Q„ P 1 to write D as a composition

a

¯

D = D —¦ (“, id). Since the covariant derivatives of the metric with respect to

the metric connection are zero, we can restrict ourselves to sub bundles in the

Ricci subspaces corresponding to the bundle S+ T — — E, which are of the form

2

S+ T — — Z k with Z k ‚ Rk’2 — E k , cf. 28.14. Let us notice that the bundles

2

Z k M involve the curvature of the Riemannian connection on M , its covariant

derivatives, and the covariant derivatives of the sections of EM . Similarly as

above, we de¬ne the inverse limits Z ∞ and D∞ and as a corollary of the non

linear Peetre theorem and 28.15 we get

Corollary. For every natural operator D : S+ T — — E

2

F there is a nat-

˜ ˜

ural transformation D : S+ T — Z ’ F such that D = D —¦ D∞ —¦ (“, id).

2— ∞

Furthermore, for every m-dimensional compact manifold M and every section

s ∈ C ∞ (S+ T — M —M EM ), there is a ¬nite order k and a neighborhood V of s

2

˜ ˜

in the C k -topology such that DM |(D∞ —¦ (“, id))M (V ) = (πk )— (Dk )M , where

∞

˜ ˜

(Dk )M : (Dk —¦ (“, id))M (V ) ’ C ∞ (Z k M ), and DM |V = (Dk )M —¦ (Dk )M —¦

(“, id)M |V .

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

33. Topics from Riemannian geometry 273

33.11. Polynomiality. Since the standard ¬ber V0 of E0 is embedded identi-

cally into Z0 Rm by the associated map to the operator Dk , we can use 28.16 and

k

add the following proposition to the statements of 33.9, or 33.10, respectively.

˜

Corollary. The operator D is polynomial if and only if the operators Dk are

polynomial. Further D is polynomial with smooth real functions on the values of

˜

E0 , or S+ T — , as coe¬cients if and only if the operators Dk are polynomial with

2

smooth real functions on the values of E0 , or S+ T — , as coe¬cients, respectively.

2

33.12. Natural operators D : S+ T — T (p,q) . According to 33.9 we ¬nd G∞ -

2

m

invariant open subsets Uk in J0 (S+ T — Rm ) forming a ¬ltration of the whole jet

∞ 2

space, such that on these subsets D factorizes through smooth Gk+1 -equivariant

m

mappings

i1 ...ip i1 ...ip

fj1 ...jq = fj1 ...jq (gij , . . . , gij 1 ... k )

de¬ned on πk Uk . For large k™s, the action of the homotheties c’1 δj on g™s is

∞ i

well de¬ned and we get

i ...i i ...i

cq’p fj1 ...jq (gij , . . . , gij ) = fj1 ...jq (c2 gij , . . . , c2+k gij

1 p 1 p

(1) ).

1 ... k 1 ... k

Now, let us add the assumption that D is homogeneous with weight », choose

the change g ’ c’2 g of the scale of the metric and insert this new metric into

(1). We get

i ...i i ...i

cq’p’» fj1 ...jq (gij , . . . , gij ) = fj1 ...jq (gij , c1 gij, 1 , . . . , ck gij

1 p 1 p

).

1 ... k 1 ... k

i ...i

1 p

This formula shows that the mappings fj1 ...jq are polynomials in all variables

except gij with functions in gij as coe¬cients.

According to 33.11 and 28.16, the map D is on Uk determined by a polynomial

mapping

i ...i i i

ω = (ωj1 ...jp (gij , Wjkl , . . . , Wjklm1 ...mk’2 ))

1 q

which is G1 -equivariant on the values of the covariant derivatives of the curva-

m

tures and the sections. If we apply once more the equivariance with respect to

the homothety x ’ c’1 x and at the same time the change of the scale of the

metric g ’ c’2 g, we get

i ...i

cq’p’» ωj1 ...jp (gij , Rjkl , . . . , Rjklm1 ...mk’2 ) =

i i

1 q

i ...i

= ωj1 ...jp (gij , c2 Rjkl , . . . , ck Rjklm1 ...mk’2 ).

i i

1 q

i ...i

This homogeneity shows that the polynomial functions ωj1 ...jp must be sums of

1 q

i

homogeneous polynomials with degrees a in the variables Rjklm1 ...m satisfying

2a0 + · · · + kak’2 = q ’ p ’ »

(2)

and their coe¬cients are functions depending on gij ™s.

