Lemma. For every compact set K ‚ Z there is an order r ∈ N and a neigh-

∞ ∞ ∞

borhood V ‚ E± ‚ Tn Q± of j0± s in the C r -topology such that χ± ¤ r on

V — K.

Proof. Let us apply theorem 19.7 to the translation invariant section s and a

compact set K — K ‚ C± — Z = G1 C± , where K is a compact neighborhood

of 0± ∈ Rn . We get an order r and a smooth function µ > 0 except for ¬nitely

many points y ∈ K where µ(y) = 0. Let us ¬x x in the interior of K with

µ(x) > 0. Hence there is a neighborhood V of s in the C r -topology on EC±

∞

and a neighborhood U ‚ C± of x in K such that χU (jy q, (y, z)) ¤ r whenever

∞

(y, z) ∈ U — K and q ∈ V . Now, let W be a neighborhood of j0 s in C r -topology

on E± such that tx — W is contained in the set of all in¬nite jets of sections from

∞

V . Since we might assume that tx acts on G1 C± = Rn — Z by G1 tx = tx — idZ

and we have assumed tx — s = s, the lemma follows from lemma 23.2.

Under the assumptions of 23.3 we get

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

23. The order of natural operators 207

23.4. Corollary. Let s ∈ EC± be a translation invariant section and K ‚ Z a

r

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

superscripts of

I id = idV — · · · — idV .

(1)

r-times

i i

In coordinates, I s = (δjs(1) . . . δjs(r) ). The tensors I s , which are clearly invariant,

1 r

are called the elementary invariant tensors of degree r. Obviously, if we replace

the permutation of superscripts in (1) by the permutation of subscripts, we

obtain the same collection of the elementary invariant tensors of degree r.

24.4. Invariant tensor theorem. Every invariant tensor B of degree r is a

linear combination of the elementary invariant tensors of degree r.

Proof. The condition for B = (bi1 ...ir ) ∈ —r Rm — —r Rm— to be invariant reads

j1 ...jr

ai11 . . . airr bk11...lrr = bi1 ...ir aj1 . . . ajr

...k 1 r

(1) j1 ...jr l

k kl l

for all ai ∈ GL(m). To delete the a™s, we rewrite (1) as

j

aj1 . . . ajr δj1 . . . δjr bk11...lrr = bi1 ...ir δl11 . . . δlrr aj1 . . . ajr .

i i ...k k k

j1 ...jr

k1 kr 1 rl k1 kr

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

24. Polynomial GL(V )-equivariant maps 215

Comparing the coe¬cients by the individual monomials in ai , we obtain the

j

following equivalent form of (1)

k ...ks(r) k k

i i

δl1s(1) . . . δlrs(r) bi1 ...i...js(r) .

δj1 . . . δjr bl1s(1)r

(2) = r

js(1)

...l

s(1) s(r)

s∈Sr s∈Sr

The case r ¤ m is very simple. Set cs = b1...r

s(1)...s(r) . If we put i1 = 1, . . . , ir = r,

j1 = 1, . . . , jr = r in (2), then the only non-zero term on the left-hand side

corresponds to s = id. This yields

k k

bk11...lrr =

...k

cs δl1s(1) . . . δlrs(r)

(3) l

s∈Sr

which is the coordinate form of our theorem.

For r > m we have to use a more complicated procedure (due to [Gurevich,

48]). In this case, the coe¬cients cs in (3) are not uniquely determined. This

follows from the fact that for r > m the system of m2r equations in r! variables

zs

i i

(4) δj1 . . . δjr zs = 0

s(1) s(r)

s∈Sr

has non-zero solutions. Indeed, in this case e.g. every tensor

i i

i1 i

cδ[j1 . . . δjm+1 ] δjm+2 . . . δjr

(5) m+2 r

m+1

(where the square bracket denotes alternation) is the zero tensor, since among

every j1 , . . . , jm+1 at least two indices coincide. Hence (5) expresses the zero

tensor as a non-trivial linear combination of the elementary invariant tensors.

±

Let zs , ± = 1, . . . , q be a basis of the solutions of (4). Consider the linear

equations

±

(6) zs zs = 0 ± = 1, . . . , q.

s∈Sr

To deduce that the rank of the system (4) and (6) is r!, it su¬ces to prove that

0

this system has the zero solution only. Let zs be a solution of (4) and (6). Since

0

zs satisfy (4), there are k± ∈ R such that

q

0 ±

(7) zs = k± zs .

±=1

0

Since zs satisfy (6) as well, they annihilate the linear combination

q

±0

k± zs zs = 0.

±=1 s∈Sr

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216 Chapter VI. Methods for ¬nding natural operators

By (7) the latter relation means s∈Sr (zs )2 = 0, so that all zs vanish.

0 0

In this situation, we can formulate a lemma:

Let r! tensors Xs ∈ —r Rm — —r Rm— , s ∈ Sr , satisfy the equations

i i

ci1 ...i...js(r) I s

(8) δj1 . . . δjr Xs = r

js(1)

s(1) s(r)

s∈Sr s∈Sr

with some real coe¬cients ci1 ...i...js(r) and

r

js(1)

±

(9) zs X s = 0 ± = 1, . . . , q

s∈Sr

Then every Xs is a linear combination of the elementary invariant tensors.

Indeed, since the system (4) and (6) has rank r! and the equations (6) are

linearly independent, there is a subsystem (4™) in (4) such that the system (4™)

and (6) has non-zero determinant. Let (8™) be the subsystem in (8) corresponding

to (4™). Then we can apply the Cramer rule for modules to the system (8™) and

(9). This yields that every Xs is a linear combination of the right-hand sides,

which are linearly generated by the elementary invariant tensors.

Now we can complete the proof of our theorem. Let B be an invariant tensor

and B s be the result of permutation s on its superscripts. Then (2) can be

rewritten as

i i

bi1 ...i...js(r) I s .

δj1 . . . δjr B s =

(10) r

js(1)

s(1) s(r)

s∈Sr s∈Sr

i1 ir ±

Contract the zero tensor s∈Sr δjs(1) . . . δjs(r) zs , ± = 1, . . . , q, with undeter-

mined xj1 ...jr . This yields the algebraic relations

±

(11) zs xis(1) ...is(r) = 0.

s∈Sr

In particular, for xi1 ...ir = bi1 ...ir with parameters j1 , . . . , jr we obtain

j1 ...jr

zs B s = 0

±

(12) ± = 1, . . . , q.

s∈Sr

Applying the above lemma to (10) and (12) we deduce that B is a linear com-

bination of the elementary invariant tensors.

