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CHAPTER VI. METHODS FOR FINDING NATURAL OPERATORS

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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

m. We _rst deduce in such a way that all natural

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 _pT_ ! _p+1T_ 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 TT_ ! 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_a operator. But we apply the method of di_erential

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 2 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

(1) kbf(x1; : : : ; xn) = f(ka1x1; : : : ; kanxn)

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

di in xi satisfying the relation

(2) a1d1 + _ _ _ + andn = b:

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) 6= 0, then the limit of the righthand

side of (1) for k ! 0+ would be f(0; : : : ; 0), while the limit of the left-hand

side would be improper.

In the case b _ 0 we write a = min(a1; : : : ; an) and r =

_ b

(=the integer

part of the ratio b

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

kb @f(x1; : : : ; xn)

@xji

= kai @f(ka1x1; : : : ; kanxn)

@xji

Hence for @f

@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 identically

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

x_1

1 : : : x_n

n of degree j_ij in xi, we have

(ka1x1)_1 : : : (kanxn)_n = ka1j_1j+___+anj_njx_1

1 : : : x_n

n :

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􀀀1y) = f(x; y)

for all k 6= 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

hav_; vi = hv_; a􀀀1vi

for all v 2 V , v_ 2 V _, a 2 G. In any linear coordinates, if av = (ai

jvj ), then

av_ = (~aj

i v_

j ), where ~ai

j denotes the inverse matrix to ai

j . Moreover, if we have

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 2 V1; : : : ; vn 2 Vn, g 2 G. The latter action is called

the tensor product of the original actions.

In particular, every tensor product rV qV _ 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 2 rV qV _ is said to be invariant, if aB = B for

all a 2 GL(V ).

The invariance of B with respect to the homotheties in GL(V ) yields kr􀀀qB =

B for all k 2 Rn f0g. This implies that for r 6= q the only invariant tensor is the

zero tensor. An invariant tensor from rV rV _ 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 Is 2 rV rV _ to be the result of the permutation s of the

superscripts of

(1) Iid = idV _ _ _ | {z }

r-times

idV :

In coordinates, Is = (_

is(1)

j1 : : : _

is(r)

). The tensors Is, which are clearly invariant,

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

j1:::jr

) 2 rRm rRm_ to be invariant reads

(1) ai1

k1 : : : air

bk1:::kr

l1:::lr

= bi1:::ir

j1:::jr

aj1

l1 : : : ajr

for all ai

2 GL(m). To delete the a's, we rewrite (1) as

aj1

k1 : : : ajr

_i1

j1 : : : _ir

bk1:::kr

l1:::lr

= bi1:::ir

j1:::jr

_k1

l1 : : : _kr

aj1

k1 : : : ajr

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

j , we obtain the

following equivalent form of (1)

(2)

s2Sr

_i1

js(1) : : : _ir

js(r)b

ks(1):::ks(r)

l1:::lr

s2Sr

ks(1)

l1 : : : _

ks(r)

bi1:::ir

js(1):::js(r) :

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

(3) bk1:::kr

l1:::lr

s2Sr

cs_

ks(1)

l1 : : : _

ks(r)

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

(4)

s2Sr

_i1

js(1) : : : _ir

js(r)zs = 0

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

(5) c_i1

[j1

: : : _im+1

jm+1]_im+2

jm+2 : : : _ir

(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 z_

s , _ = 1; : : : ; q be a basis of the solutions of (4). Consider the linear

equations

(6)

s2Sr

s zs = 0 _ = 1; : : : ; q:

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

this system has the zero solution only. Let z0

s be a solution of (4) and (6). Since

s satisfy (4), there are k_ 2 R such that

(7) z0

s =

_=1

k_z_

s :

Since z0

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

_=1

􀀀X

s2Sr

s z0

= 0:

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

216 Chapter VI. Methods for _nding natural operators

By (7) the latter relation means

s2Sr

(z0

s )2 = 0, so that all z0

s vanish.

In this situation, we can formulate a lemma:

Let r! tensors Xs 2 rRm rRm_, s 2 Sr, satisfy the equations

(8)

s2Sr

_i1

js(1) : : : _ir

js(r)Xs =

s2Sr

ci1:::ir

js(1):::js(r)Is

with some real coe_cients ci1:::ir

js(1):::js(r) and

(9)

s2Sr

s Xs = 0 _ = 1; : : : ; q

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

rewritten as

(10)

s2Sr

_i1

js(1) : : : _ir

js(r)Bs =

s2Sr

bi1:::ir

js(1):::js(r)Is:

Contract the zero tensor

s2Sr

_i1

js(1) : : : _ir

js(r)z_

s , _ = 1; : : : ; q, with undetermined

xj1:::jr . This yields the algebraic relations

(11)

s2Sr

s xis(1):::is(r) = 0:

In particular, for xi1:::ir = bi1:::ir

j1:::jr

with parameters j1; : : : ; jr we obtain

(12)

s2Sr

s Bs = 0 _ = 1; : : : ; q:

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

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_e, Carrell, 71], p. 29. But is assumed

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 elementary

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 2 W V _ given by f(v) =

hf; vi for all v 2 V . The following assertion is a direct consequence of the

de_nition.

Proposition. A linear map f : p V qV _ ! rV tV _ is GL(V )-

equivariant if and only if f 2 r+qV p+tV _ is an invariant tensor.

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

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 )-equivariant

maps of rV into itself. Having such a map f : r V ! rV , the equivariance

with respect to the homotheties in GL(V ) gives krf(x) = f(krx). 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 rV ! rV are the linear combinations of the permutations

of indices.

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

the invariant tensor theorem when taking into account that the tensor symmetrization

Sym: r V ! SrV and alternation Alt: r V ! _rV as well as the

inclusions SrV ,! rV and _rV ,! rV are equivariant maps. We determine

in such a way all smooth GL(V )-equivariant maps SrV ! SrV . Consider the

diagram

SrV

Sym

SrVu

Sym

rV w

Then ' = i _ f _ Sym: r V ! rV is an equivariant map and it holds f =

Sym _ ' _ i. Using 24.7, we deduce

(1) all smooth GL(V )-maps SrV ! SrV are the constant multiples of the

identity.

Quite similarly one obtains the following simple assertions.

All smooth GL(V )-maps

(2) _rV ! _rV are the constant multiples of the identity,

(3) rV ! SrV are the constant multiples of the symmetrization,

(4) rV ! _rV are the constant multiples of the alternation,

(5) SrV ! rV and _rV ! rV 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 fi

jk(xl

mn) be the components of such a

map f. Consider _rst the homotheties 1

k _ijin 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 3Rm 3Rm_. 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.

jk =

􀀀

a1_ij

k _n

l + a2_ij

l _n

k + a3_ik

j _n

+ a4_ik

l _n

j + a5_i

l _m

j _n

k + a6_i

l _m

k _n

a1; : : : ; a6 2 R. Thus, all smooth GL(m)-maps of Rm Rm_ Rm_ into itself

form the following 6-parameter family

jk = a1_ij

kl + a2_ij

lk + a3_ik

jl + a4_ik

lj + a5xi

jk + a6xi

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_e, Carrell, 71].

Let V and W be two _nite dimensional vector spaces. A map f : V ! W is

called polynomial, if in its coordinate expression

f(xivi) = fp(xi)wp

in a basis (vi) of V and a basis (wp) of W the functions fp(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 j_j. 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 : V1 _ : : : _ Vn ! W

of a _nite product of _nite dimensional vector spaces into W. Let xi 2 Vi and

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

x_1

1 : : : x_n

is said to be of degree (j_1j; : : : ; j_nj). 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 P1f : V _V ! R is de_ned as the coe_cient by t in Taylor's formula

(1) f(x + ty) = f(x) + t P1f(x; y) + _ _ _

The coordinate expression of P1f(x; y) is @f

@xi yi. Since f is homogeneous of

degree r, Euler's theorem implies

P1f(x; x) = rf(x):

The second polarization P2f(x; y1; y2) : V _ V _ V ! R is de_ned as the _rst

polarization of P1f(x; y1) with _xed values of y1. By induction, the i-th polarization

Pif(x; y1; : : : ; yi) of f is the _rst polarization of Pi��1f(x; y1; : : : ; yi􀀀1)

with _xed values of y1; : : : ; yi􀀀1. Obviously, the r-th polarization Prf is independent

on x and is linear and symmetric in y1; : : : ; yr. The induced linear map

Pf : SrV ! R is called the total polarization of f. An iterated application of

the Euler formula gives

r! f(x) = Pf(x_ _ _ | {z }

r-times

x):

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

respect to a basis ofW. Thus, the i-th polarization of f is a map Pif : i+1

_ V ! W

and the total polarization of f is a linear map Pf : SrV ! W. Let a group G

act linearly on both V and W.

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

degree r, then every polarization Pif : i+1

_ V ! W as well as the total polarization

Pf are also equivariant.

Proof. The _rst polarization is given by formula 24.12.(1). Since f is equivariant,

we have f(gx + tgy) = gf(x + ty) for all g 2 G. Then 24.12.(1) implies

g P1f(x; y) = P1f(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 polynomial

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

P(i1;::: ;in)f :

i1+1

_ V1 _ : : : _

in+1

_ Vn ! W:

The multipolarization P(r1;::: ;rn)f induces a linear map

Pf : Sr1V1 _ _ _ SrnVn ! 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 polynomial

map, then all its multipolarizations P(i1;::: ;in)f and its total polarization

Pf 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 ! rV . Using

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

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

polarization is an equivariant map Pf : SrV ! rV . By 24.8.(5), Pf is a

constant multiple of the inclusion SrV ,! rV . Hence all smooth GL(V )-

equivariant maps V ! rV are of the form x 7! k(x _ _ _ x), k 2 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 manifold

M into themselves. In 17.7 we denoted by QP1M the connection bundle

of the _rst order frame bundle P1M of M. This is an a_ne bundle modelled on

vector bundle TM T_M T_M. The linear connections on M coincide with

the sections of QP1M. Obviously, QP1 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 QP1 QP1. Let S be the torsion

tensor of a linear connection 􀀀 2 C1(QP1M), see 16.2, let ^ S be the contracted

torsion tensor and let I be the identity tensor of TM T_M. Then S, I ^ S

and ^ S I are three sections of TM T_M T_M.

