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runs again over all connected components of M .
Proposition. Any product preserving functor F : Mf ’ Mf which maps
embeddings to injective mappings is of the form F = Gc — . . . — Gc for product
n
1
preserving functors Gi which also map embeddings to injective mappings.
Proof. Let again Al(F ) = A = A1 • · · · • Ak be the decomposition into Weil
algebras. We conclude from 36.8.2 that χF,M : F (M ) ’ TA (M ) is injective for
each manifold M . We have to show that the set {1, . . . , k} can be divided into
equivalence classes I1 , . . . , In such that F (M ) ⊆ TA (M ) is the inverse image
under π : TA (M ) ’ Pk (M ) of the union of all N1 — . . . — Nk where the Ni run
through all connected components of M in such a way that i, j ∈ Ir for some r
implies that Ni = Nj . Then each Ir gives rise to Gc = T c .
r i∈Ir Ai
To ¬nd the equivalence classes we consider X = {1, . . . , k} as a discrete man-
ifold and consider F (X) ⊆ TA (X) = X k . Choose an element i = (i1 , . . . , ik ) ∈
F (X) with maximal number of distinct members. The classes Ir will then be
the non-empty sets of the form {s : is = j} for 1 ¤ j ¤ k. Let n be the number
of di¬erent classes.
Now let D be a discrete manifold. Then the claim says that

F (D) = {(d1 , . . . , dk ) ∈ Dk : s, t ∈ Ir implies ds = dt for all r}.

Suppose not, then there exist d = (d1 , . . . , dk ) ∈ F (D) and r, s, t with s, t ∈ Ir
and ds = dt . So among the pairs (i1 , d1 ), . . . , (ik , dk ) there are at least n + 1
distinct ones. Let f : X — D ’ X be any function mapping those pairs to
1, . . . , n + 1. Then F (f )(i, d) = (f (i1 , d1 ), . . . , f (ik , dk )) ∈ F (X) has at least
n + 1 distinct members, contradicting the maximality of n. This proves the
claim for D and also F (Rm — D) = Am — F (D) is of the right form since the
connected components of Rm — D correspond to the points of D.
Now let M be any manifold, let p : M ’ π0 (M ) be the projection of M
onto the (discrete) set of its connected components. For a ∈ F (M ) the value
F (p)(a) ∈ F (π0 (M )) just classi¬es the connected component of Pk (M ) over
which a lies, and this component of Pk (M ) must be of the right form. Let
x1 , . . . , xk ∈ M such s, t ∈ Ir implies that xs and xt are in the same connected
component Mr , say, for all r. The proof will be ¬nished if we can show that the
¬ber π ’1 (x1 , . . . , xk ) ‚ TA (M ) is contained in F (M ) ‚ TA (M ). Let m = dim M
(or the maximum of dim Mi for 1 ¤ i ¤ n if M is not a pure manifold) and let

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
318 Chapter VIII. Product preserving functors


N = Rm — {1, . . . , n}. We choose y1 , . . . , yk ∈ N and a smooth mapping g :
N ’ M with g(yi ) = xi which is a di¬eomorphism onto an open neighborhood
of the xi (a submersion for non pure M ). Then clearly TA (g)(π ’1 (y1 , . . . , yk )) =
π ’1 (x1 , . . . , xk ), and from the last step of the proof we know that F (N ) contains
π ’1 (y1 , . . . , yk ). So the result follows.
By theorem 36.10 we know the minimal data to reconstruct the action of F
on connected manifolds. For a not connected manifold M we ¬rst consider the
surjective mapping M ’ π0 (M ) onto the space of connected components of M .
Since π0 (M ) ∈ Mf , the functor F acts on this discrete set. Since F is weakly
local and maps points to points, F (π0 (M )) is again discrete. This gives us a
product preserving functor F0 on the category of countable discrete sets.
If conversely we are given a product preserving functor F0 on the category of
countable discrete sets, a formally real ¬nite dimensional algebra A consisting
of k Weil parts, and a product preserving functor G : conMf ’ groups with
k
π1 ⊆ G ⊆ π1 , then clearly one can construct a unique product preserving weakly
local functor F : Mf ’ Mf ¬tting these data.