24.5. Remark. The invariant tensor theorem follows directly from the classi¬-

cation of all relative invariants of GL(m, „¦) with p vectors in „¦m and q covectors

in „¦m— given in section 2.7 of [Dieudonn´, Carrell, 71], p. 29. But „¦ is assumed

e

to be an algebraically closed ¬eld there and the complexi¬cation procedure is

rather technical in this case. That is why we decided to present a more elemen-

tary proof, which ¬ts better to the main line of our book.

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

24. Polynomial GL(V )-equivariant maps 217

24.6. Having two vector spaces V and W , there is a canonical bijection between

the linear maps f : V ’ W and the elements f — ∈ W — V — given by f (v) =

f — , v for all v ∈ V . The following assertion is a direct consequence of the

de¬nition.

Proposition. A linear map f : —p V — —q V — ’ —r V — —t V — is GL(V )-

equivariant if and only if f — ∈ —r+q V — —p+t V — is an invariant tensor.

24.7. In several cases we can combine the use of the homogeneous function the-

orem and the invariant tensor theorem to deduce all smooth GL(V )-equivariant

maps of certain types. As an example we determine all smooth GL(V )-equivar-

iant maps of —r V into itself. Having such a map f : —r V ’ —r V , the equivari-

ance with respect to the homotheties in GL(V ) gives k r f (x) = f (k r x). Since the

only solution of rd = r is d = 1, the homogeneous function theorem implies f is

linear. Then the invariant tensor theorem and 24.6 yield that all smooth GL(V )-

equivariant maps —r V ’ —r V are the linear combinations of the permutations

of indices.

24.8. If we study the symmetric and antisymmetric tensor powers, we can ap-

ply the invariant tensor theorem when taking into account that the tensor sym-

metrization Sym : —r V ’ S r V and alternation Alt : —r V ’ Λr V as well as the

inclusions S r V ’ —r V and Λr V ’ —r V are equivariant maps. We determine

in such a way all smooth GL(V )-equivariant maps S r V ’ S r V . Consider the

diagram

w yu

uz

f

SrV SrV

ui u

Sym i Sym

w— V

•

—r V r

Then • = i —¦ f —¦ Sym : —r V ’ —r V is an equivariant map and it holds f =

Sym —¦ • —¦ i. Using 24.7, we deduce

(1) all smooth GL(V )-maps S r V ’ S r V are the constant multiples of the

identity.

Quite similarly one obtains the following simple assertions.

All smooth GL(V )-maps

(2) Λr V ’ Λr V are the constant multiples of the identity,

(3) —r V ’ S r V are the constant multiples of the symmetrization,

(4) —r V ’ Λr V are the constant multiples of the alternation,

(5) S r V ’ —r V and Λr V ’ —r V are the constant multiples of the inclusion.

24.9. In the next section we shall need all smooth GL(m)-equivariant maps

of Rm — Rm— — Rm— into itself. Let fjk (xl ) be the components of such a

i

mn

1i

map f . Consider ¬rst the homotheties k δj in GL(m). The equivariance of f

with respect to these homotheties yields kf (x) = f (kx). By the homogeneous

function theorem, f is a linear map. The corresponding tensor f — is invariant

in —3 Rm — —3 Rm— . Hence f — is a linear combination of all six permutations of

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

218 Chapter VI. Methods for ¬nding natural operators

the tensor products of the identity maps, i.e.

i imn imn imn

fjk = a1 δj δk δl + a2 δj δl δk + a3 δk δj δl

+ a4 δk δl δj + a5 δl δj δk + a6 δl δk δj xl

imn imn imn

mn

a1 , . . . , a6 ∈ R. Thus, all smooth GL(m)-maps of Rm — Rm— — Rm— into itself

form the following 6-parameter family

fjk = a1 δj xl + a2 δj xl + a3 δk xl + a4 δk xl + a5 xi + a6 xi .

i i i i i

kl lk jl lj jk kj

24.10. The invariant tensor theorem can be used for ¬nding the polynomial

equivariant maps, if we add the standard polarization technique. We present

the basic general facts according to [Dieudonn´, Carrell, 71].

e

Let V and W be two ¬nite dimensional vector spaces. A map f : V ’ W is

called polynomial, if in its coordinate expression

f (xi vi ) = f p (xi )wp

in a basis (vi ) of V and a basis (wp ) of W the functions f p (xi ) are polynomial.

One sees directly that such a de¬nition does not depend on the choice of both

bases.

We recall that for a multi index ± = (±1 , . . . , ±m ) of range m = dim V we

write

x± = (x1 )±1 . . . (xm )±m .

The degree of monomial x± is |±|. A linear combination of the monomials of the

same degree r is called a homogeneous polynomial of degree r. Every polynomial

map f : V ’ W is uniquely decomposed into the homogeneous components

f = f0 + f1 + · · · + fr .

Consider a group G acting linearly on both V and W .

Proposition. Each homogeneous component of an equivariant polynomial map

f : V ’ W is also equivariant.

Proof. This follows directly from the fact that the actions of G on both V and

W are linear.

24.11. In the same way one introduces the notion of a polynomial map

f : V 1 — . . . — Vn ’ W

of a ¬nite product of ¬nite dimensional vector spaces into W . Let xi ∈ Vi and

±i be a multi index of range mi = dim Vi , i = 1, . . . , n. A monomial

x±1 . . . x±n

n

1

is said to be of degree (|±1 |, . . . , |±n |). The multihomogeneous component

f(r1 ,... ,rn ) of degree (r1 , . . . , rn ) of a polynomial map f : V1 — . . . — Vn ’ W

consists of all monomials of this degree in f .

Having a group G acting linearly on all V1 , . . . , Vn and W , one deduces quite

similarly to 24.10

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

24. Polynomial GL(V )-equivariant maps 219

Proposition. Each multihomogeneous component of an equivariant polynomial

map f : V1 — . . . — Vn ’ W is also equivariant.