Proposition. All natural operators QP1 QP1 form the following 3-parameter

family

(1) 􀀀 + k1S + k2I ^ S + k3 ^ S I; k1; k2; k3 2 R:

Proof. In the canonical coordinates xi, xi

j on P1Rm, the equations of a principal

connection 􀀀 are

(2) dxi

j = 􀀀i

lk(x)xl

jdxk

where 􀀀i

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

m on the standard _ber F0 = (QP1Rm)0

(3) _

􀀀

jk = ail

􀀀l

mn~amj

~ank+ ai

lm~al

j~amk

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

m-equivariant maps

f : F0 ! F0. The group G2

m is a semidirect product of the kernel K of the

jet projection G2

! G1

m, the elements of which satisfy ai

j = _ij

, and of the

subgroup i(G1

m), the elements of which are characterized by ai

jk = 0. By (3),

F0 with the action of i(G1

m) coincides with Rm Rm_ Rm_ with the canonical

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

(4) fi

jk = a1_ij

kl + a2_ij

lk + a3_ik

jl + a4_ik

lj + a5xi

jk + a6xi

kj :

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

(5) ai

jk = (a1 + a2)_ij

lk + (a3 + a4)_ik

lj + (a5 + a6)ai

jk:

This is a polynomial identity in ai

jk. For m _ 2, (5) is equivalent to a1 +a2 = 0,

a3 + a4 = 0, a5 + a6 = 1. From 16.2 we _nd easily S = (􀀀i

􀀀 􀀀i

kj) =: (Si

jk), so

that I ^ S = (_ij

lk) and ^ S I = (_ik

lj ). Hence (5) implies (1). For m = 1, we

have only one quantity a1

11, so that (5) gives 1 = a1 + a2 + a3 + a4 + a5 + a6.

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

Step II. The r-th order natural operators QP1 QP1 correspond to the

Gr+2

m -equivariant maps from (JrQP1Rm)0 into F0. Denote by 􀀀s the collection

of all s-th order partial derivatives 􀀀i

jk;l1;::: ;ls

, s = 1; : : : ; r. According to 14.20,

the action of i(G1

m) _ Gr+2

m on every 􀀀s is tensorial. Using the equivariance

with respect to the homotheties in G1

m, we obtain a homogeneity condition

k f(􀀀; 􀀀1; : : : ; 􀀀r) = f(k􀀀; k2􀀀1; : : : ; kr+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 QP1 QP1

has _nite order. This completes the proof. _

25.3. Rigidity of the torsion-free connections. Let Q_P1M ! M be the

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

The symmetrization 􀀀 7! 􀀀􀀀1

2S of linear connections is a natural transformation

_ : QP1 ! Q_P1 satisfying _ _ i = idQ_P1 , where i : Q_P1 ! QP1 is the

inclusion. Hence for every natural operator A: Q_P1 Q_P1, B = i _ A _ _

is a natural operator QP1 QP1, 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_P1 Q_P1 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: _pT_

_p+1T_. The oldest result on natural operators is a theorem by Palais, who

deduced that all linear natural operators _pT_ _p+1T_ are the constant multiples

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 _pT_ _p+1T_ are the constant

multiples kd of the exterior di_erential d, k 2 R.

Proof. The canonical coordinates on _pRm_ are bi1:::ip =: b antisymmetric in all

subscripts and the action of GL(m) is

(1) _bi1:::ip = bj1:::jp~aj1

i1 : : : ~ajp

The induced coordinates on F1 = J1

0_pT_Rm are bi1:::ip;ip+1 =: b1. One evaluates

easily that the action of G2

m on F1 is given by (1) and

(2)

i1:::ip;i = bj1:::jp;j~aj1

i1 : : : ~ajp

~aj

i + bj1:::jp~aj1

i1i : : : ~ajp

_ _ _ + bj1:::jp~aj1

i1 : : : ~ajp

ipi:

The action of GL(m) on _p+1Rm_ is

(3) _ci1:::ip+1 = cj1:::jp+1~aj1

i1 : : : ~ajp+1

ip+1 :

Step I. The _rst order natural operators are in bijection with G2

m-maps

f : F1 ! _p+1Rm_. Consider _rst the equivariance of f with respect to the

homotheties in i(G1

m). This gives a homogeneity condition

(4) kp+1f(b; b1) = f(kpb; kp+1b1):

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

m) implies

(5) _ci1:::ip+1 = k b[i1:::ip;ip+1] k 2 R:

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 S2Rm_ into

_2Rm_ 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

m -map

f : Fr := Jr

0_pT_Rm ! _p+1Rm_. Denote by bs the collection of all s-th order

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

action of i(G1

m) _ Gr+1

m on every bs is tensorial. Using the equivariance with

respect to the homotheties in G1

m, we obtain

kp+1f(b; b1; : : : ; br) = f(kpb; kp+1b1; : : : ; kp+rbr):

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 _pT_

_p+1T_ 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 2 R, where ' is any smooth function of one variable. Hence all

natural operators _0T_ _1T_ are of the form g 7! '(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

Vk;l := V

k-times z }| {

_: : :_V _ V _

l-times z }| {

_: : :_V _

of k copies of a vector space V and of l copies of its dual V _. Let h ; i : V _V _ ! R

be the evaluation map hx; yi = 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(hx_; y_i):

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 2 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_ = he1; y_ie1 + _ _ _ + hem; y_iem;

f(xi; y_) = g(hxi; y_i) follows from the invariance of f. But the subset with linearly

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 mth

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

a smooth function G(zi_) : Rml ! R satisfying f(xi; y_) = G(hxi; y_i). 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

c y1; : : : ; 1

c yl)

for all 0 6= c 2 R, there exists a smooth function g : Rkl ! R such that

f(x1; : : : ; xk; y1; : : : ; yl) = g(x1y1; : : : ; xkyl). Even this is a non-trivial analytical

problem.

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224 Chapter VI. Methods for _nding natural operators

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

pV _; : : : ;qV _ 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 2 pV _; : : : ; b 2

qV _. 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

Rkp

; : : : ; zj1:::jq

2 Rkq be the canonical coordinates.

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

f : p V _ _ : : : _ qV _ _ _kV ! R there exists a smooth function

g(yi1:::ip ; : : : ; zj1:::jq ) : Rkp

_ : : : _ Rkq

! 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 multiplicative

group R+ of all positive real numbers. The formula (cx; 1

c y), c 2 R+,

(x; y) 2 R2 de_nes a linear action of R+ on R2. The rule (x; y) 7! 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

1; : : : ; xj

k) and we de_ne

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 pV _ _ : : : _ qV _ _ _mV ! 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 pV _ _ : : : _ qV _ _ _kV . Indeed, let h be a GL(V )-invariant

polynomial and Hi1:::is

A:::B 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).

Since h is GL(V )-invariant, the total polarization of each Hi1:::is

A:::B 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

(1) _a(_xi1 ; : : : ; _xip ) = a(xi1 ; : : : ; xip ); : : : ;_b(_xj1 ; : : : ; _xjq ) = b(xj1 ; : : : ; xjq )

implies

(2) f(_a; : : : ; _b; _x1; : : : ; _xk) = f(a; : : : ; b; x1; : : : ; xk):

Consider _rst the case that both m-tuples x1; : : : ; xm and _x1; : : : ; _xm are linearly

independent. Hence x_ = ci

_xi, _x_ = _ci

__xi, i = 1; : : : ;m, _ = m+1; : : : ; k. Then

the _rst collection from (1) yields, for each _ = m + 1; : : : ; k,

(3)

i=1

(ci

􀀀 _ci

_)a(xi; x1; : : : ; x1) = 0

...

i=1

(ci

􀀀 _ci

_)a(x1; : : : ; x1; xi) = 0:

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

(4) ~ f(a; : : : ; b; x1; : : : ; xm) = f(a; : : : ; b; x1; : : : ; xm; ci

_xi):

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 _kV by a product of some tensorial

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

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

respectively. The action of GL(1) is (x; y) 7! (k2x; 1

k2 y), 0 6= k 2 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 2R _ 2R_. 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 : pV __: : :_qV __

_kV _W ! R there exists a smooth function g(yi1:::ip ; : : : ; zj1:::jq ; t) : Rkp

_: : :_

Rkq

_W ! R such that

f(a; : : : ; b; x1; : : : ; xk; t) = g(a(xi1 ; : : : ; xip ); : : : ; b(xj1 ; : : : ; xjq ); t); t 2 W:

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

F(x_; y_;w) = hf(x_; y_);wi; w 2 V _:

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(hx_; y_i; hx_;wi):

Taking the partial di_erential with respect to w and setting w = 0, we obtain

f(x_; y_) =

@g(hx_; y_i; 0)

@z_

x_; _ = 1; : : : ; k:

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

_=1

g_(hx_; y_i)x_

with arbitrary smooth functions g_ : 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

_=1

g_(hx_; y_i)y_

with arbitrary smooth functions g_ : Rkl ! R.

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

operations.

26.11. Natural transformations TT_ ! T_T. Starting from some problems

in analytical mechanics, Modugno and Stefani introduced a geometrical isomorphism

between the bundles TT_M = T(T_M) and T_TM = T_(TM) for every

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

of view this is a natural equivalence between bundle functors TT_ and T_T de-

_ned on the categoryMfm. Our aim is to determine all natural transformations

TT_ ! T_T.

We _rst give a simple construction of the isomorphism sM : TT_M ! T_TM

by Modugno and Stefani. Let q : T_M ! M be the bundle projection and

_: TTM ! TTM be the canonical involution. Every A 2 TT_M is a vector tangent

to a curve (t) : R ! T_M at t = 0. If B is any vector of TTq(A)TM, then

_B is tangent to the curve _(t) : R ! TM over the curve q((t)) on M. Hence we

can evaluate h(t); _(t)i for every t and the derivative @

h(t); _(t)i =: _(A;B)

depends on A and B only. This determines a linear map TTq(A)TM ! R,

B 7! _(A;B), i.e. an element sM(A) 2 T_TM.