37. Examples and applications

37.1. The tangent bundle functor. The tangent mappings of the algebra
structural mappings of R are given by

T R = R2 ,
T (+)(a, a )(b, b ) = (a + b, a + b ),
T (m)(a, a )(b, b ) = (ab, ab + a b),
T (m» )(a, a ) = (»a, »a ).
So the Weil algebra T R = Al(T ) =: D is the algebra generated by 1 and δ with
δ 2 = 0. It is sometimes called the algebra of dual numbers or also of Study
numbers. It is also the truncated polynomial algebra of order 1 on R. We will
write (a + a δ)(b + b δ) = ab + (ab + a b)δ for the multiplication in T R.
By 35.17 we can now determine all natural transformations over the category
Mf between the following functors.
(1) The natural transformations T ’ T consist of all ¬ber scalar multipli-
cations m» for » ∈ R, which act on T R by m» (1) = 1 and m» (δ) = ».δ.
(2) The projection π : T ’ IdMf is the only natural transformation.
37.2. Lemma. Let F : Mf ’ Mf be a multiplicative functor, which is also
a natural vector bundle over IdMf in the sense of 6.14, then F (M ) = V — T M
for a ¬nite dimensional vector space V with ¬berwise tensor product. Moreover
for the space of natural transformations between two such functors we have
N at(V — T, W — T ) = L(V, W ).
Proof. A natural vector bundle is local, so by theorem 36.1 it coincides with
TA , where A is its Weil algebra. But by theorem 35.13.(1) TA is a natural

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
37. Examples and applications 319


vector bundle if and only if the nilideal of A = F (R) is nilpotent of order
2, so A = F (R) = R · 1 • V , where the multiplication on V is 0. Then by
construction 35.11 we have F (M ) = V — T M . Finally by 35.17.(2) we have
N at(V — T, W — T ) = Hom(R · 1 • V, R · 1 • W ) ∼ L(V, W ).
=
37.3. The most important natural transformations. Let F , F1 , and F2
be multiplicative bundle functors (Weil functors by theorem 36.1) with Weil
algebras A = R • N , A1 = R • N1 , and A2 = R • N2 where the N ™s denote
the maximal nilpotent ideals. We will denote by N (F ) the nilpotent ideal in the
Weil algebra of a general functor F . By 36.13 we have Al(F2 —¦ F1 ) = A1 — A2 .
Using this and 35.17 we de¬ne the following natural transformations:
(1) The projections π1 : F1 ’ Id, π2 : F2 ’ Id induced by (».1 + n) ’
» ∈ R. In general we will write πF : F ’ Id. Thus we have also
F2 π1 : F2 —¦ F1 ’ F2 and π2 F1 : F2 —¦ F1 ’ F1 .
(2) The zero sections 01 : Id ’ F1 and 02 : Id ’ F2 induced by R ’ A1 ,
» ’ ».1. Then we have F2 01 : F2 ’ F2 —¦ F1 and 02 F1 : F1 ’ F2 —¦ F1 .
(3) The isomorphism A1 — A2 ∼ A2 — A1 , given by a1 — a2 ’ a2 — a1 induces
=
the canonical ¬‚ip mapping κF1 ,F2 = κ : F2 —¦ F1 ’ F1 —¦ F2 . We have
κF1 ,F2 = κ’1,F1 .
F2
(4) The multiplication m in A is a homomorphism A—A ’ A which induces
a natural transformation µ = µF : F —¦ F ’ F .
(5) Clearly the Weil algebra of the product F1 —Id F2 in the category of
bundle functors is given by R.1 • N1 • N2 . We consider the two natural
transformations

(π2 F1 , F2 π1 ), 0F1 —Id F2 —¦ πF2 —¦F1 : F2 —¦ F1 ’ (F1 —Id F2 ).