24.12. Let f : V ’ R be a homogeneous polynomial of degree r. Its ¬rst

polarization P1 f : V —V ’ R is de¬ned as the coe¬cient by t in Taylor™s formula

f (x + ty) = f (x) + t P1 f (x, y) + · · ·

(1)

‚f i

The coordinate expression of P1 f (x, y) is ‚xi y . Since f is homogeneous of

degree r, Euler™s theorem implies

P1 f (x, x) = rf (x).

The second polarization P2 f (x, y1 , y2 ) : V — V — V ’ R is de¬ned as the ¬rst

polarization of P1 f (x, y1 ) with ¬xed values of y1 . By induction, the i-th polar-

ization Pi f (x, y1 , . . . , yi ) of f is the ¬rst polarization of Pi’1 f (x, y1 , . . . , yi’1 )

with ¬xed values of y1 , . . . , yi’1 . Obviously, the r-th polarization Pr f is inde-

pendent on x and is linear and symmetric in y1 , . . . , yr . The induced linear map

P f : S r V ’ R is called the total polarization of f . An iterated application of

the Euler formula gives

r! f (x) = P f (x — · · · — x).

r-times

The concept of polarization is extended to a homogeneous polynomial map

f : V ’ W of degree r by applying this procedure to each component of f with

i+1

respect to a basis of W . Thus, the i-th polarization of f is a map Pi f : — V ’ W

and the total polarization of f is a linear map P f : S r V ’ W . Let a group G

act linearly on both V and W .

Proposition. If f : V ’ W is an equivariant homogeneous polynomial map of

i+1

degree r, then every polarization Pi f : — V ’ W as well as the total polarization

P f are also equivariant.

Proof. The ¬rst polarization is given by formula 24.12.(1). Since f is equivari-

ant, we have f (gx + tgy) = gf (x + ty) for all g ∈ G. Then 24.12.(1) implies

g P1 f (x, y) = P1 f (gx, gy). By iteration we deduce the same result for the i-th

polarization. The equivariance of the r-th polarization implies the equivariance

of the total polarization.

24.13. The same construction can be applied to a multihomogeneous polyno-

mial map f : V1 — . . . — Vn ’ W of degree (r1 , . . . , rn ). For any (i1 , . . . , in ), i1 ¤

r1 , . . . , in ¤ rn , we de¬ne the multipolarization P(i1 ,... ,in ) f of type (i1 , . . . , in ) by

constructing the corresponding polarization of f in each component separately.

Hence

i1 +1 in +1

P(i1 ,... ,in ) f : — V1 — . . . — — Vn ’ W.

The multipolarization P(r1 ,... ,rn ) f induces a linear map

P f : S r1 V1 — · · · — S rn V n ’ W

called the total polarization of f .

Given a linear action of a group on V1 , . . . , Vn , W , the following assertion is

a direct analogy of proposition 24.12.

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

220 Chapter VI. Methods for ¬nding natural operators

Proposition. If f : V1 —. . .—Vn ’ W is an equivariant multihomogeneous poly-

nomial map, then all its multipolarizations P(i1 ,... ,in ) f and its total polarization

P f are also equivariant.

24.14. Example. The simplest example for the polarization technique is the

problem of ¬nding all smooth GL(V )-equivariant maps f : V ’ —r V . Using

the homotheties in GL(V ), we obtain k r f (x) = f (kx). By the homogeneous

function theorem, f is a homogeneous polynomial map of degree r. Its total

polarization is an equivariant map P f : S r V ’ —r V . By 24.8.(5), P f is a

constant multiple of the inclusion S r V ’ —r V . Hence all smooth GL(V )-

equivariant maps V ’ —r V are of the form x ’ k(x — · · · — x), k ∈ R.

25. Natural operators on linear connections,

the exterior di¬erential

25.1. Our ¬rst geometrical application of the general methods deals with the

natural operators transforming the linear connections on an m-dimensional man-

ifold M into themselves. In 17.7 we denoted by QP 1 M the connection bundle

of the ¬rst order frame bundle P 1 M of M . This is an a¬ne bundle modelled on

vector bundle T M — T — M — T — M . The linear connections on M coincide with

the sections of QP 1 M . Obviously, QP 1 is a second order bundle functor on the

category Mfm of all m-dimensional manifolds and their local di¬eomorphisms.

25.2. We determine all natural operators QP 1 QP 1 . Let S be the torsion

ˆ

tensor of a linear connection “ ∈ C ∞ (QP 1 M ), see 16.2, let S be the contracted

ˆ

torsion tensor and let I be the identity tensor of T M — T — M . Then S, I — S

ˆ

and S — I are three sections of T M — T — M — T — M .

Proposition. All natural operators QP 1 QP 1 form the following 3-para-

meter family

ˆ ˆ

“ + k1 S + k2 I — S + k3 S — I, k1 , k2 , k3 ∈ R.

(1)

Proof. In the canonical coordinates xi , xi on P 1 Rm , the equations of a principal

j

connection “ are

dxi = “i (x)xl dxk

(2) j lk j

where “i are any smooth functions on Rm . From (2) we obtain the action of

jk

G2 on the standard ¬ber F0 = (QP 1 Rm )0

m

¯

“i = ai “l am an + ai al am

(3) l mn ˜j ˜k lm ˜j ˜k

jk

see 17.7. The proof will be performed in 3 steps, which are typical for a wider

class of naturality problems.

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

25. Natural operators on linear connections, the exterior di¬erential 221

Step I. The zero order operators correspond to the G2 -equivariant maps

m

f : F0 ’ F0 . The group G2 is a semidirect product of the kernel K of the

m

jet projection G2 ’ G1 , the elements of which satisfy ai = δj , and of the

i

m m j

subgroup i(G1 ), the elements of which are characterized by ai = 0. By (3),

m jk

F0 with the action of i(G1 ) coincides with Rm — Rm— — Rm— with the canonical

m

action of GL(m). We have deduced in 24.9 that all GL(m)-equivariant maps of

Rm — Rm— — Rm— into itself form the 6-parameter family

fjk = a1 δj xl + a2 δj xl + a3 δk xl + a4 δk xl + a5 xi + a6 xi .

i i i i i

(4) kl lk jl lj jk kj

The equivariance of (4) with respect to K then yields

ai = (a1 + a2 )δj al + (a3 + a4 )δk al + (a5 + a6 )ai .

i i

(5) jk lk lj jk

This is a polynomial identity in ai . For m ≥ 2, (5) is equivalent to a1 + a2 = 0,

jk

a3 + a4 = 0, a5 + a6 = 1. From 16.2 we ¬nd easily S = (“i ’ “i ) =: (Sjk ), so i

jk kj

ˆ ˆ

il il

that I — S = (δj Slk ) and S — I = (δk Slj ). Hence (5) implies (1). For m = 1, we

have only one quantity a1 , so that (5) gives 1 = a1 + a2 + a3 + a4 + a5 + a6 .