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

274 Chapter VII. Further applications

Now, we shall ¬x gij = δij and use the O(m)-equivariance of the homogeneous

components of the polynomial mapping ω. For this reason, we shall switch to

a

the variables Rijklm1 ...ms = gia Rjklm1 ...ms . Using the standard polarization tech-

nique and H. Weyl™s theorem, we get that each multi homogeneous component

in question results from multiplication of variables Rijklm1 ,... ,ms , s = 0, 1, . . . , r,

and application of some O(m)-equivariant tensor operations on the target space.

Hence our operators result from a ¬nite number of the following steps.

(a) take tensor product of arbitrary covariant derivatives of the curvature

tensor

(b) tensorize by the metric or by its inverse

(c) apply arbitrary GL(m)-equivariant operation

(d) take linear combinations.

33.13. Remark. If q ’ p = » + 1, then there is no non negative integer solution

of 33.12.(2) and so all natural tensors in question are zero. The case q = 2,

p = 1, » = 0 implies that the Levi-Civit` connection is the only conformal

a

natural connection on Riemannian manifolds.

Indeed, the di¬erence of two such connections is a natural tensor twice co-

variant and once contravariant, and so zero.

33.14. Consider now Λp Rm— as the target tensor space. So in the above proce-

dure, all indices which were not contracted must be alternated at the end. Since

the metric is a symmetric tensor, we get zero whenever using the above step

(b) and alternating over both indices. But contracting over any of them has no

proper e¬ect, for δij Rjklnm1 ,... ,ms = Riklnm1 ,... ,ms . So we can omit the step (b)

at all.

The ¬rst Bianchi identity and 33.7.(1) imply Rijkl = Rklij . Then the lemma

in 33.5 and 33.7.(1) yield

Lemma. The alternation of Rijklm1 ...ms , 0 ¤ s, over arbitrary 3 indices among

the ¬rst four or ¬ve ones is zero.

Consider a monomial P in the variables Rijkl± with degrees as in Rijklm1 ...ms .

In view of the above lemma, if P remains non zero after all alternations, then we

must contract over at least two indices in each Rijkl± and so we can alternate

over at most 2a0 + · · · + kak’2 indices. This means p ¤ 2a0 + · · · + kak’2 = p ’ ».

Consequently » ¤ 0 if there is a non zero natural form with weight ». This proves

the ¬rst assertion of theorem 33.8.

Let » = 0. Since the weight of g ij is ’2, any contraction on two indices

in the monomial decreases the weight of the operator by 2. Every covariant

derivative Rijklm1 ...ms of the curvature has weight 2. So we must contract on

exactly two indices in each Rijklm1 ...ms which implies there are s + 2 of them

under alternation. But then there must appear three alternated indices among

the ¬rst ¬ve if s = 0. This proves a1 = · · · = ak’2 = 0, so that p = 2a0 . Hence

all the natural forms have even degrees and they are generated by the forms

ωq , cf. 33.5. As we deduced in 33.7, these forms are zero if their degree is not

divisible by four.

This completes the proof of the theorem 33.8.

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

33. Topics from Riemannian geometry 275

33.15. Remark. The original proof of the Gilkey theorem assumes a poly-

nomial dependence of the natural forms on a ¬nite number of the derivatives

gij,± of the metric and on the entries of the inverse matrix g ij , but also the

homogeneity in the weight, [Gilkey, 73]. Under such polynomiality assumption,

our methods apply to all natural tensors. In particular, it follows easily that

the Levi-Civit` connection is the only second order polynomial connection on

a

Riemannian manifolds. Of course, the latter is not true in higher orders, for we

can contract appropriate covariant derivatives of the curvature and so we get

natural tensors in T — T — — T — of orders higher than two.

33.16. Operations on exterior forms. The approach from 33.4“33.5 can be

easily extended to the study of all natural operators D : Q„ P 1 — T (s,r) T (q,p)

with s < r or s = r = 0. This was done in [Slov´k, 92a], we shall present only

a

the ¬nal results. If we omit the assumption on s and r, we have to assume the

polynomiality.

Theorem. All natural operators D : Q„ P 1 —T (s,r) T (q,p) , s < r, are obtained

by a ¬nite iteration of the following steps: take tensor product of arbitrary

covariant derivatives of the curvature tensor or the covariant derivatives of the

tensor ¬elds from the domain, apply arbitrary GL(m)-equivariant operation,

take linear combinations. In the case s = r = 0 we have to add one more

step, the compositions of the functions from the domain with arbitrary smooth

functions of one real variable.

ΛT — , r > 0, is

The algebra of all natural operators D : Q„ P 1 — T (0,r)

generated by the alternation, the exterior derivative d and the Chern forms cq .