In general, for every vector bundle p: E ! M, the tangent map Tp: TE !

TM de_nes another vector bundle structure on TE. Even on the cotangent

bundle T_E ! E there is another vector bundle structure _: T_E ! E_ de_ned

by the restriction of a linear map TyE ! R to the vertical tangent space, which

is identi_ed with Ep(y). This enables us to introduce a sum Y u Z for every

Y 2 T_

y TM and Z 2 T_

_(y)M as follows. We have (_(Y );Z) 2 T_M _M T_M =

V T_M ,! TT_M and we can apply sM : TT_M ! T_TM. Then Y u Z is

de_ned as the sum Y +sM(_(Y );Z) with respect to the vector bundle structure

26.12. For every X 2 TT_M we write p 2 T_M for its point of contact and

_ = Tq(X) 2 TM. Taking into account both vector bundle structures on T_TM,

we denote by Y 7! (k)1Y or Y 7! (k)2Y , k 2 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 TT_ ! T_T are of the form

(1)

􀀀

F(hp; _i)

􀀀

G(hp; _i)

2sM(X) u H(hp; _i)p

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

Proof. Since TT_ and T_T are second order bundle functors on Mfm, we have

to determine all G2

m-equivariant maps of S := TT_

0 Rm into Z := T_T0Rm. The

canonical coordinates xi on Rm induce the additional coordinates pi on T_Rm

and _i = dxi, _i = dpi on TT_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

on S are

(2) _pi = ~aj

i pj ; __i = ai

j_j ; __i = ~aj

i_j 􀀀 al

jk~aml

~aj

i pm_k:

Further, if _i are the induced coordinates on TRm, then the expression _idxi +

_id_i determines the additional coordinates _i, _i on T_TRm. Similarly to (2)

we obtain the following action of G2

m on Z

(3) __i = ai

j_j ; __i = ~aj

i_j ; __i = ~aj

i _j 􀀀 al

jk~aml

~aj

i_m_k:

Any map ': S ! Z has the form

_i = fi(p; _; _); _i = gi(p; _; _); _i = hi(p; _; _):

The equivariance of fi is expressed by

(4) ai

jfj(p; _; _) = fi(~aj

i pj ; ai

j_j ; ~aj

i_j 􀀀 al

jk~aml

~aj

i pm_k):

Setting ai

j = _ij

, we obtain fi(p; _; _) = fi(p; _; _j 􀀀 al

jkpl_k). This implies that

the fi are independent of _j . Then (4) shows that fi(p; _) is a GL(m)-equivariant

map Rm _ Rm_ ! Rm. By proposition 26.9,

(5) fi = F(hp; _i)_i

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

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

yields

(6) gi = G(hp; _i)pi

where G is another smooth function of one variable.

Consider further the di_erence ki = hi 􀀀 F(hp; _i)G(hp; _i)_i. Using the fact

that hp; _i is invariant, we express the equivariance of ki in the form

~aj

i kj(p; _; _) = ki(~aj

i pj ; ai

j_j ; ~aj

i_j 􀀀 al

jk~aml

~aj

i pm_k):

Quite similarly to (4) and (6) we then deduce ki = H(hp; _i)pi, i.e.

(7) hi = F(hp; _i)G(hp; _i)_i + H(hp; _i)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 geometrically,

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 2 TT_M is tangent

to a curve (t), then fA is tangent to (ft). For every vector B 2 TfTq(A)TM,

_B is tangent to a curve _(t) : R ! TM over the curve q((ft)) on M. Then

we de_ne an element s(f;g;h)A 2 T_TM by

(1) hs(f;g;h)A;Bi = @

h(ft); g_(t)i + hh(0); _(0)i:

The coordinate expression of (1) is (fg_i+hpi)dxi+gpid_i and our construction

implies _i = f_i. This gives 26.12.(1) with constant coe_cients. Moreover, the

general case can also be interpreted in such a way. Let _ : TT_M ! T_M

be the bundle projection. Every A 2 TT_M determines Tq(A) 2 TM and

_(A) 2 T_M over the same base point in M. Then we take the values of F, G

and H at h_(A); Tq(A)i and apply the latter construction.

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

distinguished among all natural transformations TT_ ! T_T by an interesting

geometric construction explained in [Kol_a_r, Radziszewski, 88].

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 TT_, T_T and T_T_ can be

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

of a natural equivalence between TT_ and T_T_. Consider the Liouville

1-form !: TT_M ! R de_ned by !(A) = h_(A); Tq(A)i. The exterior di_erential

d! = endows T_M with a natural symplectic structure. This de_nes

a bijection between the tangent and cotangent bundles of T_M transforming

X 2 TT_M into its inner product with . Hence the natural transformations

between any two of the functors TT_, T_T and T_T_ depend on three arbitrary

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

[Kol_a_r, Radziszewski, 88].

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 TT is de_ned on the whole categoryMf and is product preserving.

Since the natural transformations of TT into itself are essentially di_erent from

the natural transformations of T_T into itself, there is no natural equivalence

between TT 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: TT_ ! T_T

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

that any two of the third order functors TTT_, TT_T, TT_T_, T_TT, T_TT_,

T_T_T and T_T_T_ are naturally equivalent, but TTT is naturally equivalent

to none of them. All natural transformations TTT_ ! TT_T for manifolds of

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

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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 pV qV _ and of GL(W) on rW sW_ de_nes the

standard action of GL(V )_GL(W) on pV qV _ rW sW_. A tensor

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

all a 2 GL(V ) _ GL(W). The invariance of B with respect to the homotheties

in GL(V ) or GL(W) gives kp􀀀qB = B or kr􀀀sB = B, respectively. This implies

that for p 6= q or r 6= s the only generalized invariant tensor is the zero tensor.

Generalized invariant tensor theorem. Every generalized invariant tensor

B 2 qV qV _rW rW_ 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 multilinearity,

B is of the form

(1) B =

s2Sr

Bs Js with Bs 2 qV qV _,

where Js are the elementary GL(W)-invariant tensors of degree r. If we construct

the total contraction of (1) with one tensor J_, _ 2 Sr, we obtain B_􀀀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 fp

ij(xq

k; yr

sl) be the coordinate expression of such

a map. The equivariance of f with respect to the homotheties 1

k _ij

in GL(V )

gives

k2fp

ij(xq

k; yr

sl) = fp

ij(kxq

k; kyr

sl):

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)

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

q in

GL(W) yield

kfp

ij(xq

k; yr

sl) = fp

ij(kxq

k; yr

sl):

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

and yr

sl. Its total polarization corresponds to a generalized invariant tensor in

2V 2V _ 2W_ 2W_. By theorem 27.1, the coordinate form of f is

ij =

􀀀

a_p

q _r

s_k

i _lj

+ b_p

s _r

q _k

i _lj

+ c_p

q _r

s_k

j _l

i + d_p

s _r

q _k

j _l

kys

rl;

a; b; c; d 2 R. Hence all smooth equivariant maps W V _ _W W_ V _ !

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

axp

i yq

qj + bxq

i yp

qj + cxp

j yq

qi + dxq

jyp

qi:

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 ! J1Y

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 _2T_BY . The geometrical de_nition of curvature implies

that C is a natural operator between two bundle functors J1 and V _2T_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).

Proposition. All natural operators J1 V 2T_B are the constant multiples

kC of the curvature operator, k 2 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

on the standard _ber S1 = J1

0 (J1(Rn+m ! Rm) ! Rn+m) of J1J1 are yp

i ,

ij = @yp

i =@xj , yp

iq = @yp

i =@yq. Evaluating the e_ect of the isomorphisms in

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

m;n on S1

_yp

i = apq

j ~aj

i + ap

j ~aj

i (1)

_yp

iq = apr

js~asq

~aj

i + apr

syr

j ~asq

~aj

i + ap

rj~arq

~aj

i (2)

_yp

ij = apq

kl~aki

~al

j + apq

kr~arj

~aki

+ ap

qlyq

k~aki

~al

j + apq

ryq

k~arj

~aki

(3)

+ apq

k~ak

ij + ap

kl~aki

~al

j + ap

kq~aq

j~aki

+ ap

k~ak

On the other hand, the standard _ber of V 2T_B is Rn 2Rm_ with

canonical coordinates zp

ij and the following action

_zp

ij = apq

kl~aki

~al

We have to determine all G2

m;n- equivariant maps S1 ! Rn 2Rm_. Let

ij = fp

ij(yq

k; yr

`s; yt

mn) be the coordinate expression of such a map. Consider the

canonical injection of GL(m)_GL(n) into G2

m;n de_ned by 2-jets of the products

of linear transformations of Rm and Rn. The equivariance with respect to the

homotheties in GL(m) gives a homogeneity condition

k2fp

ij(yq

k; yr

`s; yt

mn) = fp

ij(kyq

k; kyr

`s; k2yt

mn):

When applying the homogeneous function theorem, we have to discuss the equation

2 = d1 + d2 + 2d3. Hence fp

ij is a sum gp

ij + hp

ij where gp

ij is a linear map

of Rn Rm_ Rm_ into itself and hp

ij is a polynomial map Rn Rm_ _ Rn

Rn_ Rm_ ! Rn Rm_ Rm_. Then we see directly that both gp

ij and hp

ij are

GL(m) _ GL(n)-equivariant. For hp

ij we have deduced in example 27.2

ij = ayp

i yq

jq + byq

i yp

jq + cyp

j yq

iq + dyq

j yp

while for gp

ij a direct use of theorem 27.1 yields

ij = eyp

ij + fyp

ji:

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

232 Chapter VI. Methods for _nding natural operators

Moreover, the equivariance with respect to the subgroup K _ G2

m;n characterized

by ai

j = _ij

, apq

= _p

q leads to the relations a = 0 = c, e = 􀀀f = 􀀀b = d.