11

But it is easy to check this leads to the same geometrical result (1).

Step II. The r-th order natural operators QP 1 QP 1 correspond to the

Gr+2 -equivariant maps from (J r QP 1 Rm )0 into F0 . Denote by “s the collection

m

of all s-th order partial derivatives “i 1 ,... ,ls , s = 1, . . . , r. According to 14.20,

jk,l

1 r+2

the action of i(Gm ) ‚ Gm on every “s is tensorial. Using the equivariance

with respect to the homotheties in G1 , we obtain a homogeneity condition

m

k f (“, “1 , . . . , “r ) = f (k“, k 2 “1 , . . . , k r+1 “r ).

By the homogeneous function theorem, f is a polynomial of degree d0 in “ and

ds in “s such that

1 = d0 + 2d1 + · · · + (r + 1)dr .

Obviously, the only possibility is d0 = 1, d1 = · · · = dr = 0. This implies that f

is independent of “1 , . . . , “r , so that we get the case I.

Step III. In example 23.6 we deduced that every natural operator QP 1 QP 1

has ¬nite order. This completes the proof.

25.3. Rigidity of the torsion-free connections. Let Q„ P 1 M ’ M be the

bundle of all torsion-free (in other words: symmetric) linear connections on M .

1

The symmetrization “ ’ “’ 2 S of linear connections is a natural transformation

σ : QP 1 ’ Q„ P 1 satisfying σ —¦ i = idQ„ P 1 , where i : Q„ P 1 ’ QP 1 is the

inclusion. Hence for every natural operator A : Q„ P 1 Q„ P 1 , B = i —¦ A —¦ σ

is a natural operator QP 1 QP 1 , i.e. one of the list 25.2.(1). By this list,

B(“) = “ for every symmetric connection. This implies that the only natural

operator Q„ P 1 Q„ P 1 is the identity.

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

222 Chapter VI. Methods for ¬nding natural operators

25.4. The exterior di¬erential of p-forms is a natural operator d : Λp T —

Λp+1 T — . The oldest result on natural operators is a theorem by Palais, who

deduced that all linear natural operators Λp T — Λp+1 T — are the constant mul-

tiples of the exterior di¬erential only, [Palais, 59]. Using a similar procedure as

in the proof of proposition 25.2, we deduce that for p > 0 even linearity follows

from naturality.

Proposition. For p > 0, all natural operators Λp T — Λp+1 T — are the constant

multiples kd of the exterior di¬erential d, k ∈ R.

Proof. The canonical coordinates on Λp Rm— are bi1 ...ip =: b antisymmetric in all

subscripts and the action of GL(m) is

¯i ...i = bj ...j aj1 . . . ajp .

(1) b ˜ ˜ i1 ip

1 p 1 p

The induced coordinates on F1 = J0 Λp T — Rm are bi1 ...ip ,ip+1 =: b1 . One evaluates

1

easily that the action of G2 on F1 is given by (1) and

m

¯i ...i ,i = bj ...j ,j aj1 . . . ajp aj + bj ...j aj1 . . . ajp +

b1 p p ˜ i1 ˜ ip ˜ i p ˜ i1 i ˜ ip

1 1

(2) j

· · · + bj1 ...jp aj1 . . . aip i .

˜p

˜i 1

The action of GL(m) on Λp+1 Rm— is

j

ci1 ...ip+1 = cj1 ...jp+1 aj1 . . . aip+1 .

˜ p+1

˜i 1

(3) ¯

Step I. The ¬rst order natural operators are in bijection with G2 -maps

m

f : F1 ’ Λp+1 Rm— . Consider ¬rst the equivariance of f with respect to the

homotheties in i(G1 ). This gives a homogeneity condition

m

k p+1 f (b, b1 ) = f (k p b, k p+1 b1 ).

(4)

For p > 0, f must be a polynomial of degrees d0 in b and d1 in b1 such that

p + 1 = pd0 + (p + 1)d1 . For p > 1 the only possibility is d0 = 0, d1 = 1, i.e. f is

linear in b1 . By 24.8.(4), the equivariance of f with respect to the whole group

i(G1 ) implies

m

k ∈ R.

(5) ci1 ...ip+1 = k b[i1 ...ip ,ip+1 ]

¯

For p = 1, there is another possibility d0 = 2, d1 = 0. But 24.8 and the

polarization technique yield that the only smooth GL(m)-map of S 2 Rm— into

Λ2 Rm— is the zero map. Thus all ¬rst order natural operators are of the form

(5), which is the coordinate expression of kd.

Step II. Every r-th order natural operator is determined by a Gr+1 -map m

f : Fr := J0 Λp T — Rm ’ Λp+1 Rm— . Denote by bs the collection of all s-th order

r

coordinates bi1 ...ip ,j1 ...js induced on Fr , s = 1, . . . , r. According to 14.20 the

action of i(G1 ) ‚ Gr+1 on every bs is tensorial. Using the equivariance with

m m

respect to the homotheties in G1 , we obtain

m

k p+1 f (b, b1 , . . . , br ) = f (k p b, k p+1 b1 , . . . , k p+r br ).

This implies that f is independent of b2 , . . . , br . Hence the r-th order natural

operators are reduced to the case I for every r > 1.

Step III. In example 23.6 we deduced that every natural operator Λp T —

Λp+1 T — has ¬nite order.

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

26. The tensor evaluation theorem 223

25.5. Remark. For p = 0 the homogeneity condition 25.4.(4) yields f =

•(b)b1 , b, b1 ∈ R, where • is any smooth function of one variable. Hence all

natural operators Λ0 T — Λ1 T — are of the form g ’ •(g)dg with an arbitrary

smooth function • : R ’ R.

26. The tensor evaluation theorem

26.1. We ¬rst formulate an important special case. Consider the product

k-times l-times

Vk,l := V — . . . — V — V — — . . . — V —

of k copies of a vector space V and of l copies of its dual V — . Let , : V —V — ’ R

be the evaluation map x, y = y(x). The following assertion gives a very useful

description of all smooth GL(V )-invariant functions

f (x± , y» ) : Vk,l ’ R, ± = 1, . . . , k, » = 1, . . . , l.

