κ’1 »’1

with ω1 = •

˜ .ω1 , ω2 = •

˜ .ω2 . After the substitution into (1), there appears

the extra summand

(c1 d•κ’1 § ω1 § •»’1 ω2 + c2 •κ’1 ω1 § d•»’1 § ω2 )˜mu’1

v

= (κ ’ 1)c1 + (’1)k (» ’ 1)c2 .d(ln•) § ω1 § ω2 .v µ’1 .

Thus (1) is a correct de¬nition of an invariant operation if and only if

(κ ’ 1)c1 + (’1)k (» ’ 1)c2 = 0.

(2)

Now take k + l ¤ n + 1. We ¬nd an operation if and only if m = k + l ’ 1

and µ = κ + ». As before, we ¬x an auxiliary volume form v and we write

the ¬elds of type Λk „ — ∆κ as a.v κ where a is a k-vector ¬eld. The usual

divergence of vector ¬elds extends to a linear operation δv on k-vector ¬elds,

k

δv (X1 § · · · § Xk ) = i=1 (’1)i+1 divXi .X1 § · · · §i · · · § Xk , where §i means that

the entry is missing. Of course, this divergence depends on the choice of v. We

have

(3)

k

(’1)i+1 Xi (•).X1 § · · · §i · · · § Xk .

δ•v (X1 § · · · § Xk ) = •.δv (X1 § · · · § Xk ) +

i=1

Let us look for a natural operator D of the form

D(a.v κ , b.v » ) = (c1 δv (a) § b + c2 a § δv (b) + c3 δv (a § b)) .v µ .

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34. Multilinear natural operators 291

Formula (3) implies that D is natural if and only if

(κ ’ 1)c1 + (κ + » ’ 1)c3 = 0, (» ’ 1)c2 + (κ + » ’ 1)c3 = 0.

(4)

The formulas (2) and (4) de¬ne the constants uniquely except the case κ =

» = 1 when we get two independent operations, see also the ¬fth class. Let

us point out that the second class involves also the Schouten-Nijenhuis bracket

Λp T • Λq T Λp+q’1 T (the case κ = » = 0, k + l ¤ n + 1), cf. 30.10, sometimes

also caled the antisymmetric Schouten concomitant, which de¬nes the structure

of a graded Lie algebra on the ¬elds in question. This bracket is given by

[X1 § · · · § Xk , Y1 § · · · § Yl ]

§ X1 § · · · §i · · · § Xk § Y1 § · · · §j · · · § Yl .

i+j

= i,j (’1) [Xi , Yj ]

The second class is invariant under the action of Σ3 .

3. The third class is represented by the so called symmetric Schouten con-

comitant. This is an operation of type (S k „, S l „ ; S k+l’1 „ ) with a nice geometric

de¬nition. The elements in S k T M can be identi¬ed with functions on T — M ¬ber-

wise polynomial of degree k. Since there is a canonical symplectic structure on

T — M , there is the Poisson bracket on C ∞ (T — M ). The bracket of two ¬berwise

polynomial functions is also ¬berwise polynomial and so the bracket gives rise

to our operation.

The action of Σ3 yields an operation of the type (S k „, S l „ — —∆; S l’k+1 „ — —∆).

If k = 1, this is the Lie derivative and if k = l, we get the lagrangian Schouten

concomitant.

4. This class involves the Fr¨licher-Nijenhuis bracket, an operation of the type

o

k— l— k+l —

(„ — Λ „ , „ — Λ „ ; „ — Λ „ ), k + l ¤ n. The tensor spaces in question are not

irreducible, „ — Λk „ — is a sum of Λk’1 „ — and an irreducible representation ρk

of highest weight (1, . . . , 1, 0, . . . , 0, ’1) where 1 appears k-times (the trace-free

vector valued forms). The Fr¨licher-Nijenhuis bracket is a sum of an operation

o

of type (ρk , ρl ; ρk+l ) and several other simpler operations.

If we apply the action of Σ3 to the Fr¨licher-Nijenhuis bracket, we get an

o

operation of the type („ — Λ „ , „ — Λ „ ; „ — Λk+m „ — ) which is expressed

m— — k— —

through contractions and the exterior derivative.