The equalizer of these two transformations will be denoted by vl : F2 —
F1 ’ F2 —¦ F1 and will be called the vertical lift. At the level of Weil
algebras one checks that the Weil algebra of F2 — F1 is given by R.1 •
(N1 — N2 ).
(6) The canonical ¬‚ip κ factors to a natural transformation κF2 —F1 : F2 —F1 ’
F1 — F2 with vl —¦ κF2 —F1 = κF2 ,F1 —¦ vl.
(7) The multiplication µ induces a natural transformation µ—¦vl : F —F ’ F .
It is clear that κ expresses the symmetry of higher derivatives. We will see that
the vertical lift vl expresses linearity of di¬erentiation.
The reader is advised to work out the Weil algebra side of all these natural
transformations.
37.4. The second tangent bundle. In the setting of 35.5 we let F1 = F2 = T
be the tangent bundle functor, and we let T 2 = T —¦ T be the second tangent
bundle. Its Weil algebra is D2 := Al(T 2 ) = D — D = R4 with generators
2 2
1, δ1 , and δ2 and with relations δ1 = δ2 = 0. Then (1, δ1 ; δ2 , δ1 δ2 ) is the
standard basis of R4 = T 2 R in the usual description, which we also used in 6.12.
From the list of natural transformations in 37.1 we get πT : (δ1 , δ2 ) ’ (δ, 0),
T π : (δ1 , δ2 ) ’ (0, δ), and µ = + —¦ (πT, T π) : T 2 ’ T, (δ1 , δ2 ) ’ (δ, δ). Then we

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
320 Chapter VIII. Product preserving functors


have T — T = T , since N (T ) — N (T ) = N (T ), and the natural transformations
from 37.3 have the following form:
κ : T 2 ’ T 2,
κ(a1 + x1 δ1 + x2 δ2 + x3 δ1 δ2 ) = a1 + x2 δ1 + x1 δ2 + x3 δ1 δ2 .
vl : T ’ T 2 , vl(a1 + xδ) = a1 + xδ1 δ2 .
m» T : T 2 ’ T 2 ,
m» T (a1 + x1 δ1 + x2 δ2 + x3 δ1 δ2 ) = a1 + x1 δ1 + »x2 δ2 + »x3 δ1 δ2 .
T m» : T 2 ’ T 2 ,
T m» (a1 + x1 δ1 + x2 δ2 + x3 δ1 δ2 ) = a1 + »x1 δ1 + x2 δ2 + »x3 δ1 δ2 .
(+T ) : T 2 —T T 2 ’ T 2 ,
(+T )((a1 + x1 δ1 + x2 δ2 + x3 δ1 δ2 ), (a1 + x1 δ1 + y2 δ2 + y3 δ1 δ2 )) =
= a1 + x1 δ1 + (x2 + y2 )δ2 + (x3 + y3 )δ1 δ2 .
(T +)((a1 + x1 δ1 + x2 δ2 + x3 δ1 δ2 ), (a1 + y1 δ1 + x2 δ2 + y3 δ1 δ2 )) =
= a1 + (x1 + y1 )δ1 + x2 δ2 + (x3 + y3 )δ1 δ2 .
The space of all natural transformations Nat(T, T 2 ) ∼ Hom(D, D2 ) turns out to
=
2 2
be the real algebraic variety R ∪R R consisting of all homomorphisms δ ’ x1 δ1 +
x2 δ2 + x3 δ1 δ2 with x1 x2 = 0, since δ 2 = 0. The homomorphism δ ’ xδ1 + yδ1 δ2
corresponds to the natural transformation (+T ) —¦ (vl —¦ my , 0T —¦ mx ), and the
homomorphism δ ’ xδ2 + yδ1 δ2 corresponds to (T +) —¦ (vl —¦ my , T 0 —¦ mx ). So any
element in Nat(T, T 2 ) can be expressed in terms of the natural transformations
{0T, T 0, (T +), (+T ), T π, πT, vl, m» for » ∈ R}.
Similarly Nat(T 2 , T 2 ) ∼ Hom(D2 , D2 ) turns out to be the real algebraic vari-
=
ety (R ∪R R ) — (R ∪R R2 ) consisting of all
2 2 2