Proposition. For every smooth GL(V )-invariant function f : Vk,l ’ R there

exists a smooth function g(z±» ) : Rkl ’ R such that

(1) f (x± , y» ) = g( x± , y» ).

We remark that this result can easily be proved in the case k ¤ m = dimV (or

l ¤ m by duality). Consider ¬rst the case k = m. Let e1 , . . . , em be a basis of

V and e1 , . . . , em be the dual basis of V — . Write Z» = z1» e1 + · · · + zk» ek ∈ V —

and de¬ne

g(z11 , . . . , zkl ) = f (e1 , . . . , ek , Z1 , . . . , Zl ).

Assume x1 , . . . , xm are linearly independent vectors. Hence there is a linear

isomorphism transforming e1 , . . . , ek into x1 , . . . , xk . Since we have

y» = e1 , y» e1 + · · · + em , y» em ,

f (xi , y» ) = g( xi , y» ) follows from the invariance of f . But the subset with lin-

early independent x1 , . . . , xm is dense in Vm,l and f and g are smooth functions,

so that the latter relation holds everywhere. In the case k < m, f : Vk,l ’ R

can be interpreted as a function Vm,l ’ R independent of (k + 1)-st up to m-

th vector components. This function is also GL(V )-invariant. Hence there is

a smooth function G(zi» ) : Rml ’ R satisfying f (xi , y» ) = G( xi , y» ). Put

g(zi» ) = G(zi» , 0). Since f is independent of xk+1 , . . . , xm , we can set xk+1 =

0, . . . , xm = 0. This implies (1).

However, in the case m < min(k, l), the function g need not to be uniquely

determined. For example, in the extreme case m = 1 our proposition asserts

that for every smooth function f (x1 , . . . , xk , y1 , . . . , yl ) of k + l scalar variables

satisfying

f (x1 , . . . , xk , y1 , . . . , yl ) = f (cx1 , . . . , cxk , 1 y1 , . . . , 1 yl )

c c

for all 0 = c ∈ R, there exists a smooth function g : Rkl ’ R such that

f (x1 , . . . , xk , y1 , . . . , yl ) = g(x1 y1 , . . . , xk yl ). Even this is a non-trivial ana-

lytical problem.

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

224 Chapter VI. Methods for ¬nding natural operators

26.2. In general, consider k copies of V and a ¬nite number of tensor products

—p V — , . . . , —q V — of V — . (Proposition 26.1 corresponds to the case p = 1, . . . , q =

1.) Write xi for the elements of the i-th copy of V and a ∈ —p V — , . . . , b ∈

—q V — . Denote by a(xi1 , . . . , xip ) or . . . or b(xj1 , . . . , xjq ) the full contraction of

a with xi1 , . . . , xip or . . . or of b with xj1 , . . . , xjq , respectively. Let yi1 ...ip ∈

p q

Rk , . . . , zj1 ...jq ∈ Rk be the canonical coordinates.

Tensor evaluation theorem. For every smooth GL(V )-invariant function

f : —p V — — . . . — —q V — — —k V ’ R there exists a smooth function

p q

g(yi1 ...ip , . . . , zj1 ...jq ) : Rk — . . . — Rk ’ R

such that

(1) f (a, . . . , b, x1 , . . . , xk ) = g(a(xi1 , . . . , xip ), . . . , b(xj1 , . . . , xjq )).

To prove this, we shall use a general result by D. Luna.

26.3. Luna™s theorem. Consider a completely reducible action of a group G

on Rn , see 13.5. Let P (Rn ) be the ring of all polynomials on Rn and P (Rn )G

be the subring of all G-invariant polynomials. By the classical Hilbert theorem,

P (Rn )G is ¬nitely generated. Consider a system p1 , . . . , ps of its generators

(called the Hilbert generators) and denote by p : Rn ’ Rs the mapping with

components p1 , . . . , ps . Luna deduced the following theorem, [Luna, 76], which

we present without proof.

Theorem. For every smooth function f : Rn ’ R which is constant on the

¬bers of p there exists a smooth function g : Rs ’ R satisfying f = g —¦ p.

We remark that in the category of sets it is trivial that constant values of f

on the pre-images of p form a necessary and su¬cient condition for the existence

of a map g such that f = g —¦ p. If some pre-images are empty, then g is not

uniquely determined. The proper meaning of the above result by Luna is that

smoothness of f implies the existence of a smooth g.

26.4. Remark. In the real analytic case [Luna, 76] deduced an essentially

stronger result: If f is a real analytic G-invariant function on Rn , then there

exists a real analytic function g de¬ned on a neighborhood of p(Rn ) ‚ Rs such

that f = g —¦ p. But the following example shows that the smooth case is really

di¬erent from the analytic one.

Example. The connected component of unity in GL(1) coincides with the mul-

tiplicative group R+ of all positive real numbers. The formula (cx, 1 y), c ∈ R+ ,

c

2 + 2

(x, y) ∈ R de¬nes a linear action of R on R . The rule (x, y) ’ sgnx is a

non-smooth R+ -invariant function on R2 . Take a smooth function •(t) of one

variable with in¬nite order zero at t = 0. Then (sgnx)•(xy) is a smooth R+ -

invariant function on R2 . Using homogeneity one ¬nds directly that the ring of

R+ -invariant polynomials on R2 is generated by xy. But (sgnx)•(xy) cannot

be expressed as a function of xy, since it changes sign when replacing (x, y) by

(’x, ’y).

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

26. The tensor evaluation theorem 225

26.5. Theorem 26.2 can easily be proved in the case k ¤ m. Assume ¬rst

k = m. Let ai1 ...ip , . . . , bj1 ...jq be the coordinates of a, . . . , b. Hence f =

f (ai1 ...ip , . . . , bj1 ...jq , xi , . . . , xj ) and we de¬ne

1 k

g(yi1 ...ip , . . . , zj1 ...jq ) = f (yi1 ...ip , . . . , zj1 ...jq , e1 , . . . , ek ).

