming Riemannian metrics into linear

connections is the Levi-Civit` operator. But we apply the method of di¬erential

a

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

24. Polynomial GL(V )-equivariant maps 213

equations only in the ¬rst part of the proof, while in the ¬nal step a direct

geometric consideration is used.

24. Polynomial GL(V )-equivariant maps

24.1. We ¬rst deduce a result on the globally de¬ned smooth homogeneous

functions, which is useful in the theory of natural operators.

Consider a product V1 — . . . — Vn of ¬nite dimensional vector spaces. Write

xi ∈ Vi , i = 1, . . . , n.

Homogeneous function theorem. Let f (x1 , . . . , xn ) be a smooth function

de¬ned on V1 — . . . — Vn and let ai > 0, b be real numbers such that

k b f (x1 , . . . , xn ) = f (k a1 x1 , . . . , k an xn )

(1)

holds for every real number k > 0. Then f is a sum of the polynomials of degree

di in xi satisfying the relation

a1 d1 + · · · + an dn = b.

(2)

If there are no non-negative integers d1 , . . . , dn with the property (2), then f is

the zero function.

Proof. First we remark that if f satis¬es (1) with b < 0, then f is the zero

function. Indeed, if there were f (x1 , . . . , xn ) = 0, then the limit of the right-

hand side of (1) for k ’ 0+ would be f (0, . . . , 0), while the limit of the left-hand

side would be improper.

b

In the case b ≥ 0 we write a = min(a1 , . . . , an ) and r = a (=the integer

b

part of the ratio a ). Consider some linear coordinates xji on each Vi . We claim

that all partial derivatives of the order r + 1 of every function f satisfying (1)

vanish identically. Di¬erentiating (1) with respect to xji , we obtain

‚f (k a1 x1 , . . . , k an xn )

‚f (x1 , . . . , xn )

kb = k ai .

‚xji ‚xji

‚f

Hence for ‚xji we have (1) with b replaced by b ’ ai . This implies that every

partial derivative of the order r + 1 of f satis¬es (1) with a negative exponent

on the left-hand side, so that it is the zero function by the above remark.

Since all the partial derivatives of f of order r + 1 vanish identically, the

remainder in the r-th order Taylor expansion of f at the origin vanishes identi-

cally as well, so that f is a polynomial of order at most r. For every monomial

x±1 . . . x±n of degree |±i | in xi , we have

n

1

(k a1 x1 )±1 . . . (k an xn )±n = k a1 |±1 |+···+an |±n | x±1 . . . x±n .

n

1

Since k is an arbitrary positive real number, a non-zero polynomial satis¬es (1)

if and only if (2) holds.

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

214 Chapter VI. Methods for ¬nding natural operators

24.2. Remark. The assumption ai > 0, i = 1, . . . , n in the homogeneous

function theorem is essential. We shall see in section 26 that e.g. all smooth

functions f (x, y) of two independent variables satisfying f (kx, k ’1 y) = f (x, y)

for all k = 0 are of the form •(xy), where •(t) is any smooth function of one

variable. In this case we have a1 = 1, a2 = ’1, b = 0.

24.3. Invariant tensors. Consider a ¬nite dimensional vector space V with

a linear action of a group G. The induced action of G on the dual space V — is

given by

av — , v = v — , a’1 v

for all v ∈ V , v — ∈ V — , a ∈ G. In any linear coordinates, if av = (ai v j ), then

j

j—

— i i

av = (˜i vj ), where aj denotes the inverse matrix to aj . Moreover, if we have

a ˜

some linear actions of G on vector spaces V1 , . . . , Vn , then there is a unique linear

action of G on the tensor product V1 — · · · — Vn satisfying g(v1 — · · · — vn ) =

(gv1 ) — · · · — (gvn ) for all v1 ∈ V1 , . . . , vn ∈ Vn , g ∈ G. The latter action is called

the tensor product of the original actions.

In particular, every tensor product —r V — —q V — is considered as a GL(V )-

space with respect to the tensor product of the canonical action of GL(V ) on V

and the induced action of GL(V ) on V — .

De¬nition. A tensor B ∈ —r V — —q V — is said to be invariant, if aB = B for

all a ∈ GL(V ).

The invariance of B with respect to the homotheties in GL(V ) yields k r’q B =

B for all k ∈ R \ {0}. This implies that for r = q the only invariant tensor is the

zero tensor. An invariant tensor from —r V — —r V — will be called an invariant

tensor of degree r. For every s from the group Sr of all permutations of r

letters we de¬ne I s ∈ —r V — —r V — to be the result of the permutation s of the