δ1 x1 δ1 + x2 δ2 + x3 δ1 δ2

δ2 y 1 δ 1 + y2 δ 2 + y 3 δ 1 δ 2
with x1 x2 = y1 y2 = 0. Again any element of Nat(T 2 , T 2 ) can be written in
terms of {0T, T 0, (T +), (+T ), T π, πT, κ, m» T, T m» for » ∈ R}. If for example
x2 = y1 = 0 then the corresponding transformation is
(+T ) —¦ (my2 T —¦ T mx1 , (T +) —¦ (vl —¦ + —¦ (mx3 —¦ πT, my3 —¦ T π), 0T —¦ mx1 —¦ πT )).
Note also the relations T π —¦ κ = πT , κ —¦ (T +) = (+T ) —¦ (κ — κ), κ —¦ vl = vl,
κ —¦ T m» = m» T ; so κ interchanges the two vector bundle structures on T 2 ’ T ,
namely ((+T ), m» T, πT ) and ((T +), T m» , T π), and vl : T ’ T 2 is linear for
both of them. The reader is advised now to have again a look at 6.12.
37.5. In the situation of 37.3 we let now F1 = F be a general Weil functor and
F2 = T . So we consider T —¦ F which is isomorphic to F —¦ T via κF,T . In general
we have (F1 —Id F2 ) — F = F1 — F —Id F2 — F , so + : T —Id T ’ T induces a ¬ber
addition (+ — F ) : T — F —Id T — F ’ T — F , and m» — F : T — F ’ T — F is a
¬ber scalar multiplication. So T — F is a vector bundle functor on the category
Mf which can be described in terms of lemma 37.2 as follows.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
37. Examples and applications 321


Lemma. In the notation of lemma 37.2 we have T — F ∼ N — T , where N is
=¯ ¯
the underlying vector spaces of the nilradical N (F ) of F .
Proof. The Weil algebra of T — F is R.1 • (N (F ) — N (T )) by 37.3.(5). We have
¯
N (F ) — N (T ) = N (F ) — R.δ = N as vector space, and the multiplication on
N (F ) — N (T ) is zero.
37.6. Sections and expansions. For a Weil functor F with Weil algebra
A = R.1 • N and for a manifold M we denote by XF (M ) the space of all smooth
sections of πF,M : F (M ) ’ M . Note that this space is in¬nite dimensional in
general. Recall from theorem 35.14 that
·M,A
F (M ) = TA (M ) ← ’ Hom(C ∞ (M, R), A)
’’

is an isomorphism. For f ∈ C ∞ (M, R) we can decompose F (f ) = TA (f ) :
F (M ) ’ F (R) = A = R.1 • N into

F (f ) = TA (f ) = (f —¦ π) • N (f ),
N (f ) : F (M ) ’ N.

Lemma.
(1) Each Xx ∈ F (M )x = π ’1 (x) for x ∈ M de¬nes an R-linear mapping

DXx : C ∞ (M, R) ’ N,
DXx (f ) := N (f )(Xx ) = F (f )(Xx ) ’ f (x).1,

which satis¬es

DXx (f.g) = DXx (f ).g(x) + f (x).DXx (g) + DXx (f ).DXx (g).

We call this the expansion property at x ∈ M .
(2) Each R-linear mapping ξ : C ∞ (M, R) ’ N which satis¬es the expansion
property at x ∈ M is of the form ξ = DXx for a unique Xx ∈ F (M )x .
(3) The R-linear mappings ξ : C ∞ (M, R) ’ C ∞ (M, N ) = N — C ∞ (M, R)
which have the expansion property

f, g ∈ C ∞ (M, R),
(a) ξ(f.g) = ξ(f ).g + f.ξ(g) + ξ(f ).ξ(g),

are exactly those induced (via 1 and 2) by the smooth sections of π :
F (M ) ’ M .

Linear mappings satisfying the expansion property 1 will be called expansions:
if N is generated by δ with δ k+1 = 0, so that F (M ) = J0 (R, M ), then these
k

are parametrized Taylor expansions of f to order k (applied to a k-jet of a
curve through each point). For X ∈ XF (M ) we will write DX : C ∞ (M, R) ’

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
322 Chapter VIII. Product preserving functors


C ∞ (M, N ) = N — C ∞ (M, R) for the expansion induced by X. Note the de¬ning
equation

F (f ) —¦ X = f.1 + DX (f ) = (f.1, DX (f )) or
(b)
’1
f (x).1 + DX (f )(x) = F (f )(X(x)) = ·M,A (X(x))(f ).