Obviously, g is a smooth function. Then 26.2.(1) holds on the set of all linearly

independent vector k-tuples of V by invariance of f . But the latter set is dense,

so that 26.2.(1) holds everywhere by the continuity. In the case k < m we

interpret f as a function —p V — — . . . — —q V — — —m V ’ R independent of the

(k + 1)-st up to m-th vector component and we proceed in the same way as in

26.1.

26.6. In the case m < k we have to apply Luna™s theorem. First we claim that

the set of all contractions a(xi1 , . . . , xip ), . . . , b(xj1 , . . . , xjq ) form the Hilbert

generators on —p V — — . . . — —q V — — —k V . Indeed, let h be a GL(V )-invariant

i1 ...i

polynomial and HA...Bs be its component linearly generated by all monomials of

degree A in the components of a, . . . , of degree B in the components of b and with

simple entries of the components of xi1 , . . . , xis (repeated indices being allowed).

i1 ...i

Since h is GL(V )-invariant, the total polarization of each HA...Bs corresponds to

an invariant tensor. By the invariant tensor theorem, the latter tensor is a linear

combination of the elementary invariant tensors in the case Ap + · · · + Bq = s

and vanishes otherwise. But the elementary invariant tensors induce just the

contractions we mentioned in our claim.

Then we have to prove that

a(¯i1 , . . . , xip ) = a(xi1 , . . . , xip ), . . . , ¯ xj1 , . . . , xjq ) = b(xj1 , . . . , xjq )

(1) ¯x ¯ b(¯ ¯

implies

f (¯, . . . , ¯ x1 , . . . , xk ) = f (a, . . . , b, x1 , . . . , xk ).

(2) a b, ¯ ¯

Consider ¬rst the case that both m-tuples x1 , . . . , xm and x1 , . . . , xm are linearly

¯ ¯

i i

independent. Hence x» = c» xi , x» = c» xi , i = 1, . . . , m, » = m + 1, . . . , k. Then

¯ ¯¯

the ¬rst collection from (1) yields, for each » = m + 1, . . . , k,

m

(ci ’ ci )a(xi , x1 , . . . , x1 ) = 0

¯»

»

i=1

.

.

(3) .

m

(ci ’ ci )a(x1 , . . . , x1 , xi ) = 0.

¯»

»

i=1

We restrict ourselves to the subset, on which the determinant of linear system (3)

does not vanish. (This determinant does not vanish identically, as for xi = ei it

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226 Chapter VI. Methods for ¬nding natural operators

is a polynomial in the components of the tensor a, whose coe¬cient by (a1...1 )m

is 1.) Then (3) yields ci = ci . Consider now the functions

¯»

»

˜

f (a, . . . , b, x1 , . . . , xm ) = f (a, . . . , b, x1 , . . . , xm , ci xi ).

(4) »

˜

By the ¬rst part of the proof, f can be expressed in the form 26.2.(1). This

implies (2).

Thus, we have deduced that a dense subset of the solutions of (1) is formed

by the solutions of (2). Since both solution sets are closed, this completes the

proof of the tensor evaluation theorem.

26.7. Remark. We remark that there are some obstructions to obtain a general

result of such a type if we replace the product —k V by a product of some tensorial

powers of V . Consider the simpliest case of the smooth GL(1)-invariant functions

on —2 R — —2 R— . Let x or y be the canonical coordinate on —2 R or —2 R— ,

1

respectively. The action of GL(1) is (x, y) ’ (k 2 x, k2 y), 0 = k ∈ R. But this is

the situation of example 26.4, so that e.g. (sgnx)•(xy), where •(t) is a smooth

function on R with in¬nite zero at t = 0, is a smooth GL(1)-invariant function

on —2 R — —2 R— . Here the smooth case is essentially di¬erent from the analytic

one.

26.8. Tensor evaluation theorem with parameters. Analyzing the proof

of theorem 26.2, one can see that the result depends smoothly on ˜constant™

parameters in the following sense. Let W be another vector space endowed with

the identity action of GL(V ).

Theorem. For every smooth GL(V )-invariant function f : —p V — —. . .——q V — —

p

—k V —W ’ R there exists a smooth function g(yi1 ...ip , . . . , zj1 ...jq , t) : Rk —. . .—

q

Rk — W ’ R such that

t ∈ W.

f (a, . . . , b, x1 , . . . , xk , t) = g(a(xi1 , . . . , xip ), . . . , b(xj1 , . . . , xjq ), t),

The proof is left to the reader.

26.9. Smooth GL(V )-equivariant maps Vk,l ’ V . As the ¬rst application

of the tensor evaluation theorem we determine all smooth GL(V )-equivariant

maps f : Vk,l ’ V . Let us construct a function F : Vk,l — V — ’ R by

w ∈ V —.

F (x± , y» , w) = f (x± , y» ), w ,

This is a GL(V )-invariant function, so that there is a smooth function

g(z±» , z± ) : Rk(l+1) ’ R

such that

F (x± , y» , w) = g( x± , y» , x± , w ).

Taking the partial di¬erential with respect to w and setting w = 0, we obtain

‚g( x± , y» , 0)

f (x± , y» ) = xβ , β = 1, . . . , k.

‚zβ

β

This proves

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

26. The tensor evaluation theorem 227

Proposition. All GL(V )-equivariant maps Vk,l ’ V are of the form

k

gβ ( x± , y» )xβ

β=1

with arbitrary smooth functions gβ : Rkl ’ R.

If we replace vectors and covectors, we obtain

26.10. Proposition. All GL(V )-equivariant maps Vk,l ’ V — are of the form

l

gµ ( x± , y» )yµ

µ=1

with arbitrary smooth functions gµ : Rkl ’ R.

Next we present a simple application of this result in the theory of natural

operations.

26.11. Natural transformations T T — ’ T — T . Starting from some problems

in analytical mechanics, Modugno and Stefani introduced a geometrical isomor-

phism between the bundles T T — M = T (T — M ) and T — T M = T — (T M ) for every

manifold M , [Tulczyjew, 74], [Modugno, Stefani, 78]. From the categorical point

of view this is a natural equivalence between bundle functors T T — and T — T de-

¬ned on the category Mfm . Our aim is to determine all natural transformations

T T — ’ T —T .