5. Finally, there are the natural operations which reduce to compositions of

wedge products and exterior di¬erentiation. Such operations are always de¬ned

if at least one of the representations ρ1 , ρ2 , or one of the irreducible components

of ρ1 — ρ2 coincides with Λk „ — . Since Λk „ — = Λn’k „ — , this class is also invariant

under the action of Σ3 .

In [Grozman, 80b] we ¬nd the next theorem. Unfortunately its proof based

on the Rudakov™s algebraic methods is not available in the literature. In an

earlier paper, [Grozman, 80a], he classi¬ed the bilinear operations in dimension

two, including the unimodular case.

34.18. Theorem. All natural bilinear operators between natural bundles cor-

responding to irreducible representations of GL(n) are exhausted by the zero

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292 Chapter VII. Further applications

order operators, the ¬ve classes of ¬rst order operators described in 34.17, the

operators of second and third order obtained by the composition of the ¬rst and

zero order operators and one exceptional operation in dimension n = 1, see the

example below.

In particular, there are no bilinear natural operations of order greater then

three.

34.19. Example. A tensor density on the real line is determined by one com-

plex number », we write f (x)(dx)’» ∈ C ∞ (E» R) for the corresponding ¬elds of

geometric objects. There is a natural bilinear operator D : E2/3 • E2/3 E’5/3

f g df /dx dg/dx

D(f (dx)’2/3 , g(dx)’2/3 ) = 2 .(dx)5/3

+3

d3 f /dx3 d3 g/dx3 d2 f /dx2 d2 g/dx2

This is a third order operation which is not a composition of lower order ones.

34.20. The multilinear natural operators are also related to the cohomology

theory of Lie algebras of formal vector ¬elds. In fact these operators express

zero dimensional cohomologies with coe¬cients in tensor products of the spaces

of the ¬elds in question, see [Fuks, 84]. The situation is much further analyzed

in dimension n = 1 in [Feigin, Fuks, 82]. In particular, they have described all

skew symmetric operations E» • · · · • E» Eµ . They have deduced

Theorem. For every » ∈ C, m > 0, k ∈ Z, there is at most one skew symmetric

operation D : Λm C ∞ E» ’ C ∞ Eµ with µ = m»’ 2 m(m’1)’k, up to a constant

1

multiple. A necessary and su¬cient condition for its existence is the following:

either k=0, or 0 < k ¤ m and » satis¬es the quadratic equation

(» + 1 )(k2 + 1) ’ m = 1 (k2 ’ k1 )2

1

(» + 2 )(k1 + 1) ’ m 2 2

with arbitrary positive k1 ∈ Z, k2 ∈ Z, k1 .k2 = k.

The operator corresponding to the ¬rst possibility k = 0, D : Λm C ∞ (E» R) ’

C ∞ (Em»’ 1 m(m’1) R), admits a simple expression

2

(m’1)

f1 f1 ... f1

(m’1) 1

f1 (dx)’» § · · · § fm (dx)’» ’ (dx)’m»+ 2 m(m’1)

f2 f2 ... f2

.............

(m’1)

fm fm ... fm

Grozman™s operator from 34.19 corresponds to the choice m = 2, k = 2,

» = 2/3, k1 = 2, k2 = 1. The proof of this theorem is rather involved. It

is based on the structure of projective representations of the algebra of formal

vector ¬elds on the one-dimensional sphere.

34.21. The problem of ¬nding all natural m-linear operations has been also for-

mulated for super manifolds. As far as we know, only the linear operations were

classi¬ed, see [Bernstein, Leites, 77], [Leites, 80], [Shmelev, 83], but their results

include also the unimodular, and Hamiltonian cases. Some more information is

also available in [Kirillov, 80].

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34. Multilinear natural operators 293

34.22. The linear natural operations on conformal manifolds. As we

have seen, the description of the linear natural operators is heavily based on the

structure of the subalgebra in the algebra of formal vector ¬elds which corre-

sponds to the jet groups in the category in question. If the category involves

very few morphisms, these algebras become small. In particular, they might

have ¬nite dimensions like in the case of Riemannian manifolds or conformal

Riemannian manifolds. The former example is not so interesting for the follow-

ing reasons: Since all irreducible representations of the orthogonal groups are

O(m, R)-invariant irreducible subspaces in tensor spaces, we can work in the

whole category of manifolds in the way demonstrated in section 33. On the

other hand, if we include the so called spinor representations of the orthogonal

group, we get serious problems with the whole setting. However, the second

example is of highest interest for many reasons coming both from mathematics

and physics and it is treated extensively nowadays. Let us conclude this section

with a very short overview of the known results, for more information see the

survey [Baston, Eastwood, 90] or the papers [Baston, 90], [Branson, 85].

