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compact set. Assume that for every order r ∈ N, every neighborhood V of j0± s
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

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