Hence fp

ij = e(yp

􀀀 yp

􀀀 yq

i yp

qj + yq

j yp

qi), which is the coordinate expression of

eC, e 2 R.

Step II. Assume we have an r-th order natural operator A: J1 V 2T_B.

It corresponds to a Gr+1

m;n-equivariant map from the standard _ber Sr of JrJ1

into RnRm_Rm_. Denote by yp

i__ the partial derivative of yp

i with respect to a

multi index _ in xi and _ in yp. Any map f : Sr ! RnRm_Rm_ is of the form

f(yp

i__), _+_ _ r. Similarly to the _rst part of the proof, GL(m)_GL(n) can

be considered as a subgroup of Gr+1

m;n. One veri_es easily that the transformation

law of yp

i__ with respect to GL(m) _ GL(n) is tensorial. Using the homotheties

in GL(m), we obtain a homogeneity condition k2f(yp

i__) = f(kj_j+1yp

i__). This

implies that f is a polynomial linear in the coordinates with j_j = 1 and bilinear

in the coordinates with j_j = 0. Using the homotheties in GL(n), we _nd

kf(yp

i__) = f(k1􀀀j_jyp

i__). This yields that f is independent of all coordinates

with j_j + j_j > 1. Hence A is a _rst order operator.

Step III. Using 23.7 we conclude that every natural operator J1 V 2T_B

has _nite order. This completes the proof. _

27.4. Curvature-like operators on pairs of connections. The Frolicher-

Nijenhuis bracket [􀀀; _] =: _(􀀀; _) of two general connections 􀀀 and _ on Y is

a section Y ! V Y _2T_BY , which may be called the mixed curvature of 􀀀

and _. Since the pair 􀀀, _ can be interpreted as a section Y ! J1Y _Y J1Y ,

_ is a natural operator _: J1 _ J1 V _2T_B between two bundle functors

on FMm;n. Let C1 : J1 _ J1 V _2T_B or C2 : J1 _ J1 V _2T_B

denote the curvature operator of the _rst or the second connection, respectively.

The following assertion can be deduced in the same way as proposition 27.3, see

[Kol_a_r, 87a].

Proposition. All natural operators J1 _ J1 V 2T_B form the following

3-parameter family

k1C1 + k2C2 + k3_; k1; k2; k3 2 R:

From a general point of view, this result enlightens us on the fact that the

mixed curvature of two general connections can be de_ned in an `essentially

unique' way, i.e. the possibility of de_ning the mixed curvature is limited by the

above 3-parameter family with trivial terms C1 and C2.

27.5. Remark. [Kurek, 91] deduced that the only natural operator J1

V _3T_B is the zero operator. This result presents an interesting point of

view to the Bianchi identity for general connections.

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

28. The orbit reduction 233

28. The orbit reduction

We are going to explain another general procedure used in the theory of natural

operators. From the computational point of view, the orbit reduction is an

almost self-evident assertion about independence of the maps in question on some

variables. This was already used e.g. for the simpli_cation of (4) in 26.12. But

the explicit formulation of such a procedure presented below is useful in several

problems. First we discuss a concrete example, in which we obtain a Utiyamalike

theorem for general connections. Then we present a complete treatment of

the `classical' reduction theorems from the theory of linear connections and from

Riemannian geometry.

28.1. Let p: G ! H be a Lie group homomorphism with kernel K, M be a Gspace,

Q be an H-space and _ : M ! Q be a p-equivariant surjective submersion,

i.e. _(gx) = p(g)_(x) for all x 2 M, g 2 G. Having p, we can consider every

H-space N as a G-space by gy = p(g)y, g 2 G, y 2 N.

Proposition. If each _􀀀1(q), q 2 Q is a K-orbit in M, then there is a bijection

between the G-maps f : M ! N and the H-maps ': Q ! N given by f = '__.

Proof. Clearly, ' _ _ is a G-map M ! N for every H-map ': Q ! N. Conversely,

let f : M ! N be a G-map. Then we de_ne ': Q ! N by '(_(x)) =

f(x). This is a correct de_nition, since _(_x) = _(x) implies _x = kx with k 2 K

by the orbit condition, so that '(_(_x)) = f(kx) = p(k)f(x) = ef(x). We have

f = ' _ _ by de_nition and ' is smooth, since _ is a surjective submersion. _

28.2. Example. We continue in our study of the standard _ber

S1 = J1

0 (J1(Rm+n ! Rm) ! Rm+n)

corresponding to the _rst order operators on general connections from 27.3. If

we replace the coordinates yp

ij by

(1) Y p

ij = yp

ij + yp

iqyq

j ;

we _nd easily that the action of G2

m;n on S1 is given by 27.3.(1), 27.3.(2) and

(2)

_ Y p

ij = apq

Y q

kl~aki

~al

j + apr

syrk

l ~aki

~al

j + ap

qlyq

k~al

i~akj

+ ap

qlyq

k~aki~al

+ apq

k~ak

ij + ap

kl~aki

~al

j + ap

k~ak

ij :

De_ne further

(3) Sp

ij =

(Y p

ij + Y p

ji); Rp

ij =

(Y p

􀀀 Y p

ji):

Since the right-hand side of (2) except the _rst term is symmetric in i and j, we

obtain the action formula for _ Sp

ij by replacing Y q

kl by Sq

kl on the right-hand side

of (2). On the other hand,

ij = apq

kl~aki

~al

j :

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

234 Chapter VI. Methods for _nding natural operators

The map : S1 ! Rn _2Rm_, (yq

k; yr

`s; yt

mn) = Rp

ij will be called the formal

curvature map.

Let Z be any (G2

_ G2

n)-space. The canonical projection G2

m;n

! G2

m and

the group homomorphism G2

m;n

! G2

n determined by the restriction of local

isomorphisms of Rm+n ! Rm to f0g _ Rn _ Rm+n de_ne a map p: G2

m;n

_ G2

n. The kernel K of p is characterized by ai

j = _ij

, ai

jk = 0, apq

= _p

q ,

apq

r = 0. The group G2

m;n acts on Rn_2Rm_ by means of the jet homomorphism

1 into G1

_G1

n. One sees directly, that the curvature map satis_es the orbit

condition with respect to K. Indeed, on K we have

(5) _yp

i = yp

i + ap

i ; _yp

iq = yp

iq + ap

iq; _ Sp

ij = Sp

ij + ap

qiyq

j + ap

qjyq

i + ap

ij :

Using ap

i , aq

jr, as

k`, we can transform every (yp

i ; yq

jr; Ss

k`) into (0; 0; 0). In this

situation, proposition 28.1 yields directly the following assertion.

Proposition. Every G2

m;n-map S1 ! Z factorizes through the formal curvature

map : S1 ! Rn _2Rm_.

28.3. The Utiyama theorem and general connections. In general, an r-th

order Lagrangian on a _bered manifold Y ! M is de_ned as a base-preserving

morphism JrY ! _mT_M, m = dimM. Roughly speaking, the Utiyama theorem

reads that every invariant _rst order Lagrangian on the connection bundle

QP ! M of an arbitrary principal _ber bundle P ! M factorizes through the

curvature map. This assertion will be formulated in a precise way in the framework

of the theory of gauge natural operators in chapter XII. At this moment we

shall apply proposition 28.2 to deduce similar results for the general connections

on an arbitrary _bered manifold Y ! M.

Since the action 28.2.(5) is simply transitive, proposition 28.2 reects exactly

the possibilities for formulating Utiyama-like theorems for general connections.

But the general interpretation of proposition 28.2 in terms of natural operators

is beyond the scope of this example and we restrict ourselves to one special case

only.

If we let the group G2

_G2

n act on a manifold S by means of the _rst product

projection, we obtain a G2

m-space, which corresponds to a second order bundle

functor F on Mfm. (In the classical Utiyama theorem we have the _rst order

bundle functor _mT_, which is allowed to be viewed as a second order functor

as well.) Obviously, F can be interpreted as a bundle functor on FMm;n, if

we compose it with the base functor B: FM ! Mf and apply the pullback

construction. If we interpret proposition 28.2 in terms of natural operators

between bundle functors on FMm;n, we obtain immediately

Proposition. There is a bijection between the _rst order natural operators

A: J1 F and the zero order natural operators A0 : V _2T_B F given by

A = A0 _ C, where C : J1 V _2T_B is the curvature operator.

28.4. The general Ricci identity. Before treating the classical tensor _elds

on manifolds, we deduce a general result for arbitrary vector bundles. Consider a

linear connection 􀀀 on a vector bundle E ! M and a classical linear connection

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

28. The orbit reduction 235

_ on M, i.e. a linear connection on TM ! M. The absolute di_erential rs of

a section s: M ! E is a section M ! E T_M. Hence we can use the tensor

product 􀀀 __ of connection 􀀀 and the dual connection __ of _, see 47.14, to

construct the absolute di_erential of rs. This is a section r2

_s: ET_MT_M

called the second absolute di_erential of s with respect to 􀀀 and _. We describe

the alternation Alt(r2

_s) : M ! E _2T_M. Let R: M ! E E_ _2T_M

be the curvature of 􀀀 and S : M ! TM _2T_M be the torsion of _. Then

the contractions hR; si and hS;rsi are sections of E _2T_M.

Proposition. It holds

(1) Alt(r2

_s) = 􀀀hR; si + hS;rsi:

Proof. This follows directly from the coordinate formula for r2

@xj

􀀀@sp

@xi

􀀀 􀀀p

qisq_

􀀀 􀀀p

􀀀@sr

@xi

􀀀 􀀀rq

isq_

+ _k

rksp: _

The coordinate form of (1) will be called the general Ricci identity of E. If

E is a vector bundle associated to P1M and 􀀀 is induced from a principal connection

on P1M, we take for _ the connection induced from the same principal

connection. In this case we write r2s only. For the classical tensor _elds on M

our proposition gives the classical Ricci identity, see e.g. [Lichnerowicz, 76, p.

69].

