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