ΛT — is generated

The algebra of all natural operators D : Q„ P 1 — T (0,0)

by the compositions with arbitrary smooth functions of one real variable, the

exterior derivative d and the Chern forms cq .

The proof of these results follows the lines of 33.4“33.5 using two more lemmas:

First, the alternation on all indices of the second covariant derivative 2 t of an

arbitrary tensor t ∈ C ∞ (—s Rm— ) is zero (which is proved easily using the Bianchi

and Ricci identities) and , second, the alternation of the ¬rst covariant derivative

of an arbitrary tensor t ∈ C ∞ (—s Rm— ) coincides with the exterior di¬erential of

the alternation of t (this well known fact is proved easily in normal coordinates).

33.17. Operations on exterior forms on Riemannian manifolds. A mod-

i¬cation of our proof of the Gilkey theorem for operations on exterior forms on

Riemannian manifolds, which is also based on the two lemmas mentioned above,

appeared in [Slov´k, 92a]. The equality 33.7.(2) on the Riemannian curvatures

a

can be expressed as Rijkl = Rjikl , and this holds for curvatures of metrics

with arbitrary signatures. This observation extends our considerations to pseu-

doriemannian manifolds, see [Slov´k, 92b]. In particular, our proof of the Gilkey

a

theorem extends to the classi¬cation of natural forms on pseudoriemannian man-

ifolds. Let us write Sreg T — for the bundle functor of all non degenerate symmetric

2

two-forms. The de¬nition of the weight of the operators depending on metrics

and the de¬nition of the Pontryagin forms extend obviously to the pseudorie-

mannian case. All the considerations go also through for metrics with any ¬xed

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

276 Chapter VII. Further applications

signature.

Theorem. All natural operators D : Sreg T — — T (s,r)

2

T (q,p) , s < r, homoge-

neous in weight are obtained by a ¬nite iteration of the following steps: take

tensor product of arbitrary covariant derivatives of the curvature tensor or the

covariant derivatives of the tensor ¬elds from the domain, tensorize by the metric

or its inverse, apply arbitrary GL(m)-equivariant operation, take linear combi-

nations. In the case s = r = 0 we have to add one more step, the compositions

of the functions from the domain with arbitrary smooth functions of one real

variable.

There are no non zero operators D : Sreg T — — T (0,r) ΛT — , r ≥ 0, with a

2

positive weight. The algebra of all conformal natural operators Sreg T — —T (0,r)

2

ΛT — , r > 0, is generated by the alternation, the exterior derivative d and the

Pontryagin forms pq .

The algebra of all conformal natural operators D : Sreg T — — T (0,0) ΛT —

2

is generated by the compositions with arbitrary smooth functions of one real

variable, the exterior derivative d and the Pontryagin forms pq .

The discussion from the proof of these results can be continued for every ¬xed

negative weight. In particular, the situation is interesting for » = ’2 and linear

operators D : Λp T — Λp T — depending on the metric. Beside the compositions

d —¦ δ and δ —¦ d of the exterior di¬erential d and the well known codi¬erential

δ : Λp Λp’1 (the Laplace-Beltrami operator is ∆ = δ —¦ d + d —¦ δ), there are

only three other generators: the multiplication by the scalar curvature, the con-

traction with the Ricci curvature and the contraction with the full Riemmanian

curvature. This classi¬cation was derived under some additional assumptions in

[Stredder, 75], see also [Slov´k, 92b].

a

33.18. Oriented pseudoriemannian manifolds. It is also quite important

in Riemannian geometry to know what are the operators natural with respect to

the orientation preserving local isometries. We shall not go into details here since

this would require to extend the description from 33.2 to all SO(m)-invariant

linear maps and then to repeat some steps of the proof of the Gilkey theo-

rem. This was done in [Stredder, 75] (for the polynomial forms and Riemannian

manifolds), and in [Slov´k, 92b]. Let us only remark that on oriented pseu-

a

doriemannian manifolds we have a natural volume form ω : Sreg T — Λm T — and

2

natural transformations — : Λp T — ’ Λm’p T — . All natural operators on oriented

pseudoriemannian manifolds homogeneous in the weight are generated by those

described above, the volume form ω, and the natural transformations —.

As an example, let us draw a diagram which involves all linear natural con-

formal operators on exterior forms on oriented pseudoriemannian manifolds of

even dimensions which do not vanish on ¬‚at pseudoriemannian manifolds, up to

the possible omitting of the d™s on the sides in the operators indicated by the

long arrows. (More explicitely, we do not consider any contribution from the

curvatures.) The symbols „¦p refer, as usual, to the p-forms, the plus and minus

subscripts indicate the splitting into the selfdual and anti-selfdual forms in the

1

degree 2 m.