28.5. Curvature subspaces. We are going to describe some properties of

the absolute derivatives of curvature tensors of linear symmetric connections on

m-manifolds. Let Q = (Q_P1Rm)0 denote the standard _ber of the connection

bundle in question, see 25.3, let W = Rm Rm_ _2Rm_, Wr = W rRm_,

Wr = W _ W1 _ : : : _ Wr. The formal curvature is a map C : T1m

Q ! W,

its formal r-th absolute di_erential is Cr = rrC : Tr+1

m Q ! Wr. We write

Cr = (C;C1; : : : ;Cr) : Tr+1

m Q ! Wr, where the jet projections Tr+1

m Q ! Tsm

s < r + 1, are not indicated explicitly. (Such a slight simpli_cation of notation

will be used even later in this section.)

We de_ne the r-th order curvature equations Er on Wr as follows.

i) E0 are the _rst Bianchi identity

(1) Wi

jkl +Wik

lj +Wi

ljk = 0

ii) E1 are the absolute derivatives of (1)

(2) Wi

jklm +Wik

ljm +Wi

ljkm = 0

and the second Bianchi identity

(3) Wi

jklm +Wi

jlmk +Wi

jmkl = 0

iii) Es, s > 1, are the absolute derivatives of Es􀀀1 and the formal Ricci

identity of the product vector bundle Ws􀀀2 _Rm. By 28.4, the latter equations

are of the form

(4) Wi

jklm1___[ms􀀀1ms] = bilin(W;Ws􀀀2)

where the right-hand sides are some bilinear functions on W _Ws􀀀2.

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

236 Chapter VI. Methods for _nding natural operators

De_nition. The r-th order curvature subspace Kr _ Wr is de_ned by

E0 = 0; : : : ;Er = 0:

We write K = K0 _ W. For r = 1 we denote by K1 _ W1 the subspace

de_ned by E1 = 0. Hence K1 = K _ K1.

Lemma. Kr is a submanifold of Wr, it holds Kr = Cr(Tr+1

m Q) and the restricted

map Cr : Tr+1

m Q ! Kr is a submersion.

Proof. To prove Kr is a submanifold we proceed by induction. For r = 0 we

have a linear subspace. Assume Kr􀀀1 _ Wr􀀀1 is a submanifold. Consider the

product bundle Kr􀀀1 _Wr. Equations Er consist of the following 3 systems

(5) fjklgm1:::mr = 0

jfklm1gm2:::mr (6) = 0

jklm1___[ms􀀀1ms]___mr (7) + polyn(Wr􀀀2) = 0

where f: : : g denotes the cyclic permutation and polyn(Wr􀀀2) are some polynomials

on Wr􀀀2. The map de_ned by the left-hand sides of (5){(7) represents

an a_ne bundle morphism Kr􀀀1 _ Wr ! Kr􀀀1 _ RN of constant rank,

N = the number of equations (5){(7). Analogously to 6.6 we _nd that its kernel

Kr is a subbundle of Kr􀀀1 _Wr.

To prove Kr = Cr(Tr+1

m Q) we also proceed by induction.

Sublemma. It holds K = C(T1m

Q) and K1 = C1(T2m

Q).

Proof. The coordinate form of C is

(8) Wi

jkl = 􀀀i

jk;l

􀀀 􀀀i

jl;k + 􀀀i

ml􀀀mj

􀀀 􀀀i

mk􀀀mj

l :

This is an a_ne bundle morphism of a_ne bundle T1m

Q ! Q into W of constant

rank. We know that the values of C lie in K, so that it su_ces to prove that the

image is the whole K at one point 0 2 Q. The restricted map _ C : RmS2Rm_

Rm_ ! W is of the form

(9) Wi

jkl = 􀀀i

jk;l

􀀀 􀀀i

jl;k:

Denote by dimE0 the number of independent equations in E0, so that dimK =

dimW 􀀀 dimE0. From linear algebra we know that K is the image of _ C if

(10) dimW 􀀀 dimE0 = dimRm S2Rm_ Rm_ 􀀀 dim Ker _ C:

Clearly, dimW = m3(m􀀀1)=2 and dimRm S2Rm_ Rm_ = m3(m+1)=2. By

(9) we have Ker _ C = Rm S3Rm_, so that dim Ker _ C = m2(m + 1)(m + 2)=6.

One _nds easily that (1) represents one equation on W for any i and mutually

di_erent j, k, l, while (1) holds identically if at least two subscripts coincide.

Hence dimE0 = m2(m􀀀1)(m􀀀2)=6. Now (10) is veri_ed by simple evaluation.

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

28. The orbit reduction 237

The absolute di_erentiation of (8) yields that C1 is an a_ne morphism of

a_ne bundle T2m

Q ! T1m

Q into W1 of constant rank. We know that the values

of C1 lie in K1 so that it su_ces to prove that the image is the whole K1 at one

point 0 2 T1m

Q. The restricted map _ C1 : Rm S2Rm_ S2Rm_ ! W1 is

(11) Wi

jklm = 􀀀i

jk;lm

􀀀 􀀀i

jl;km:

Analogously to (10) we shall verify the dimension condition

(12) dimW1 􀀀 dimE1 = dimRm S2Rm_ S2Rm_ 􀀀 dim Ker _ C1:

Clearly, dimW1 = m4(m􀀀 1)=2, dimRm 2S2Rm_ = m3(m+ 1)2=4. We have

Ker _ C1 = Rm S4Rm_, so that dim Ker _ C1 = m2(m + 1)(m + 2)(m + 3)=24.

For any i and mutually di_erent j, k, l, m, (2) and (3) represent 8 equations,

but one _nds easily that only 7 of them are linearly independent. This yields

7m2(m􀀀1)(m􀀀2)(m􀀀3)=24 independent equations. If exactly two subscripts

coincide, (2) and (3) represent 2 independent equations. This yields another

m2(m􀀀1)(m􀀀2) equations. In the remaining cases (2) and (3) hold identically.

Now a direct evaluation proves our sublemma. _

Assume by induction Cr􀀀1 : Trm

Q ! Kr􀀀1 is a surjective submersion. The

iterated absolute di_erentiation of (8) yields the following coordinate form of Cr

(13) Wi

jklm1:::mr = 􀀀i

j[k;l]m1:::mr + polyn(Trm

where polyn(Trm

Q) are some polynomials on Trm

Q. This implies Cr is an a_ne

bundle morphism

Tr+1

m Q w Cr

Trm

Q w Cr􀀀1

Kr􀀀1

of constant rank. Hence it su_ces to prove at one point 0 2 Trm

Q that the

image is the whole _ber of Kr ! Kr􀀀1. The restricted map _ Cr : Rm S2Rm_

Sr+1Rm_ ! Wr is of the form

(14) Wi

jklm1:::mr = 􀀀i

jk;lm1:::mr

􀀀 􀀀i

jl;km1:::mr :

By (7) the values of _ Cr lie in W SrRm_. Then (5) and (6) characterize

(K SrRm_) \ (K1 Sr􀀀1Rm_). Consider an element X = (Xi

jklm1:::mr

) of the

latter space. Since _ C1(RmS2Rm_S2Rm_) = K1 by the sublemma, the tensor

product _ C1 idSr􀀀1Rm_ : Rm S2Rm_ S2Rm_ Sr􀀀1Rm_ ! K1 Sr􀀀1Rm_ is

a surjective map. Hence there is a Y 2 Rm S2Rm_ S2Rm_ Sr􀀀1Rm_ such

that

(15) Xi

jklm1:::mr = Y i

jklm1:::mr

􀀀 Y i

jlkm1:::mr :

Consider the symmetrization _ Y = (Y i

jkl(m1m2)___mr

) 2 Rm S2Rm_ Sr+1Rm_.

The second condition X 2 K SrRm_ implies X is symmetric in m1 and m2,

so that _ Cr( _ Y ) = X.

Finally, since Cr􀀀1 : Trm

Q ! Kr􀀀1 is a submersion and Cr : Tr+1

m Q ! Kr is

an a_ne bundle morphism surjective on each _ber, Cr is also a submersion. _

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

238 Chapter VI. Methods for _nding natural operators

28.6. Linear symmetric connections. A fundamental result on the r-th

order natural operators on linear symmetric connections with values in a _rst

order natural bundle is that they factorize through the curvature operator and

its absolute derivatives up to order r 􀀀 1. We present a formal version of this

result, which involves a precise description of the factorization.

Let F be a G1

m-space, which is considered as a Gr+2

m -space by means of the

jet homomorphism Gr+2

! G1

Theorem. For every Gr+2

m -map f : Trm

Q ! F there exists a unique G1

m-map

g : Kr􀀀1 ! F satisfying f = g _ Cr􀀀1.

Proof. We use a recurrence procedure, in the _rst step of which we apply the

orbit reduction with respect to the kernel Br+2

r+1 of the jet projection Gr+2

Gr+1

m . Let Sr : Trm

Q ! Rm Sr+2Rm_ =: S1

r+2 be the symmetrization

(1) Si

j1:::jr+2 = 􀀀i

(j1j2;j3:::jr+2)

and _r

r􀀀1 : Trm

Q ! Tr􀀀1

m Q be the jet projection. De_ne

'r = (Sr; _r

r􀀀1;Cr􀀀1) : Trm

Q ! S1

r+2

_ Tr􀀀1

m Q _Wr􀀀1:

The map Cr􀀀1 is of the form

(2) Wi

jkl1:::lr = 􀀀i

jk;l1:::lr

􀀀 􀀀i

jl1;kl2:::lr + polyn(Tr􀀀1

m Q):

One sees easily that in the formula

(3) 􀀀i

jk;l1:::lr = Si

jkl1:::lr + (􀀀i

jk;l1:::lr

􀀀 􀀀i

(jk;l1:::lr))

the expression in brackets can be rewritten as a linear combination of terms of

the form 􀀀i

mn;p1:::pr

􀀀 􀀀i

mp1;np2:::pr . If we replace each of them by Wi

mnp1:::pr

􀀀

polyn(Tr􀀀1

m Q) according to (2), we obtain a map (not uniquely determined)

r : S1

r+2

_ Tr􀀀1

m Q _Wr􀀀1 ! Trm

Q over idTr􀀀1

m Q satisfying

(4) r _ 'r = idTrm

Consider the canonical action of Abelian group Br+2

r+1 = S1

r+2 on itself, which

is simply transitive. From the transformation laws of 􀀀i

jk it follows that r is

a Br+2

r+1-map. Thus the composed map f _ r : S1

r+2

_ Tr􀀀1

m Q _ Wr􀀀1 ! F

satis_es the orbit condition for Br+2

r+1 with respect to the product projection

pr : S1

r+2

_ Tr􀀀1

m Q _ Wr􀀀1 ! Tr􀀀1

m Q _ Wr􀀀1. By 28.1 there is a Gr+1

m -map

gr : Tr􀀀1

m Q _Wr􀀀1 ! F satisfying f _ r = gr _ pr. Composing both sides with

'r, we obtain f = gr _ (_r

r􀀀1;Cr􀀀1).