Proof. (1) and (2). For • ∈ Hom(C ∞ (M, R), A) = F (M ) we consider the foot
point π(·M,A (•)) = ·M,R (π(•)) = x ∈ M and ·M,A (•) = Xx ∈ F (M )x . Then
we have •(f ) = TA (f )(Xx ) and the expansion property for DXx is equivalent to
•(f.g) = •(f ).•(g).
(3) For each x ∈ M the mapping f ’ ξ(f )(x) ∈ N is of the form DX(x) for a
unique X(x) ∈ F (M )x by 1 and 2, and clearly X : M ’ F (M ) is smooth.
37.7. Theorem. Let F be a Weil functor with Weil algebra A = R.1 • N .
Using the natural transformations from 37.3 we have:
(1) XF (M ) is a group with multiplication X Y = µF —¦F (Y )—¦X and identity
0F .
(2) XT —F (M ) is a Lie algebra with bracket induced from the usual Lie bracket
on XT (M ) and the multiplication m : N —N ’ N by [a—X, b—Y ]T —F =
a.b — [X, Y ].
(3) There is a bijective mapping exp : XT —F (M ) ’ XF (M ) which expresses
the multiplication by the Baker-Campbell-Hausdor¬ formula.
(4) The multiplication , the Lie bracket [ , ]T —F , and exp are natural
in F (with respect to natural operators) and M (with respect to local
di¬eomorphisms).

Remark. If F = T , then XT (M ) is the space of all vector ¬elds on M , the
multiplication is X Y = X + Y , and the bracket is [X, Y ]T —T = 0, and exp is
the identity. So the multiplication in (1), which is commutative only if F is a
natural vector bundle, generalizes the linear structure on X(M ).
37.8. For the proof of theorem 37.7 we need some preparation. If a ∈ N and
X ∈ X(M ) is a smooth vector ¬eld on M , then by lemma 37.5 we have a — X ∈
XT —F (M ) and for f ∈ C ∞ (M, R) we use T f (X) = f.1 + df (X) to get

(T — F )(f )(a — X) = (IdN —T f )(a — X)
= f.1 + a.df (X) = f.1 + a.X(f )
= f.1 + Da—X (f ) by 37.6.(b). Thus
T —F
(a) Da—X (f ) = Da—X (f ) = a.X(f ) = a.df (X).

So again by 37.5 we see that XT —F (M ) is isomorphic to the space of all R-linear
mappings ξ : C ∞ (M, R) ’ N — C ∞ (M, R) satisfying

ξ(f.g) = ξ(f ).g + f.ξ(g).

These mappings are called derivations.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
37. Examples and applications 323


Now we denote L := LR (C ∞ (M, R), N — C ∞ (M, R)) for short, and for ξ,
· ∈ L we de¬ne

(b) ξ • · := (m — IdC ∞ (M,R) ) —¦ (IdN —ξ) —¦ · : C ∞ (M, R) ’
’ N — C ∞ (M, R) ’ N — N — C ∞ (M, R) ’ N — C ∞ (M, R),

where m : N — N ’ N is the (nilpotent) multiplication on N . Note that
DF : XF (M ) ’ L and DT —F : XT —F (M ) ’ L are injective linear mappings.

37.9. Lemma. 1. L is a real associative nilpotent algebra without unit under
the multiplication •, and it is commutative if and only if m = 0 : N — N ’ N .
F F F F F
(1) For X, Y ∈ XF (M ) we have DX Y = DX • DY + DX + DY .
F F F F F
(2) For X, Y ∈ XT —F (M ) we have D[X,Y ]T —F = DX • DY ’ DY • DX .
(3) For ξ ∈ L de¬ne


1 •i
exp(ξ) := ξ
i!
i=1

(’1)i’1 •i
log(ξ) := ξ.
i
i=1


Then exp, log : L ’ L are bijective and inverse to each other. exp(ξ) is
an expansion if and only if ξ is a derivation.

