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.

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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.

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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

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

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 ’