Let us write C for the category of manifolds with a conformal Riemannian

structure, i.e. with a distinguished line bundle in S+ T — M , and the morphisms

2

keeping this structure. More explicitely, two metrics g, g on M are called confor-

ˆ

mal if there is a positive smooth function f on M such that g = f 2 g. A conformal

ˆ

structure is an equivalence class with respect to this equivalence relation. The

conformal structure on M can also be described as a reduction of the ¬rst order

frame bundle P 1 M to the conformal group CO(m, R) = R O(m, R), and the

conformal morphisms • are just those local di¬eomorphisms which preserve this

reduction under the P 1 •-action. Thus, each linear representation of CO(m, R)

on a vector space V de¬nes a bundle functor on C. The category C is not locally

homogeneous, but it is local.

The main di¬erence from the situations typical for this book is that there

are new natural bundles in the category C. In fact, we can take any linear

representation of O(m, R) and a representation of the center R ‚ GL(m, R)

and combine them together. The representations of the center are of the form

(t.id)(v) = t’w .v with an arbitrary real number w, which is called the confor-

mal weight of the representation or of the corresponding bundle functor. Each

tensorial representation of GL(m, R) induces a representation of CO(m, R) with

the conformal weight equal to the di¬erence of the number of covariant and

contravariant indices. In particular, the convention for the weight is chosen in

such a way that the bundle of metrics has conformal weight two. If we restrict

our considerations to the tensorial representations, we exclude nearly all natural

linear operators.

Each isometry of a conformal manifold with respect to an arbitrary metric

from the distinguished class is a conformal morphism. Thus, the Riemannian

natural operators described in section 33 can be taken for candidates in the

classi¬cation. But the remaining problems are still so di¬cult that a general

solution has not been found yet.

Let us mention at least two possibilities how to treat the problem. The

¬rst one is to restrict ourselves to locally conformally ¬‚at manifolds, i.e. we

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294 Chapter VII. Further applications

consider only a subcategory in C which is admissible in our sense. Thus, the

classi¬cation problem for linear operators reduces to a (di¬cult) problem from

the representation theory. But what remains then is to distinguish those natural

operators on the conformally ¬‚at manifolds which are restrictions of natural

operators on the whole category, and to ¬nd explicite formulas for them. For

general reasons, there must be a universal formula in the terms of the covariant

derivatives, curvatures and their covariant derivatives. The best known example

is the conformal Laplace operator on functions in dimension 4

1

a

D= +R

a

6

where a a means the operator of the covariant di¬erentiation applied twice

and followed by taking trace, and R is the scalar curvature. The proper confor-

mal weights ensuring the invariance are ’1 on the source and ’3 on the target.

The ¬rst summand D0 = a a of D is an operator which is natural on the

functions with the speci¬ed weights on conformally ¬‚at manifolds and the sec-

ond summand is a correction for the general case. In view of this example, the

question is how far we can modify the natural operators (homogeneous in the

order and acting between bundles corresponding to irreducible representations of

CO(m, R)) found on the ¬‚at manifolds by adding some corrections. The answer

is rather nice: with some few exceptions this is always possible and the order

of the correction term is less by two (or more) than that of D0 . Moreover, the

correction involves only the Ricci curvature and its covariant derivatives. This

was deduced in [Eastwood, Rice, 87] in dimension four, and in [Baston, 90] for

dimensions greater than two (the complex representations are treated explicitely

and the authors assert that the real analogy is available with mild changes). In

particular, there are no corrections necessary for the ¬rst order operators, which

where completely classi¬ed by [Fegan, 76]. Nevertheless, the concrete formu-

las for the operators (¬rst of all for the curvature terms) are rarely available.

