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

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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

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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.

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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) ’

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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).

= = =

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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