Note that i = 0 lacks in the de¬nitions of exp and log, since L has no unit.

Proof. (1) We use that m is associative in the following computation.

ξ • (· • ζ) = (m — IdC ∞ (M,R) ) —¦ (IdN —ξ) —¦ (· • ζ)
= (m — Id) —¦ (IdN —ξ) —¦ (m — Id) —¦ (IdN —·) —¦ ζ
= (m — Id) —¦ (m — IdN — Id) —¦ (IdN —N —ξ) —¦ (IdN —·) —¦ ζ
= (m — Id) —¦ (IdN —m — Id) —¦ (IdN —N —ξ) —¦ (IdN —·) —¦ ζ
= (m — Id) —¦ (IdN — (m — Id) —¦ (IdN —ξ) —¦ · —¦ ζ
= (ξ • ·) • ζ.

So • is associative, and it is obviously R-bilinear. The order of nilpotence equals
that of N .
(2) Recall from 36.13 and 37.3 that

F (F (R)) = A — A = (R.1 — R.1) • (R.1 — N ) • (N — R.1) • (N — N )
∼ A • F (N ) ∼ F (R.1) — F (N ) ∼ F (R.1 — N ) = F (A).
= = =

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
324 Chapter VIII. Product preserving functors


We will use this decomposition in exactly this order in the following computation.

F
(f ) = F (f ) —¦ (X
f.1 + DX Y) by 37.6.(b)
Y
= F (f ) —¦ µF,M —¦ F (Y ) —¦ X by 37.7(1)
= µF,R —¦ F (F (f )) —¦ F (Y ) —¦ X since µ is natural
= m —¦ F (F (f ) —¦ Y ) —¦ X
F
= m —¦ F (f.1, DY (f )) —¦ X by 37.6.(b)
F
= m —¦ (1 — F (f ) —¦ X) • (F (DY (f )) —¦ X)
F F F F
= m —¦ 1 — (f.1 + DX (f )) + DY (f ) — 1 + (IdN —DX )(DY (f ))
F F F F
= f.1 + DX (f ) + DY (f ) + (DX • DY )(f ).

(3) For vector ¬elds X, Y ∈ X(M ) on M and a, b ∈ N we have

T —F
D[a—X,b—Y ]T —F (f ) = Da.b—[X,Y ] (f )
= a.b.[X, Y ](f ) by 37.8.(a)
= a.b.(X(Y (f )) ’ Y (X(f )))
T —F T —F
= (m — IdC ∞ (M,R) ) —¦ (IdN —Da—X ) —¦ Db—Y (f ) ’ . . .
T —F T —F T —F T —F
= (Da—X • Db—Y ’ Db—Y • Da—X )(f ).

(4) After adjoining a unit to L we see that exp(ξ) = eξ ’ 1 and log(ξ) =
log(1 + ξ). So exp and log are inverse to each other in the ring of formal power
series of one variable. The elements 1 and ξ generate a quotient of the power
series ring in R.1 • L, and the formal expressions of exp and log commute with
’1
taking quotients. So exp = log . The second assertion follows from a direct
formal computation, or also from 37.10 below.

37.10. We consider now the R-linear mapping C of L in the ring of all R-linear
endomorphisms of the algebra A — C ∞ (M, R), given by

Cξ := m —¦ (IdA —ξ) : A — C ∞ (M, R) ’
’ A — N — C ∞ (M, R) ‚ A — A — C ∞ (M, R) ’ A — C ∞ (M, R),

where m : A — A ’ A is the multiplication. We have Cξ (a — f ) = a.ξ(f ).

Lemma.
(1) Cξ•· = Cξ —¦ C· , so C is an algebra homomorphism.
(2) ξ ∈ L is an expansion if and only if Id +Cξ is an automorphism of the
commutative algebra A — C ∞ (M, R).
(3) ξ ∈ L is a derivation if and only if Cξ is a derivation of the algebra
A — C ∞ (M, R).

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

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