We ¬rst give a simple construction of the isomorphism sM : T T — M ’ T — T M

by Modugno and Stefani. Let q : T — M ’ M be the bundle projection and

κ : T T M ’ T T M be the canonical involution. Every A ∈ T T — M is a vector tan-

gent to a curve γ(t) : R ’ T — M at t = 0. If B is any vector of TT q(A) T M , then

κB is tangent to the curve δ(t) : R ’ T M over the curve q(γ(t)) on M . Hence we

‚

can evaluate γ(t), δ(t) for every t and the derivative ‚t 0 γ(t), δ(t) =: σ(A, B)

depends on A and B only. This determines a linear map TT q(A) T M ’ R,

B ’ σ(A, B), i.e. an element sM (A) ∈ T — T M .

In general, for every vector bundle p : E ’ M , the tangent map T p : T E ’

T M de¬nes another vector bundle structure on T E. Even on the cotangent

bundle T — E ’ E there is another vector bundle structure ρ : T — E ’ E — de¬ned

by the restriction of a linear map Ty E ’ R to the vertical tangent space, which

is identi¬ed with Ep(y) . This enables us to introduce a sum Y Z for every

Y ∈ Ty T M and Z ∈ Tπ(y) M as follows. We have (ρ(Y ), Z) ∈ T M —M T — M =

— — —

V T — M ’ T T — M and we can apply sM : T T — M ’ T — T M . Then Y Z is

de¬ned as the sum Y + sM (ρ(Y ), Z) with respect to the vector bundle structure

ρ.

26.12. For every X ∈ T T — M we write p ∈ T — M for its point of contact and

ξ = T q(X) ∈ T M . Taking into account both vector bundle structures on T — T M ,

we denote by Y ’ (k)1 Y or Y ’ (k)2 Y , k ∈ R, the scalar multiplication with

respect to the ¬rst or second one, respectively.

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228 Chapter VI. Methods for ¬nding natural operators

Proposition. All natural transformations T T — ’ T — T are of the form

(1) F ( p, ξ ) G( p, ξ ) 2 sM (X) H( p, ξ )p

1

where F (t), G(t), H(t) are three arbitrary smooth functions of one variable.

Proof. Since T T — and T — T are second order bundle functors on Mfm , we have

to determine all G2 -equivariant maps of S := T T0 Rm into Z := T — T0 Rm . The

—

m

canonical coordinates xi on Rm induce the additional coordinates pi on T — Rm

and ξ i = dxi , πi = dpi on T T — Rm . If we evaluate the e¬ect of a di¬eomorphism

on Rm and pass to 2-jets, we ¬nd easily that the equations of the action of G2 m

on S are

pi = aj pj , πi = aj πj ’ al am aj pm ξ k .

¯

ξ i = ai ξ j ,

(2) ¯ ˜i ¯ ˜i jk ˜l ˜i

j

Further, if · i are the induced coordinates on T Rm , then the expression ρi dxi +

σi d· i determines the additional coordinates ρi , σi on T — T Rm . Similarly to (2)

we obtain the following action of G2 on Z

m

σi = aj σj , ρi = aj ρj ’ al am aj σm · k .

· i = ai · j ,

(3) ¯ ¯ ˜i ¯ ˜i jk ˜l ˜i

j

Any map • : S ’ Z has the form

· i = f i (p, ξ, π), σi = gi (p, ξ, π), ρi = hi (p, ξ, π).

The equivariance of f i is expressed by

ai f j (p, ξ, π) = f i (˜j pj , ai ξ j , aj πj ’ al am aj pm ξ k ).

(4) ai ˜i jk ˜l ˜i

j j

Setting ai = δj , we obtain f i (p, ξ, π) = f i (p, ξ, πj ’ al pl ξ k ). This implies that

i

j jk

i i

the f are independent of πj . Then (4) shows that f (p, ξ) is a GL(m)-equivariant

map Rm — Rm— ’ Rm . By proposition 26.9,

f i = F ( p, ξ )ξ i

(5)

where F is an arbitrary smooth function of one variable. Using the same pro-

cedure we obtain that the gi are independent of πj . Then proposition 26.10

yields

(6) gi = G( p, ξ )pi

where G is another smooth function of one variable.

Consider further the di¬erence ki = hi ’ F ( p, ξ )G( p, ξ )πi . Using the fact

that p, ξ is invariant, we express the equivariance of ki in the form

aj kj (p, ξ, π) = ki (˜j pj , ai ξ j , aj πj ’ al am aj pm ξ k ).

˜i ai ˜i jk ˜l ˜i

j

Quite similarly to (4) and (6) we then deduce ki = H( p, ξ )pi , i.e.

(7) hi = F ( p, ξ )G( p, ξ )πi + H( p, ξ )pi .

One veri¬es easily that (5), (6) and (7) is the coordinate form of (1).

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

26. The tensor evaluation theorem 229

26.13. To interpret all natural transformations of proposition 26.12 geometri-

cally, we ¬rst show that for any constant values F = f , G = g, H = h, 26.12.(1)

can be determined by a simple modi¬cation of the above mentioned construction

of s (s corresponds to the case f = 1, g = 1, h = 0). If A ∈ T T — M is tangent

to a curve γ(t), then f A is tangent to γ(f t). For every vector B ∈ Tf T q(A) T M ,

κB is tangent to a curve δ(t) : R ’ T M over the curve q(γ(f t)) on M . Then

we de¬ne an element s(f,g,h) A ∈ T — T M by

‚

(1) s(f,g,h) A, B = γ(f t), gδ(t) + h γ(0), δ(0) .

‚t 0

The coordinate expression of (1) is (f gπi +hpi )dxi +gpi d· i and our construction

implies · i = f ξ i . This gives 26.12.(1) with constant coe¬cients. Moreover, the

general case can also be interpreted in such a way. Let π : T T — M ’ T — M

be the bundle projection. Every A ∈ T T — M determines T q(A) ∈ T M and

π(A) ∈ T — M over the same base point in M . Then we take the values of F , G

and H at π(A), T q(A) and apply the latter construction.