In the second step we de_ne analogously

'r􀀀1 = (Sr􀀀1; _r􀀀1

r􀀀2;Cr􀀀2) : Tr􀀀1

m Q ! S1

r+1

_ Tr􀀀2

m Q _Wr􀀀2

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

28. The orbit reduction 239

and construct r􀀀1 : S1

r+1

_ Tr􀀀2

m Q _Wr􀀀2 ! Tr􀀀1

m Q satisfying r􀀀1 _ 'r􀀀1 =

idTr􀀀1

m Q. The composed map gr _ ( r􀀀1 _ idWr􀀀1 ) : S1

r+1

_ Tr􀀀2

m Q _ Wr􀀀2 _

Wr􀀀1 ! F is equivariant with respect to the kernel Br+1

r of the jet projection

Gr+1

! Gr

m. The product projection of S1

r+1

_ Tr􀀀2

m Q _ Wr􀀀2 _

Wr􀀀1 omitting the _rst factor satis_es the orbit condition for Br+1

r . This

yields a Gr

m-map gr􀀀1 : Tr􀀀2

m Q _ Wr􀀀2 _ Wr􀀀1 ! F such that gr = gr􀀀1 _ 􀀀

(_r􀀀1

r􀀀2;Cr􀀀2) _ idWr􀀀1

,i.e. f = gr􀀀1 _ (_r

r􀀀2;Cr􀀀2;Cr􀀀1).

In the last but one step we construct a G2

m-map g1 : Q_W _: : :_Wr􀀀1 ! F

such that f = g1 _(_r

0;C; : : : ;Cr􀀀1). The product projection p1 of Q_W _: : :_

Wr􀀀1 omitting the _rst factor satis_es the orbit condition for the kernel B2

1 of the

jet projection G2

! G1

m. By 28.1 there is a G1

m-map g0 : W _ : : : _Wr􀀀1 ! F

satisfying g1 = g0 _ p1. Hence f = g0 _ Cr􀀀1. Since Kr􀀀1 = Cr􀀀1(Trm

Q), the

restriction g = g0jKr􀀀1 is uniquely determined.

Trm

r􀀀1_Cr􀀀1

_ f

r+2

_ Tr􀀀1

m Q _Wr􀀀1

_______________

w pr Tr􀀀1

m Q _Wr􀀀1 w gr

_r􀀀1

r􀀀2

_Cr􀀀2_idWr􀀀1

r+1

_ Tr􀀀2

m Q _Wr􀀀2 _Wr􀀀1

AAAAAAAAAAAC

r􀀀1_idWr􀀀1

w pr􀀀1 Tr􀀀2

m Q _Wr􀀀2 _Wr􀀀1 w gr􀀀1

...

Q _Wr􀀀1

\\]

w p1 Wr􀀀1

28.7. Example. We determine all natural operators Q_P1 T_ T_. By

23.5, every such operator has a _nite order r. Let

u = f(􀀀0; 􀀀1; : : : ; 􀀀r)

􀀀s 2 Rm S2Rm_ SsRm_, be its associated map. The equivariance of f with

respect to the homotheties in G1

_ Gr+2

m yields

k2f(􀀀0; 􀀀1; : : : ; 􀀀r) = f(k􀀀0; k2􀀀1; : : : ; kr+1􀀀r):

By the homogeneous function theorem, f is a _rst order operator. According to

28.6, the _rst order operators are in bijection with G1

m-maps K ! Rm_ Rm_.

Let u = g(W) be such a map. The equivariance with respect to the homotheties

yields k2g(W) = g(k2W), so that g is linear. Consider the injection i : K !

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

240 Chapter VI. Methods for _nding natural operators

Rm 3Rm_. Since Rm 3Rm_ is a completely reducible GL(m)-module,

there is an equivariant projection p: Rm 3Rm_ ! K satisfying p _ i = idK.

Hence we can proceed analogously to 24.8. By the invariant tensor theorem, all

linear G1

m-maps Rm 3Rm_ ! Rm_ Rm_ form a 6-parameter family. Its

restriction to K gives the following 2-parameter family

k1Wk

kij + k2Wk

ikj :

Let R1 and R2 be the corresponding contractions of the curvature tensor. By

theorem 28.6, all natural operators Q_P1 T_T_ form a two parameter family

linearly generated by two contractions R1 and R2 of the curvature tensor.

28.8. Ricci subspaces. Let V = Rn be a GL(m)-module and ~ V denote

the corresponding _rst order natural vector bundle over m-manifolds. Write

Vr = V rRm_, V r = V _ V1 _ : : : _ Vr. The formal r-th order absolute

di_erentiation de_nes a map DV

r = rr : Tr􀀀1

m Q _ Trm

V ! Vr, DV

0 = idV . If vp,

i ; : : : ; vp

i1:::ir

are the jet coordinates on Trm

V (symmetric in all subscripts) and

V p

i1:::ir

are the canonical coordinates on Vr, then DV

r is of the form

(1) V p

i1:::ir

= vp

i1:::ir

+ polyn(Tr􀀀1

m Q _ Tr􀀀1

m V ):

Set DrV

= (DV

0 ;DV

1 ; : : : ;DV

r ) : Tr􀀀1

m Q _ Trm

V ! V r.

We de_ne the r-th order Ricci equations EV

r , r _ 2, as follows. For r = 2,

2 are the formal Ricci identities of ~ V (Rm). By 28.4, they are of the form

(2) V p

[ij]

􀀀 bilin(W; V ) = 0:

For r > 2, EV

r are the absolute derivatives of EV

r􀀀1 and the formal Ricci identities

of ~ V (Rm) r􀀀2T_Rm. These equations are of the form

(3) V p

i1___[is􀀀1is]___ir

􀀀 bilin(Wr􀀀2; V r􀀀2) = 0:

De_nition. The r-th order Ricci subspace ZrV

_ Kr􀀀2_V r is de_ned by EV

2 =

0; : : : ;EV

r = 0, r _ 2. For r = 0; 1 we set Z0V

= V and Z1V

= V 1.

Lemma. ZrV

is a submanifold of Kr􀀀2_V r, it holds ZrV

= (Cr􀀀2;DrV

)(Tr􀀀1

m Q_

Trm

V ) and the restricted map (Cr􀀀2;DrV

) : Tr􀀀1

m Q_Trm

V ! ZrV

is a submersion.

Proof. For r = 0 we have Z0V

= V and D0V

= idV . For r = 1, D1V

: Q _ T1m

V !

V 1 = Z1V

is of the form

V p = vp; V p

i = vp

i + bilin(Q; V )

so that our claim is trivial. Assume by induction Zr􀀀1

V is a submanifold and the

restriction of the _rst product projection of Kr􀀀3 _V r􀀀1 to Zr􀀀1

V is a surjective

submersion. Consider the _ber product Kr􀀀2_

Kr��3Zr􀀀1

V and the product vector

bundle (Kr􀀀2 _

Kr􀀀3 Zr􀀀1

V )_Vr. By (3) ZrV

is characterized by a_ne equations

of constant rank. This proves ZrV

is a subbundle and ZrV

! Kr􀀀2 is a surjective

submersion.

Assume by induction (Cr􀀀3;Dr􀀀1

V ) : Tr􀀀2

m Q _ Tr􀀀1

m V ! Zr􀀀1

V is a surjective

submersion. We have Trm

V = Tr􀀀1

m V _ V SrRm_. By (1) and (3),

(Cr􀀀2;DrV

) : (Tr􀀀1

m Q _ Tr􀀀1

m V ) _ V SrRm_ ! (ZrV

! Kr􀀀2 _

Kr􀀀3 Zr􀀀1

V ) is

bijective on each _ber. This proves our lemma. _

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

28. The orbit reduction 241

28.9. The following result is of technical character, but it covers the core of

several applications. Let F be a G1

m-space.

Proposition. For every Gr+1

m -map f : Tr􀀀1

m Q_Trm

V ! F there exists a unique

m-map g : ZrV

! F satisfying f = g _ (Cr􀀀2;DrV

Proof. First we deduce a lemma.

Lemma. If y, _y 2 Tr􀀀1

m Q satisfy Cr􀀀2(y) = Cr􀀀2(_y), then there is an element

h 2 Br+1

1 of the kernel Br+1

1 of the jet projection Gr+1

! G1

m such that h(_y) = y.

Indeed, consider the orbit set Tr􀀀1

m Q=Br+1

1 . (We shall not need a manifold

structure on it, as one checks easily that 28.1 and 28.6 work at the set-theoretical

level as well.) This is a G1

m-set under the action a(Br+1

1 y) = aBr+1

1 (y), y 2

Tr􀀀1

m Q, a 2 G1

_ Gr+1

m . Clearly, the factor projection

p: Tr􀀀1

m Q ! Tr􀀀1

m Q=Br+1

is a Gr+1

m -map. By 28.6 there is a map g : Kr􀀀2 ! Tr􀀀1

m Q=Br+1

1 satisfying

p = g _ Cr􀀀2. If Cr􀀀2(y) = Cr􀀀2(_y) = x, then p(y) = p(_y) = g(x). This proves

our lemma.

