Proof. In dimension one, we deal with the well known assertion that each even

polynomial, i.e. h(x) = h(’x), is a polynomial in x2 . However in higher dimen-

sions, the proof is quite non trivial. We present only the main ideas and refer

the reader to our source, [Atiyah, Bott, Patodi, 73, p. 323], for more details.

First notice that it su¬ces to prove the lemma for non singular matrices, for

then the assertion follows by continuity. Next, if AT A = P with P non singular

and if there is a symmetric Q, Q2 = P , then A lies in the O(m)-orbit of Q.

Indeed, Q is also non singular and B = AQ’1 satis¬es B T B = Q’1 AT AQ’1 =

Im . So it su¬ces to restrict ourselves to symmetric matrices.

Hence we want to ¬nd a polynomial map g satisfying h(Q) = g(Q2 ) for all

symmetric matrices. For every symmetric matrix P , there is the square root

√

P = Q if we extend the ¬eld of scalars to its algebraic closure. This can be

computed easily if we express √ = B T DB with an√

P orthogonal matrix B and

√

T

diagonal matrix D, since then P = B DB and D is the diagonal matrix

with the square roots of the eigen values of P on its diagonal. But we should

express Q as a universal polynomial in the elements pij of the matrix P . Let us

assume that all eigenvalues »i of P are di¬erent. Then we can write

m

P ’ »j

Q= »i .

»i ’ » j

i=1 j=i

Notice that the eigen values »i are given by rational functions of the elements

pij of P . Thus, in order to make this to a polynomial expression, we have ¬rst to

extend the ¬eld of complex numbers to the ¬eld K of rational functions (i.e. the

elements are ratios of polynomials in pij ™s). So for matrices with entries from K,

all eigen values depend polynomially on pij ™s. We also need their square roots

√

to express Q, but next we shall prove that after inserting Q = P into h(Q) all

square roots will factor out. For any ¬xed P , let us consider the splitting ¬eld

√

L over K with respect to the roots of the equation det(P ’ »2 ) = 0. So P is

polynomial over L. As a polynomial map, h extends to gl(m, L) and the next

sublemma shows that it is in fact O(m, L)-invariant.

Sublemma. Let L be any algebraic extension of R and let f : O(m, L) ’ L be

a rational function. If f vanishes on O(m, R) then f is zero.

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

33. Topics from Riemannian geometry 267

Proof. The Cayley map C : o(m, R) ’ O(m, R) is a birational isomorphism of

the orthogonal group with an a¬ne space. Hence there are ˜enough real points™

to make zero all coe¬cients of the rational map. For more details see [Atiyah,

Bott, Patodi, 73]

Now the basic fact is, that for any automorphism σ : L ’ L of the Galois

group of L over K we have (σQ)2 = σP = P and since both Q and σQ are sym-

metric, B = σQQ’1 is orthogonal. Hence we get σh(Q) = h(σQ) = h(BQ) =

h(Q). Since this holds for all σ, h(Q) lies in K and so h(Q) = g(Q2 ) for a

rational function g.

The latter equality remains true if P is a real symmetric matrix such that all

its eigen values are distinct and the denominator of g(P ) is non zero. If g = F/G

for two polynomials F and G, we get F (AT A) = h(A)G(AT A). If we choose A

so that G(AT A) = 0, we get F (AT A) = 0. Hence g is a globally de¬ned rational

function without poles and so a polynomial.

Thus, we have found a polynomial F on the space of symmetric matrices

such that h(A) = F (AT A) holds for a Zariski open set in gl(m). This proves

our lemma.

Let us continue in the proof of the Weyl™s theorem. By the lemma, every

fx satis¬es fx (A) = gx (AT A) for certain polynomial gx and so we get a poly-

nomial mapping g : S 2 Rm— — V ’ R linear in V . For all B, A ∈ GL(m, C) we

have g((B ’1 )T AT AB ’1 , Bx) = f (AB ’1 , Bx) = f (A, x) = g(AT A, x) and so

g : S 2 Rm— — V ’ R is GL(m)-invariant. Then the composition of g with the

symmetrization yields a polynomial GL(m)-invariant map —2 Rm— ——k Rm ’ R,

linear in the second entry. Each multi homogeneous component of degree s+1 in

the sense of 24.11 is also GL(m)-invariant and so its total polarization is a linear

GL(m)-invariant map H : —2s Rm— ——k Rm ’ R. Hence, by the Invariant tensor

theorem, k = 2s and H is a sum of complete contractions over possible permu-

tations of indices. Since the original mapping • is given by •(x) = g(Im , x),

Weyl™s theorem follows.