We remark that the natural transformation s by Modugno and Stefani can be

distinguished among all natural transformations T T — ’ T — T by an interesting

geometric construction explained in [Kol´ˇ, Radziszewski, 88].

ar

26.14. The functor T — T — . The iterated cotangent functor T — T — is also a

second order bundle functor on Mfm . The problem of ¬nding of all natural

transformations between any two of the functors T T — , T — T and T — T — can be

reduced to proposition 26.12, if we take into account a classical geometrical con-

struction of a natural equivalence between T T — and T — T — . Consider the Liouville

1-form ω : T T — M ’ R de¬ned by ω(A) = π(A), T q(A) . The exterior di¬er-

ential dω = „¦ endows T — M with a natural symplectic structure. This de¬nes

a bijection between the tangent and cotangent bundles of T — M transforming

X ∈ T T — M into its inner product with „¦. Hence the natural transformations

between any two of the functors T T — , T — T and T — T — depend on three arbitrary

smooth functions of one variable. Their coordinate expressions can be found in

[Kol´ˇ, Radziszewski, 88].

ar

26.15. Non-existence of natural symplectic structure on the tangent

bundles. We shall see in 37.4 that the natural transformations of the iterated

tangent functor into itself depend on four real parameters. This is related with

the fact that T T is de¬ned on the whole category Mf and is product preserving.

Since the natural transformations of T T into itself are essentially di¬erent from

the natural transformations of T — T into itself, there is no natural equivalence

between T T and T — T . This implies that there is no natural symplectic structure

on the tangent bundles.

26.16. Remark. Taking into account the natural isomorphism s : T T — ’ T — T

and the canonical symplectic structure on the cotangent bundles, one sees easily

that any two of the third order functors T T T — , T T — T , T T — T — , T — T T , T — T T — ,

T — T — T and T — T — T — are naturally equivalent, but T T T is naturally equivalent

to none of them. All natural transformations T T T — ’ T T — T for manifolds of

dimension at least two are determined in [Doupovec, to appear].

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

230 Chapter VI. Methods for ¬nding natural operators

27. Generalized invariant tensors

To study the natural operators on FMm,n , we need a modi¬cation of the

Invariant tensor theorem.

27.1. Consider two vector spaces V and W . The tensor product of the standard

actions of GL(V ) on —p V — —q V — and of GL(W ) on —r W — —s W — de¬nes the

standard action of GL(V ) — GL(W ) on —p V — —q V — — —r W — —s W — . A tensor

B of the latter space is said to be a generalized invariant tensor, if aB = B for

all a ∈ GL(V ) — GL(W ). The invariance of B with respect to the homotheties

in GL(V ) or GL(W ) gives k p’q B = B or k r’s B = B, respectively. This implies

that for p = q or r = s the only generalized invariant tensor is the zero tensor.

Generalized invariant tensor theorem. Every generalized invariant tensor

B ∈ —q V — —q V — — —r W — —r W — is a linear combination of the tensor products

I — J, where I is an elementary GL(V )-invariant tensor of degree q and J is an

elementary GL(W )-invariant tensor of degree r.

Proof. Contracting B with q vectors of V and q covectors of V — , we obtain a

GL(W )-invariant tensor. By the invariant tensor theorem 24.4 and by multilin-

earity, B is of the form

with Bs ∈ —q V — —q V — ,

Bs — J s

(1) B=

s∈Sr

where J s are the elementary GL(W )-invariant tensors of degree r. If we con-

struct the total contraction of (1) with one tensor J σ , σ ∈ Sr , we obtain Bσ’1 .

Hence every Bs is a GL(V )-invariant tensor. Using theorem 24.4 once again, we

prove our assertion.

27.2. Example. We determine all smooth equivariant maps W — V — — W —

W — — V — ’ W — V — — V — . Let fij (xq , ysl ) be the coordinate expression of such

p r

k

1i

a map. The equivariance of f with respect to the homotheties k δj in GL(V )

gives

k 2 fij (xq , ysl ) = fij (kxq , kysl ).

p p

r r

k k

By the homogeneous function theorem, we have to discuss the condition 2 =

d1 + d2 . There are three possibilities: a) d1 = 2, d2 = 0, b) d1 = 1, d2 = 1, c)

p

d1 = 0, d2 = 2. In each case f is a polynomial map. The homotheties kδq in

GL(W ) yield

kfij (xq , ysl ) = fij (kxq , ysl ).

p p

r r

k k

This condition is compatible with the case b) only, so that f is bilinear in xq k

r

and ysl . Its total polarization corresponds to a generalized invariant tensor in

—2 V — —2 V — — —2 W — — —2 W — . By theorem 27.1, the coordinate form of f is

fij = aδq δs δi δj + bδs δq δi δj + cδq δs δj δi + dδs δq δj δi xq yrl ,

p prkl prkl prkl prkl s

k

a, b, c, d ∈ R. Hence all smooth equivariant maps W — V — — W — W — — V — ’

W — V — — V — form the following 4-parameter family

axp yqj + bxq yqj + cxp yqi + dxq yqi .

q p q p

i i j j

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

27. Generalized invariant tensors 231

27.3. Curvature like operators. Consider a general connection “ : Y ’ J 1 Y

on an arbitrary ¬bered manifold Y ’ BY , where B : FM ’ Mf denotes

the base functor. In 17.1 we have deduced that the curvature of “ is a map

CY “ : Y ’ V Y — Λ2 T — BY . The geometrical de¬nition of curvature implies

that C is a natural operator between two bundle functors J 1 and V — Λ2 T — B

de¬ned on the category FMm,n . In the following assertion we may replace the

second exterior power by the second tensor power (so that the antisymmetry of

the curvature operator is a consequence of its naturality).

V — —2 T — B are the constant multi-

Proposition. All natural operators J 1

ples kC of the curvature operator, k ∈ R.

Proof. We shall proceed in three steps as in the proof of proposition 25.2.

Step I. We ¬rst determine the ¬rst order operators. The canonical coordinates

p

on the standard ¬ber S1 = J0 (J 1 (Rn+m ’ Rm ) ’ Rn+m ) of J 1 J 1 are yi ,

1

p p p p

yij = ‚yi /‚xj , yiq = ‚yi /‚y q . Evaluating the e¬ect of the isomorphisms in

FMm,n and passing to 2-jets, we obtain the following action of G2 on S1

m,n

yi = ap yj aj + ap aj

¯p q

(1) ˜i j ˜i

q

yiq = ap yjs as aj + ap yj as aj + ap ar aj

¯p r r

(2) ˜q ˜i rs ˜q ˜i rj ˜q ˜i

r

yij = ap ykl ak al + ap ykr ar ak + ap yk ak al + ap yk ar ak

¯p q q q q

(3) ˜i ˜j ˜j ˜i ˜i ˜j ˜j ˜i

q q qr

ql