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

33. Topics from Riemannian geometry 277

In the even dimensional case, there are no natural conformal operators be-

tween exterior forms beside the exterior derivatives. For the proofs see [Slov´k,

a

92b].

We should also remark that the name ˜conformal™ is rather misleading in the

context of the natural operators on conformal (pseudo-) Riemannian manifolds

since we require the invariance only with respect to constant rescaling of the

metric (cf. the end of the next section). On the other hand, each natural operator

on the conformal manifolds must be conformal in our sense.

9 9 hh

Aj „¦p

d+ + d

w„¦ w ··· w„¦ w ··· w„¦ u w „¦u

e e „¦ u

eg ¢

d d d d d d

„¦0 1 p’1 p+1 m’1 m

d’ d

„¦p

’

Dp’1 =d—d=d—¦d+ ’d—¦d’

D1 =d—¦(—d)m’3

D0 =d—¦(—d)m’1

33.19. First order operators. The whole situation becomes much easier if

we look for ¬rst order natural operators D : S+ T —

2

(F, G), where F and G are

arbitrary natural bundles, say of order r. Namely, every metric g on a manifold

‚g

M satis¬es gij = δij and ‚xij = 0 at the center of any normal coordinate chart.

k

¯ are two such operators and if their values DRm (g), DRm (g)

¯

Therefore, if D, D

on the canonical Euclidean metric g on Rm coincide on the ¬ber over the origin,

¯

then D = D. Hence the whole classi¬cation problem reduces to ¬nding maps

between the standard ¬bers which are equivariant with respect to the action of

the subgroup O(m) B1 ‚ G1r

B1 = Gr . In fact we used this procedure in

r

m m

section 29.

Let us notice that the natural operators on oriented Riemannian manifolds

r r

are classi¬ed on replacing O(m) B1 by SO(m) B1 . If we modify 29.7 in such

a way, we obtain (cf. [Slov´k, 89])

a

Proposition. All ¬rst order natural connections on oriented Riemannian man-

ifolds are

(1) The Levi-Civit` connection “, if m > 3 or m = 2

a

(2) The one parametric family “ + kD1 where D1 means the scalar product

and k ∈ R, if m = 1

(3) The one parametric family “ + kD3 where D3 means the vector product

and k ∈ R, if m = 3.

33.20. Natural metrics on the tangent spaces of Riemannian mani-

folds. At the end of this section, we shall describe all ¬rst order natural opera-

tors transforming metrics into metrics on the tangent bundles. The results were

proved in [Kowalski, Sekizava, 88] by the method of di¬erential equations. Let

us start with some notation.

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

278 Chapter VII. Further applications

We write πM : T M ’ M for the natural projection and F for the natural

bundle with F M = πM (T — —T — )M ’ M , F f (Xx , gx ) = (T f.Xx , (T — —T — )f.gx )

—

for all manifolds M , local di¬eomorphisms f , Xx ∈ Tx M , gx ∈ (T — — T — )x M .

The sections of the canonical projection F M ’ T M are called the F-metrics

in literature. So the F-metrics are mappings T M • T M • T M ’ R which are

linear in the second and the third summand. We ¬rst show that it su¬ces to

describe all natural F-metrics, i.e. natural operators S+ T —

2

(T, F ).

There is the natural Levi-Civit` connection “ : T M • T M ’ T T M and the

a

natural equivalence ν : T M •T M ’ V T M . There are three F-metrics, naturally

derived from sections G : T M ’ (S 2 T — )T M . Given such G on T M , we de¬ne

γ1 (G)(u, X, Y ) = G(“(u, X), “(u, Y ))

γ2 (G)(u, X, Y ) = G(“(u, X), ν(u, Y ))

(1)

γ3 (G)(u, X, Y ) = G(ν(u, X), ν(u, Y )).

Since G is symmetric, we know also G(ν(u, X), “(u, Y )) = γ2 (G)(u, Y, X). No-

tice also that γ1 and γ3 are symmetric.

The connection “ de¬nes the splitting of the second tangent space into the

h v

vertical and horizontal subspaces. We shall write Xx,u = Xu + Xu for each

Xx,u ∈ Tu T M , π(u) = x. Since for every Xx,u there are unique vectors X h ∈

Tx M , X v ∈ Tx M such that “(u, X h ) = Xu and ν(u, X v ) = Xu , we can recover

h v

the values of G from the three F-metrics γi ,