33.3. To explain the coordinate form of 33.2, it is useful to consider an ar-

bitrary metric G = (gij ) ∈ S+ Rm— . Let O(G) ‚ GL(m) be the subgroup

2

of all linear isomorphisms preserving G, so that O(m) = O(Im ). Clearly,

theorem 33.2 holds for O(G)-invariant tensors as well. Every O(G)-invariant

i1 ...i

tensor B = (Bj1 ...jp ) ∈ —p Rm — —q Rm— induces an O(G)-invariant tensor

p

k ...k

gi1 k1 . . . gip kp Bj11...jqp ∈ —p+q Rm— . Hence theorem 33.2 implies that all O(G)-

invariant tensors in —p Rm — —q Rm— with p + q even are linearly generated by

g i1 k1 . . . g ip kp gσ(k1 )σ(k2 ) . . . gσ(jq’1 )σ(jq )

where g ik gjk = δj , g ij = g ji , for all permutations σ of p + q letters.

i

Consequently, all O(G)-equivariant tensor operations are generated by: ten-

˜

sorizing by the metric tensor G : Rm ’ Rm— or by its inverse G : Rm— ’ Rm ,

applying contractions and permutations of indices, and taking linear combina-

tions.

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

268 Chapter VII. Further applications

33.4. Our next main goal is to prove the famous Gilkey theorem on natural

exterior forms on Riemannian metrics, i.e. to determine all natural operators

S+ T — Λp T — . This will be based on 33.2 and on the reduction theorems from

2

section 28. But since the resulting forms come from the Levi-Civit` connection

a

via the Chern-Weil construction, we ¬rst determine all natural operators trans-

forming linear symmetric connections into exterior forms. This will help us to

describe easily the metric operators later on.

Let us start with a description of natural tensors depending on symmetric

linear connections, i.e. natural operators Q„ P 1 T (p,q) , where T (p,q) Rm =

Rm — —p Rm — —q Rm— . Each covariant derivative of the curvature R(“) ∈

C ∞ (T M — T — M — Λ2 T — M ) of the connection “ on M is natural. Further every

tensor multiplication of two natural tensors and every contraction on one covari-

ant and one contravariant entry of a natural tensor give new natural tensors.

Finally we can tensorize any natural tensor with a GL(m)-invariant tensor, we

can permute any number of entries in the tensor products and we can repeat

each of these steps and take linear combinations.

Lemma. All natural operators Q„ P 1 T (p,q) are obtained by this procedure.

In particular, there are no non zero operators if q ’ p = 1 or q ’ p < 0.

Proof. By 23.5, every such operator has some ¬nite order r and so it is deter-

mined by a smooth Gr+2 -equivariant map f : Tm Q ’ V , where Q is the standard

r

m

¬ber of the connection bundle and V = —p Rm ——q Rm— . By the proof of the the-

orem 28.6, there is a G1 -equivariant map g : W r’1 ’ V such that f = g —¦ C r’1 .

m

Here W r’1 = W — . . . — Wr’1 , W = Rm — Rm— — Λ2 Rm— , Wi = W — —i Rm— ,

i = 1, . . . , r ’ 1. Therefore the coordinate expression of a natural tensor is given

by smooth maps

i ...i i i

ωj1 ...jp (Wjkl , . . . , Wjklm1 ...mr’1 ).

1 q

Hence we can apply the Homogeneous function theorem (see 24.1). The action

of the homotheties c’1 δj ∈ G1 gives

i

m

i ...i i ...i

cq’p ωj1 ...jp (Wjkl , . . . , Wjklm1 ...mr’1 ) = ωj1 ...jp (c2 Wjkl , . . . , cr+1 Wjklm1 ...mr’1 ).

i i i i

1 q 1 q

Hence the ω™s must be sums of homogeneous polynomials of degrees ds in the

i

variables Wjklm1 ...ms satisfying

2d0 + · · · + (r + 1)dr’1 = q ’ p.