Another disadvantage of this approach is that we have no information on the

operators which vanish on the conformally ¬‚at manifolds, even we do not know

how far the extension of a given operator to the whole category is determined.

The description of all linear natural operators on the conformally ¬‚at mani-

folds is based on the general ideas as presented at the begining of this section.

This means we have to ¬nd the morphisms of g-modules W — ’ (Tn V )— , where

∞

g is the algebra of formal vector ¬elds on Rn with ¬‚ows consisting of conformal

morphisms. One can show that g = o(n + 1, 1), the pseudo-orthogonal algebra,

with grading g = g’1 • g0 • g1 = Rn • co(n, R) • Rn— . The lemmas 34.5 and 34.6

remain true and we see that (Tn V )— is the so called generalized Verma module

∞

corresponding to the representation of CO(n, R) on V . Each homomorphism

W — ’ (Tn V )— extends to a homomorphism of the generalized Verma modules

∞

(Tn W )— ’ (Tn V )— and so we have to classify all morphisms of generalized

∞ ∞

Verma modules. These were described in [Boe, Collingwood, 85a, 85b]. In par-

ticular, if we start with usual functions (i.e. with conformal weight zero), then

all conformally invariant operators which form a ˜connected pattern™ involving

the functions are drawn in 33.18. (The latter means that there are no more

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Remarks 295

operators having one of the bundles indicated on the diagram as the source or

target.) A very interesting point is a general principal coming from the repre-

sentation theory (the so called Jantzen-Zuzkermann functors) which asserts that

once we have got such a ˜connected pattern™ all other ones are obtained by a

general procedure. Unfortunately this ˜translation procedure™ is not of a clear

geometric character and so we cannot get the formulas for the corresponding

operators in this way, cf. [Baston, 90]. The general theory mentioned above

implies that all the operators from the diagram in 33.18 admit the extension to

the whole category of conformal manifolds, except the longest arrow „¦0 ’ „¦m .

By the ˜translation procedure™, the same is ensured for all such patterns, but

the question whether there is an extension for the exceptional ˜long arrows™ is

not solved in general. Some of them do extend, but there are counter examples

of operators which do not admit any extension, see [Branson, 89], [Graham, to

appear].

Another more direct approach is used by [Branson, 85, 89] and others. They

write down a concrete general formula in terms of the Riemannian invariants

and they study the action of the conformal rescaling of the metric. Since it is

su¬cient to study the in¬nitesimal condition on the invariance with respect to

the rescaling of the metric, they are able to ¬nd series of conformally invariant

operators. But a classi¬cation is available for the ¬rst and second order operators

only.

Remarks

Proposition 30.4 was proved by [Kol´ˇ, Michor, 87]. Proposition 31.1 was

ar

deduced in [Kol´ˇ, 87a]. The natural transformations J r ’ J r were determined

ar

in [Kol´ˇ, Vosmansk´, 89]. The exchange map eΛ from 32.4 was introduced by

ar a

[Modugno, 89a].

The original proof of the Gilkey theorem on the uniqueness of the Pontryagin

forms, [Gilkey, 73], was much more combinatorial and had not used H. Weyl™s

theorem. Our approach is similar to [Atiyah, Bott, Patodi, 73], but we do

not need their polynomiality assumption. The Gilkey theorem was generalized

in several directions. For the case of Hermitian bundles and connections see

[Atiyah, Bott, Patodi, 73], for oriented Riemannian manifolds see [Stredder, 75],

the metrics with a general signature are treated in [Gilkey, 75]. The uniqueness of

the Levi-Civit` connection among the polynomial conformal natural connections

a

on Riemannian manifolds was deduced by [Epstein, 75]. The classi¬cation of the

¬rst order liftings of Riemannian metrics to the tangent bundles covers the results

due to [Kowalski, Sekizawa, 88], who used the so called method of di¬erential

equations in their much longer proof. Our methods originate in [Slov´k, 89] and

a

an unpublished paper by W. M. Mikulski.

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

296

CHAPTER VIII.

PRODUCT PRESERVING FUNCTORS

We ¬rst present the theory of those bundle functors which are determined by

local algebras in the sense of A. Weil, [Weil, 51]. Then we explain that the Weil

functors are closely related to arbitrary product preserving functors Mf ’ Mf .