Consider the map (idTr􀀀1

m Q;DrV

) : Tr􀀀1

m Q _ Trm

V ! Tr􀀀1

m Q _ V r and denote

by ~ V r _ Tr􀀀1

m Q _ V r its image. By 28.8.(1), the restricted map DrV

: Tr􀀀1

m Q _

Trm

V ! ~ V r is bijective for every y 2 Tr􀀀1

m Q, so that DrV

is an equivariant di_eomorphism.

De_ne ~ Cr􀀀2 : ~ V r ! ZrV

, ~ Cr􀀀2(y; z) = (Cr􀀀2(y); z), y 2 Tr􀀀1

m Q,

z 2 V r. By lemma 28.5, ~ Cr􀀀2 is a surjective submersion. By de_nition,

~ Cr􀀀2(y; z) = ~ Cr􀀀2(_y; _z) means Cr􀀀2(y) = Cr􀀀2(_y) and z = _z. Thus, the above

lemma implies ~ Cr􀀀2 satis_es the orbit condition for Br+1

1 . By 28.1 there is a

m-map g : ZrV

! F satisfying f _ (DrV

)􀀀1 = g _ ~ Cr􀀀2. Composing both sides

with DrV

, we _nd f = g _ (Cr􀀀2;DrV

). _

28.10. Remark. The idea of the proof of proposition 28.9 can be applied

to suitable invariant subspaces of V as well. We shall need the case P =

RegS2Rm_ _ S2Rm_ of the standard _ber of the bundle of pseudoriemannian

metrics over m-manifolds. In this case we only have to modify the de_nition

of Pr to Pr = S2Rm_ rRm_, but the rest of 28.8 and 28.9 remains to be

unchanged. Thus, for every Gr+1

m -map f : Tr􀀀1

m Q _ Trm

P ! F there exists a

unique G1

m-map g : ZrP

! F satisfying f = g _ (Cr􀀀2;DrP

28.11. Linear symmetric connection and a general vector _eld. Let ~ F

denote the _rst order natural bundle over m-manifolds determined by G1

m-space

F. Consider an r-th order natural operator Q_P1 _ ~ V ~ F with associated

Gr+2

m -map f : Trm

Q _ Trm

V ! F. Let _ ZrV

_ Kr􀀀1 _ V r be the pre-image of

ZrV

_ Kr􀀀2 _ V r with respect to the canonical projection Kr􀀀1 ! Kr􀀀2.

Take the map r : S1

r+2

_ Tr􀀀1

m Q _ Wr􀀀1 ! Trm

Q from 28.6 and construct

r _ idTrm

V : S1

r+2

_ Tr􀀀1

m Q _ Wr􀀀1 _ Trm

V ! Trm

Q _ Trm

V . If we apply the

orbit reduction to f _ ( r _ idTrm

V ) in the previous way, we obtain a Gr+1

m -

map h: Tr􀀀1

m Q _Wr􀀀1 _ Trm

V ! F such that f = h _

􀀀

(_r

r􀀀1;Cr􀀀1) _ idTrm

Applying proposition 28.9 (with `parameters' from Wr􀀀1) to h, we obtain

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

242 Chapter VI. Methods for _nding natural operators

Proposition. For every Gr+2

m -map f : Trm

Q _ Trm

V ! F there exists a unique

m-map g : _ ZrV

! F satisfying f = g _ (Cr􀀀1;DrV

Roughly speaking, every r-th order natural operator Q_P1_ ~ V ~ F factorizes

through the curvature operator and its absolute derivatives up to order r 􀀀 1

and through the absolute derivatives on vector bundle ~ V up to order r.

28.12. Linear non-symmetric connections. An arbitrary linear connection

on TM can be uniquely decomposed into its symmetrization and its torsion

tensor. In other words, QP1M = Q_P1M _ TM _2T_M. Hence we have

the situation of 28.11, in which the role of standard _ber V is played by Rm

_2Rm_ =: H . This proves

Corollary. For every Gr+2

m -map f : Jr

0 (QP1Rm) ! F there exists a unique

m-map g : _ ZrH

! F satisfying f = g _ (Cr􀀀1;DrH

28.13. Example. We determine all natural operators QP1 T_ T_. In

the same way as in 28.7 we deduce that such operators are of the _rst order.

By 28.12 we have to _nd all G1

m-maps f : K _ H _ H1 ! Rm_ Rm_. The

equivariance with respect to the homotheties yields the homogeneity condition

k2f(W;H;H1) = f(k2W; kH; k2H1):

Hence f is linear in W and H1 and quadratic in H. The term linear in W was

determined in 28.7. By the invariant tensor theorem, the term quadratic in H

is generated by the permutations of m, n, p, q in

i _n

j _p

k_q

l Hk

mnHl

pq:

This yields the 3 di_erent double contractions Sk

ikSlj

l, Sk

ijSlk

l, Sk

ilSlj

k of the tensor

product S S of the torsion tensor with itself. Finally, the term linear in H1

corresponds to the permutations of l, m, n in

i_m

j _n

kHk

lmn:

This gives 3 generators

(1) Hk

ijk; Hk

ikj ; Hk

jki:

Thus, all natural operators QP1 ! T_T_ form an 8-parameter family linearly

generated by 2 di_erent contractions of the curvature tensor of the symmetrized

connection, by 3 di_erent double contractions of S S and by 3 operators

constructed from the covariant derivatives of the torsion tensor with respect

to the symmetrized connection according to (1).

We remark that the _rst author determined all natural operators QP1 ! T_

T_ by direct evaluation in [Kol_a_r, 87b]. Some of his generators are geometrically

di_erent of our present result, but both 8-parameter families are, of course,

linearly equivalent.

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

28. The orbit reduction 243

28.14. Pseudoriemannian metrics. Using the notation of 28.10, we deduce

a reduction theorem for natural operators on pseudoriemannian metrics. Let

_ Pr = ZrP

\ (Kr􀀀2 _ P _ f0g _ : : : _ f0g) be the subspace determined by 0 2

P1; : : : ; 0 2 Pr.

Lemma. _ Pr is a submanifold of ZrP

Proof. By [Lichnerowicz, 76, p. 69], the Ricci identity in the case of the bundle

of pseudoriemannian metrics has the form

(1) Pij[kl] +Wm

iklPmj +Wm

jklPim = 0:

Thus, for r = 2, _ P2 _ W _ P2 is characterized by the curvature equations E2,

by Pijk = 0, Pijkl = 0 and by

(2) Wm

iklPmj +Wm

jklPim = 0:

Equations (2) are G1

m-equivariant. We know that P is divided into m+1 components

P_ according to the signature _ of the metric in question. Every element

in each component can be transformed by a linear isomorphism into a canonical

form __ij . This implies that _ P2 is characterized by linear equations of constant

rank over each component P_. Assume by induction _ Pr􀀀1 _ Wr􀀀3 _ Pr􀀀1 is a

submanifold. Then _ Pr _ _ Pr􀀀1 _ f0g _Wr􀀀2 is characterized by the curvature

equations Er and by

(3) Wn

iklm1:::mr􀀀2Pnj +Wn

jklm1:::mr􀀀2Pin = 0:

By the above argument we deduce that this is a system of a_ne equations of

constant rank over each P_. _

Consider a Gr+1

m -map f : Trm

P ! F. Applying 28.10 to f _ p2 = Tr􀀀1

m Q _

Trm

P ! F, where p2 is the second product projection, we obtain a G1

m-map

h: ZrP

! F satisfying

(4) f _ p2 = h _ (Cr􀀀2;DrP

Let _r : Trm

P ! Tr􀀀1

m Q be the map determined by constructing the r-jets of the

Levi-Civit_a connection. Composing (4) with (_r; id) : TrmP ! Tr􀀀1

m Q _ Trm

we _nd

(5) f = h _ (Cr􀀀2;DrP

) _ (_r; id):

Let g be the restriction of h to _ Pr. Since the Levi-Civit_a connection is characterized

by the fact that the absolute di_erential of the metric tensor vanishes,

the values of (Cr􀀀2;DrP

) _ (_r; id) lie in _ Pr. Write Lr􀀀2 = (Cr􀀀2;DrP

) _

(_r; id) : Trm

P ! _ Pr. Then we can summarize by

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

244 Chapter VI. Methods for _nding natural operators

Proposition. For every Gr+1

m -map f : Trm

P ! F there exists a G1

m-map

g : _ Pr ! F satisfying f = g _ Lr􀀀2. _

This is the classical assertion that every r-th order natural operator on pseudoriemannian

metrics with values in an arbitrary _rst order natural bundle factorizes

through the metric itself and through the absolute derivatives of the

curvature tensor of the Levi-Civit_a connection up to order r 􀀀 2.

We remark that each component P_ of P can be treated separately in course

of the proof of the above proposition. Hence the result holds for any kind of

pseudoriemannian metrics (in particular for the proper Riemannian metrics).

28.15. Pseudoriemannian metric and a general vector _eld. A simple

modi_cation of 28.11 and 28.14 leads to a reduction theorem for the r-th order

natural operators transforming a pseudoriemannian metric and a general vector

_eld into a section of a _rst order natural bundle. In the notation from 28.11 and

28.14, let f : Trm

P _Trm

V ! F be a Gr+1

m -map. Consider the product projection

p: Tr􀀀1

m Q _ Trm

P _ Trm

V ! Trm

P _ Trm

V . Then we can apply 28.9 and 28.10 to

the product P _ V . Hence there exists a G1

m-map h: ZrP

! F satisfying

(1) f _ p = h _ (Cr􀀀2;DrP

_V ):

Denote by _ Pr

_ ZrP

P _ V _ Kr􀀀2 _ Pr _ V r the subspace determined by

0 2 P1; : : : ; 0 2 Pr. Analogously to 28.14 we deduce that _ Pr

V is a submanifold.

Write Lr􀀀2

V = (_r; idTrm

P ) _ idTrm

V : Trm

P _ Trm

V ! _ Pr

V , i.e. Lr􀀀2

V (u; v) =

(Cr􀀀2(_r(u)); u0; 0; : : : ; 0;DrV

(_r(u); v)), u 2 Trm

P, v 2 Trm

V , u0 = _r

0(u). Then

(1) implies f = h _Lr􀀀2

V . If we denote by g the restriction of h to _ Pr

V , we obtain

the following assertion.