(1)

Now we can consider the total polarization of each multi homogeneous compo-

nent and we obtain linear mappings

S d0 W — · · · — S dr’1 Wr’1 ’ V.

According to the Invariant tensor theorem, all the polynomials in question are

linearly generated by monomials obtained by multiplying an appropriate number

i

of variables Wjkl± and applying some of the GL(m)-equivariant operations.

If q = p, then the polynomials would be of degree zero, and so only the

GL(m)-invariant tensors can appear. If q ’ p = 1 or q ’ p < 0, there are no non

negative integers solving (1).

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

33. Topics from Riemannian geometry 269

33.5. Natural forms depending on linear connections. To determine

Λq T — , we have to consider the case p = 0 and

the natural operators Q„ P 1

apply the alternation to the subscripts. It is well known that the Chern-Weil

construction associates a natural form to every polynomial P which is de¬ned

on Rm — Rm— and invariant under the action of GL(m). This natural form is

obtained by substitution of the entries of the matrix valued curvature 2-form

R for the variables and taking the wedge product for multiplication. So if P

is homogeneous of degree j, then P (R) is a natural 2j-form. Let us denote by

ωq the form obtained from the tensor product of q copies of the curvature R

by taking its trace and alternating over the remaining entries. In coordinates,

kq kq’1

k1

ωq = (Rk1 ab Rk2 cd . . . Rkq ef ), where we alternate over all indices a, . . . , f . One

¬nds easily that the polynomials Pq depending on the entries of the matrix 2-

form R correspond to the homogeneous components of degree q in det(Im + R)

and so the forms ωq equal the Chern forms cq up to the constant factor (i/(2π))q .

The wedge product on the linear space of all natural forms depending on

connections de¬nes the structure of a graded algebra.

•m Λp T — is gener-

Theorem. The algebra of all natural operators Q„ P 1 p=0

ated by the Chern forms cq .

In particular, there are no natural forms with odd degrees and consequently

all natural forms are closed.

Proof. We have to continue our discussion from the proof of the lemma 33.4.

i

However, we need some relations on the absolute derivatives Rjklm1 ...ms of the

curvature tensor. First recall the antisymmetry, the ¬rst and the second Bianchi

identity, cf. 28.5

i i

Rjkl = ’Rjlk

(1)

i i i

(2) Rjkl + Rklj + Rljk = 0

i i i

(3) Rjklm + Rjlmk + Rjmkl = 0

i

Lemma. The alternation of Rjklm1 ...ms over any 3 indices among the ¬rst four

subscripts is zero.

Proof. Since the covariant derivative commutes with the tensor operations like

i i

the permutation of indices, it su¬ces to discuss the variables Rjkl and Rjklm .

i

By (2), the alternation over the subscripts in Rjkl is zero and (3) yields the same

i

for the alternation over k, l, m in Rjklm . In view of (1), it remains to discuss

i i i

the alternation of Rjklm over j, l, m. (1) implies Rjkml = ’Rjmkl and so we

can rewrite this alternation as follows

i i i i

Rjklm + Rjmkl + Rjlmk ’ Rjlmk

i i i i

+Rmkjl + Rmlkj + Rmjlk ’ Rmjlk

i i i i

+Rlkmj + Rljkm + Rlmjk ’ Rlmjk .

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

270 Chapter VII. Further applications

The ¬rst three entries on each row form a cyclic permutation and hence give

zero. The same applies to the last column.

Now it is easy to complete the proof of the theorem. Consider ¬rst a monomial

i

containing at least one quantity Rjklm1 ...ms with s > 0. Then there exists

one term of the product with three free subscripts among the ¬rst four ones

i

or one term Rjkl with all free subscripts, so that the monomial vanishes after

i i i

alternation. Further, (1) and (2) imply Rjkl ’ Rlkj = ’Rklj . Hence we can

restrict ourselves to contractions with the ¬rst subscripts and so all the possible

kq kq’1

k1

natural forms are generated by the expressions Rk1 ab Rk2 cd . . . Rkq ef where the

indices a, . . . , f remain free for alternation. But these are coordinate expressions

of the forms ωq .