In particular, every product preserving bundle functor on Mf is a Weil functor

and the natural transformations between two such functors are in bijection with

the homomorphisms of the local algebras in question.

In order to motivate the development in this chapter we will tell ¬rst a math-

ematical short story. For a smooth manifold M , one can prove that the space

of algebra homomorphisms Hom(C ∞ (M, R), R) equals M as follows. The ker-

nel of a homomorphism • : C ∞ (M, R) ’ R is an ideal of codimension 1 in

C ∞ (M, R). The zero sets Zf := f ’1 (0) for f ∈ ker • form a ¬lter of closed

sets, since Zf © Zg = Zf 2 +g2 , which contains a compact set Zf for a function

f which is unbounded on each non compact closed subset. Thus f ∈ker • Zf is

not empty, it contains at least one point x0 . But then for any f ∈ C ∞ (M, R)

the function f ’ •(f )1 belongs to the kernel of •, so vanishes on x0 and we have

f (x0 ) = •(f ).

An easy consequence is that Hom(C ∞ (M, R), C ∞ (N, R)) = C ∞ (N, M ). So

the category of algebras C ∞ (M, R) and their algebra homomorphisms is dual to

the category Mf of manifolds and smooth mappings.

But now let D be the algebra generated by 1 and µ with µ2 = 0 (sometimes

called the algebra of dual numbers or Study numbers, it is also the truncated

polynomial algebra of degree 1). Then it turns out that Hom(C ∞ (M, R), D) =

T M , the tangent bundle of M . For if • is a homomorphism C ∞ (M, R) ’ D,

then π —¦ • : C ∞ (M, R) ’ D ’ R equals evx for some x ∈ M and •(f ) ’

f (x).1 = X(f ).µ, where X is a derivation over x since • is a homomorphism.

So X is a tangent vector of M with foot point x. Similarly we may show that

Hom(C ∞ (M, R), D — D) = T T M .

Now let A be an arbitrary commutative real ¬nite dimensional algebra with

unit. Let W (A) be the subalgebra of A generated by the idempotent and nilpo-

tent elements of A. We will show in this chapter, that Hom(C ∞ (M, R), A) =

Hom(C ∞ (M, R), W (A)) is a manifold, functorial in M , and that in this way we

have de¬ned a product preserving functor Mf ’ Mf for any such algebra. A

will be called a Weil algebra if W (A) = A, since in [Weil, 51] this construc-

tion appeared for the ¬rst time. We are aware of the fact, that Weil algebras

denote completely di¬erent objects in the Chern-Weil construction of character-

istic classes. This will not cause troubles, and a serious group of mathematicians

has already adopted the name Weil algebra for our objects in synthetic di¬er-

ential geometry, so we decided to stick to this name. The functors constructed

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

35. Weil algebras and Weil functors 297

in this way will be called Weil functors, and we will also present a covariant

approach to them which mimics the construction of the bundles of velocities,

due to [Morimoto, 69], cf. [Kol´ˇ, 86].

ar

We will discuss thoroughly natural transformations between Weil functors and

study sections of them, a sort of generalized vector ¬elds. It turns out that the

addition of vector ¬elds generalizes to a group structure on the set of all sections,

which has a Lie algebra and an exponential mapping; it is in¬nite dimensional

but nilpotent.

Conversely under very mild conditions we will show, that up to some covering

phenomenon each product preserving functor is of this form, and that natural

transformations between them correspond to algebra homomorphisms. This has

been proved by [Kainz-Michor, 87] and independently by [Eck, 86] and [Luciano,

88].

Weil functors will play an important role in the rest of the book, and we will

frequently compare results for other functors with them. They can be much

further analyzed than other types of functors.

35. Weil algebras and Weil functors

35.1. A real commutative algebra A with unit 1 is called formally real if for any

a1 , . . . , an ∈ A the element 1 + a2 + · · · + a2 is invertible in A. Let E = {e ∈

n

1

2

A : e = e, e = 0} ‚ A be the set of all nonzero idempotent elements in A. It is

not empty since 1 ∈ E. An idempotent e ∈ E is said to be minimal if for any

e ∈ E we have ee = e or ee = 0.

Lemma. Let A be a real commutative algebra with unit which is formally real

and ¬nite dimensional as a real vector space.