Proposition. For every Gr+1

m -map f : Trm

P _Trm

V ! F there exists a G1

m-map

g : _ Pr

! F satisfying f = g _ Lr􀀀2

V .

Hence every r-th order natural operator transforming a pseudoriemannian

metric and a general vector _eld into a section of a _rst order natural bundle

factorizes through the metric itself, through the absolute derivatives of the curvature

tensor of the Levi-Civit_a connection up to the order r 􀀀 2 and through

the absolute derivatives with respect to the Levi-Civit_a connection of the general

vector _eld up to the order r.

28.16. Remark. Since Q_P1M ! M is an a_ne bundle, the standard _ber

Trm

Q of its r-th jet prolongation is an a_ne space by 12.17. In other words,

Gr+2

m acts on Trm

Q by a_ne isomorphisms. Consider an a_ne action of G1

of F (with the linear action as a special case). Then we can introduce the

concept of a polynomial map Trm

Q ! F analogously to 24.10. Analyzing the

proof of theorem 28.6, we observe that all the maps r and 'r are polynomial.

This implies that for every polynomial Gr+2

m equivariant map f : Trm

Q ! F,

the unique G1

m-equivariant map g : Kr􀀀1 ! F from the theorem 28.6 is the

restriction of a polynomial map _g : Wr􀀀1 ! F.

Consider further a G1

m-module V as in 28.8 or an invariant open subset of such

a module as in 28.10. Then we also have de_ned the concept of a polynomial

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

29. The method of di_erential equations 245

map of Tr􀀀1

m Q _ Trm

V into an a_ne G1

m-space F. Quite similarly to the _rst

part of this remark we deduce that for every polynomial Gr+1

m -equivariant map

f : Tr􀀀1

m Q _ Trm

V ! F the unique G1

m-equivariant map g : ZrV

! F from the

proposition 28.9 is the restriction of a polynomial map _g : Wr􀀀2 _ V r ! F.

29. The method of di_erential equations

29.1. In chapter IV we have clari_ed that the _nite order natural operators

between any two bundle functors are in a canonical bijection with the equivariant

maps between certain G-spaces. We recall that in 5.15 we deduced the following

in_nitesimal characterization of G-equivariance. Given a connected Lie group G

and two G-spaces S and Z we construct the induced fundamental vector _eld

_SA

and _Z

A on S and Z for every element A 2 g of the Lie algebra of G. Then

f : S ! Z is a G-equivariant map if and only if vector _elds _SA

and _Q

A are

f-related for every A 2 g, i.e.

(1) Tf _ _SA

= _Z

_ f for all A 2 g.

The coordinate expression of (1) is a system of partial di_erential equations

for the coordinate components of f. If we can _nd the general solution of this

system, we obtain all G-equivariant maps. This procedure is sometimes called

the method of di_erential equations.

29.2. Remark. If G is not connected and G+ denotes its connected component

of unity, then the solutions of 29.1.(1) determine all G+-equivariant maps S ! Z.

Obviously, there is an algebraic procedure how to decide which of these maps

are G-equivariant. We select one element ga in each connected component of

G and we check which solutions of 29.1.(1) are invariant with respect to all

ga. However, one usually interprets the solutions of 29.1.(1) geometrically. In

practice, if we succeed in _nding the geometrical constructions of all solutions

of 29.1.(1), it is clear that all of them determine the G-equivariant maps and we

are not obliged to discuss the individual connected components of G.

29.3. From 5.12 we have that for each left G-space S the map of the fundamental

vector _elds A 7! _SA

, A 2 g, is a Lie algebra antihomomorphism, i.e. _S

[A;B] =

􀀀[_SA

; _SB

] for all A, B 2 g, where on the left-hand side is the Lie bracket in g

and on the right-hand side we have the bracket of vector _elds. Hence if some

vectors A_, _ = 1; : : : ; q _ dimG generate g as a Lie algebra, i.e. A_ with all

their iterated brackets generate g as a vector space, then the equations

Tf _ _SA

_ = _Z

_ f _ = 1; : : : ; q

imply Tf _ _SA

= _Z

_ f for all A 2 g. In particular, for the group Gr

m the

generators of its Lie algebra are described in 13.9 and 13.10.

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

246 Chapter VI. Methods for _nding natural operators

29.4. The Levi-Civit_a connection. We are going to determine all _rst order

natural operators transforming pseudoriemannian metrics into linear connections.

We denote by RegS2T_M the bundle of all pseudoriemannian metrics

over an m-manifold M, so that the standard _ber of the corresponding natural

bundle over m-manifolds is the subset RegS2Rm_ _ S2Rm_ of all elements gij

satisfying det(gij) 6= 0. Since the zero of S2Rm_ does not lie in RegS2Rm_, the

homogeneous function theorem is of no use for our problem. (Of course, this

analytical fact is deeply reected in the geometry of pseudoriemannian manifolds.)

Hence we shall try to apply the method of di_erential equations. In the

canonical coordinates gij = gji, gij;k on the standard _ber S = J1

0 RegS2T_Rm,

the action of G2

m has the following form

_gij = gkl~aki

~al

j (1)

_gij;k = glm;n~al

i~amj

~ank

+ glm(~al

ik~amj

+ ~al

i~amj

(2) k):

Since we deal with a classical problem, we shall use the classical Christo_el's on

the standard _ber Z = (QP1Rm)0. In this case we have the following action of

(3) _􀀀i

jk = ail

􀀀l

mn~amj

~ank

+ ail

~al

see 17.15.

We shall not need all di_erential equations of our problem, since we shall

proceed in another way in the _nal step. It is su_cient to deduce the fundamental

vector _elds Si

jk on S and Zi

jk on Z corresponding to the one-parameter

subgroups ai

j = _ij

, ~ai

jk = t for j 6= k and ai

j = _ij

, ai

jj = 2t. From (1){(3) we

deduce easily

jk = 2gil

@glj;k

+ @

@glk;j

(4)

and

jk = @

@􀀀i

+ @

@􀀀i

(5)

Hence the corresponding part of the di_erential equations for a G2

m-equivariant

map 􀀀: S ! Z with components 􀀀i

jk(glm; glm;n) is

(6) 2glp

@􀀀i

@gpm;n

@􀀀i

@gpn;m

= _i

􀀀

j _n

k + _n

j _m

Multiplying by glq and replacing q by l, we _nd

(7)

@􀀀i

@glm;n

@􀀀i

@gln;m

2gil 􀀀

j _n

k + _m

k _n

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

29. The method of di_erential equations 247

Let (7') or (7") be the equations derived from (7) by the permutation (l; m; n) 7!

(m; n; l) or (l; m; n) 7! (n; l;m), respectively. Then the sum (7)+(70)􀀀(700) yields

(8)

@􀀀i

@glm;n

gil(_m

j _n

k + _m

k _n

j ) + gim(_n

j _lk

+ _n

k _lj

)

􀀀 gin(_lj

k + _lk

j )

The right-hand sides are independent on gij;k. Since we meet such a situation

frequently, it is useful to formulate a simple lemma of general character.

29.5. Lemma. Let U be an open subset in Ra with coordinates z_ and let

f(z_;w_) be a smooth function on U_Rb, (w_) 2 Rb, satisfying @f(z;w)

@w_ = g_(z).

Then

(1) f(z;w) =

_=1

g_(z)w_ + h(z)

where h(z) is a smooth function on U.

Proof. Notice that the di_erence F(z;w) = f(z;w) 􀀀

_=1 g_(z)w_ satis_es

@w_ = 0. _

Applying lemma 29.5 to 29.4.(8), we _nd

􀀀i

jk =

2gil(glj;k + glk;j 􀀀 gjk;l) + i

jk(glm):

For i

jk = 0 we obtain the coordinate expression of the Levi-Civit_a connection _,

which is natural by its standard geometric interpretation. Hence the di_erence

􀀀 􀀀 _ is a GL(m)-equivariant map RegS2Rm_ ! Rm Rm_ Rm_.

29.6. Lemma. The only GL(m)-equivariant map f : RegS2Rm_ ! RmRm_

Rm_ is the zero map.

Proof. Let Is be the matrix gii = 1 for i _ s, gjj = 􀀀1 for j > s and gij = 0 for

i 6= j. Since every g 2 RegS2Rm_ can be transformed into some Is, it su_ces to

deduce fi

jk(Is) = 0 for all i, j, k. If j 6= i 6= k or j = i = k, the equivariance with

respect to the change of orientation on the i-th axis gives fi

jk(Is) = 􀀀fi

jk(Is). If

j = i 6= k, we obtain the same result by changing the orientation on both the

i-th and k-th axes. _

Lemma 29.6 implies 􀀀 􀀀 _ = 0. This proves

29.7. Proposition. The only _rst order natural operator transforming pseudoriemannian

metrics into linear connections is the Levi-Civit_a operator.

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

248 Chapter VI. Methods for _nding natural operators

Remarks

The _rst version of our systematical approach to the problem of _nding natural

operators was published in [Kol_a_r, 87b]. In the same paper both geometric

results from section 25 are deduced. The smooth version of the tensor evaluation

theorem is _rst presented in this book. Proposition 26.12 was proved by [Kol_a_r,

Radziszewski, 88]. The generalized invariant tensor theorem was _rst used in

[Kol_a_r, 87b]. We remark that the natural equivalence s: TT_ ! T_T from 26.11

was _rst studied in [Tulczyjew, 74].

The reduction theorems for symmetric linear connections and pseudoriemannian

metrics are classical, see e.g. [Schouten, 54]. Some extensions or reformulations

of them are presented in [Lubczonok, 72] and [Krupka, 82]. The method of

di_erential equations is used systematically e.g. in the book [Krupka, Jany_ska,

90]. The complete version of proposition 29.7 was deduced in [Slov_ak, 89].

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

249

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