33.6. Characteristic classes. The dimension of the homogeneous component

of the algebra of natural forms of degree 2s equals the number π(s) of the parti-

tions of s into sums of positive integers. Since all natural forms are closed, they

determine cohomology classes in the De Rham cohomologies of the underlying

manifolds. It is well known from the Chern-Weil theory that these classes do

not depend on the connection. This can be deduced as follows.

¯ ¯

Consider two linear connections “, “ expressed locally by “i , “i ∈ (T — M —

j j

i ¯i

— — 2—

T M )—T M , and their curvatures Rj , Rj ∈ (T M —T M )—Λ T M . Write “t =

¯

t“ + (1 ’ t)“ and analogously Rt for the curvatures. Let Pq be the polynomial

¯

de¬ning the form ωq and Q be its total polarization. We de¬ne „q (“, “) =

1 ¯ d d

q 0 Q(“ ’ “, Rt , . . . , Rt )dt. The structure equation yields dt Rt = dt (d“t ) ’

¯

d d

dt “t § “t ’ “t § dt “t = d(“ ’ “) and we calculate easily in normal coordinates

1 1

d d

¯

ωq (“) ’ ωq (“) = Q(Rt , . . . , Rt )dt = q Q( Rt , Rt , . . . , Rt )dt

dt dt

0 0

1

¯ ¯

dQ(“ ’ “, Rt , . . . , Rt )dt = d„q (“, “).

=q

0

Λ2q’1 T — and the

In fact, „q is one of many natural operators Q„ P 1 — Q„ P 1

integration helps us to ¬nd the proper linear combination of more elementary

operators which are obtained by a procedure similar to that from 33.4“33.5. The

¯

form „q (“, “) is called the transgression.

33.7. Natural forms on Riemannian manifolds. Since there is the natural

Levi-Civit` connection, we can evaluate the natural forms from 33.5 using the

a

curvature of this connection. In this case 28.14.(3) holds, i.e.

n n

gin Wjklm1 ...mr = ’gjn Wiklm1 ...mr .

(1)

For gij = δij , r = 0, this implies

j

i

Rjkl = ’Rikl

(2)

k k

k1

q q’1

and so the contractions in a monomial Rk1 ab Rk2 cd . . . Rkq ef yield zero if q is

odd. The natural forms pj = (2π)’2j ω2j are called the Pontryagin forms. The

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

33. Topics from Riemannian geometry 271

dimension of the homogeneous component of degree 4s of the algebra of forms

generated by the Pontryagin forms is π(s), cf. 33.6.

If we assume the dependence of the natural operators on the metric, then

every two indices of any tensor can be contracted. In particular, the complete

contractions of covariant derivatives of the curvature of the Levi-Civit` connec-

a

tion give rise to natural functions of all even orders grater then one. Composing

k natural functions with any ¬xed smooth function Rk ’ R, we get a new natu-

ral function. Since every natural form can be multiplied by any natural function

without loosing naturality, we see that there is no hope to describe all natural

forms in a way similar to 33.5. However, in Riemannian geometry we often meet

operations with a sort of homogeneity with respect to the change of the scale of

the metric and these can be described in more details.

Our operators will have several arguments as a rule and we shall use the

following brief notation in this section: Given several natural bundles Fa , . . . , Fb ,

we write Fa — . . . — Fb for the natural bundle associating to each m-manifold M

the ¬bered product Fa M —M . . .—M Fb M and similarly on morphisms. (Actually,

this is the product in the category of functors, cf. 14.11.) Hence D : F1 —F2 G

∞

means a natural operator transforming couples of sections from C (F1 M ) and

C ∞ (F2 M ) to sections from C ∞ (GM ) (which is also denoted by D : F1 •F2 G

in this book). Analogously, given natural operators D1 : F1 G1 and D2 : F2

G2 , we use the symbol D1 — D2 : F1 — F2 G1 — G2 .

De¬nition. Let E and F be natural bundles over m-manifolds. We say that a

natural operator D : S+ T — — E

2

F is conformal, if D(c2 g, s) = D(g, s) for all

metrics g, sections s, and all positive c ∈ R. If F is a natural vector bundle and

D satis¬es D(c2 g) = c» D(g), then » is called the weight of D.

