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˜v ,
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

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

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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.

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