Let us notice that the weight of the metric gij is 2 (we consider the inclusion

g : S+ T — S 2 T — ), that of its inverse g ij is ’2, while the curvature and all its

2

covariant derivatives are conformal.

33.8. Gilkey theorem. There are no non zero natural forms on Riemannian

manifolds with a positive weight. The algebra of all conformal natural forms on

Riemannian manifolds is generated by the Pontryagin forms.

33.9. Let us start the proof with a discussion on the reduction procedure de-

veloped in section 28. Even if we have no estimate on the order, we can get

an analogous result. Consider an arbitrary natural operator Q„ P 1 — E F.

By the non-linear Peetre theorem, D is of order in¬nity and so it is determined

by the restriction D of its associated mapping J ∞ ((Q„ P 1 — E)Rm ) ’ F Rm

to the ¬ber over the origin. Moreover, we obtain an open ¬ltration of the

whole ¬ber J0 ((Q„ P 1 — E)Rm ) consisting of maximal G∞ -invariant open sub-

∞

m

∞

sets Uk where the associated mapping D factorizes through Dk : πk (Uk ) ‚

J0 ((Q„ P 1 — E)Rm ) ’ F0 Rm . Now, we can apply the same procedure as in

k

∞

the section 28 to this invariant open submanifolds πk (Uk ).

Let F be a ¬rst order bundle functor on Mfm , E be an open natural sub

¯

bundle of a vector bundle functor E on Mfm . The curvature and its covariant

derivatives are natural operators ρk : Q„ P 1 Rk , with values in tensor bundles

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

272 Chapter VII. Further applications

Rk , Rk Rm = Rm —Wk , W0 = Rm —Rm— —Λ2 Rm— , Wk+1 = Wk —Rm— . Similarly,

the covariant di¬erentiation of sections of E forms natural operators dk : Q„ P 1 —

¯

Ek , where E0 = E, E0 Rm =: Rm —V0 , d0 is the inclusion, Ek Rm = Rm —Vk ,

E

Vk+1 = Vk — Rm— . Let us write Dk = (ρ0 , . . . , ρk’2 , d0 , . . . , dk ) : Q„ P 1 — E

Rk’2 — E k , where Rl = R0 — . . . — Rl , E l = E0 — . . . — El . All Dk are natural

operators. In 28.8 we de¬ned the Ricci sub bundles Z k ‚ Rk’2 — E k and we

know Dk : Q„ P 1 — E Zk.

Let us further de¬ne the functor Z ∞ as the inverse limit of Z k , k ∈ N, with

respect to the obvious natural transformations (projections) pk : Z k ’ Z , k > ,

and similarly D∞ : Q„ P 1 — E Z ∞ . As a corollary of 28.11 and the non linear

Peetre theorem we get

Proposition. For every natural operator D : Q„ P 1 — E F there is a unique

˜ ˜

∞ ∞

natural transformation D : Z ’ F such that D = D—¦D . Furthermore, for ev-

ery m-dimensional compact manifold M and every section s ∈ C ∞ (Q„ P 1 M —M

EM ), there is a ¬nite order k and a neighborhood V of s in the C k -topology

˜ ˜ ˜

such that DM |(D∞ )M (V ) = (πk )— (Dk )M , for some (Dk )M : (Dk )M (V ) ’

∞

˜

C ∞ (Z k M ), and DM |V = (Dk )M —¦ (Dk )M |V .

In words, a natural operator D : Q„ — E F is determined in all coordinate

charts of an arbitrary m-dimensional manifold M by a universal smooth mapping

de¬ned on the curvatures and all their covariant derivatives and on the sections

of EM and all their covariant derivatives, which depends ˜locally™ only on ¬nite

number of these arguments.

33.10. The Riemannian case. In section 28, we also applied the reduction

procedure to operators depending on Riemannian metrics and general vector

¬elds. In fact we have viewed the operators D : S+ T — — E

2

F as operators

¯ 2—

1

D : Q„ P — (S+ T — E) F independent of the ¬rst argument and we have

used the Levi-Civit` connection “ : S+ T —