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CHAPTER VIII. PRODUCT PRESERVING FUNCTORS

Back

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 functorsMf !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 mathematical

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

of algebra homomorphisms Hom(C1(M;R);R) equals M as follows. The kernel

of a homomorphism ' : C1(M;R) ! R is an ideal of codimension 1 in

C1(M;R). The zero sets Zf := f􀀀1(0) for f 2 ker ' form a _lter of closed

sets, since Zf \ Zg = Zf2+g2 , which contains a compact set Zf for a function

f which is unbounded on each non compact closed subset. Thus

f2ker ' Zf is

not empty, it contains at least one point x0. But then for any f 2 C1(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(C1(M;R);C1(N;R)) = C1(N;M). So

the category of algebras C1(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 " with "2 = 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(C1(M;R);D) =

TM, the tangent bundle of M. For if ' is a homomorphism C1(M;R) ! D,

then _ _ ' : C1(M;R) ! D ! R equals evx for some x 2 M and '(f) 􀀀

f(x):1 = X(f):", 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(C1(M;R);D D) = TTM.

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 nilpotent

elements of A. We will show in this chapter, that Hom(C1(M;R);A) =

Hom(C1(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 construction

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 characteristic

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_erential

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_a_r, 86].

We will discuss thoroughly natural transformations betweenWeil 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 2 A the element 1 + a21

+ _ _ _ + a2

n is invertible in A. Let E = fe 2

A : e2 = e; e 6= 0g _ A be the set of all nonzero idempotent elements in A. It is

not empty since 1 2 E. An idempotent e 2 E is said to be minimal if for any

e0 2 E we have ee0 = e or ee0 = 0.

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

and _nite dimensional as a real vector space.

Then there is a decomposition 1 = e1 +_ _ _+ek into all minimal idempotents.

Furthermore A = A1 _ _ _ _ _ Ak, where Ai = eiA = R _ ei _ Ni, and Ni is a

nilpotent ideal.

Proof. First we remark that every system of nonzero idempotents e1; : : : ; er

satisfying eiej = 0 for i 6= j is linearly independent over R. Indeed, if we multiply

a linear combination k1e1 + _ _ _ + krer = 0 by ei we obtain ki = 0. Consider a

non minimal idempotent e 6= 0. Then there exists e0 2 E with e 6= ee0 =: _e 6= 0.

Then both _e and e􀀀_e are nonzero idempotents and _e(e􀀀_e) = 0. To deduce the

required decomposition of 1 we proceed by recurrence. Assume that we have a

decomposition 1 = e1 + _ _ _ + er into nonzero idempotents satisfying eiej = 0

for i 6= j. If ei is not minimal, we decompose it as ei = _ei + (ei 􀀀 _ei) as above.

The new decomposition of 1 into r + 1 idempotents is of the same type as the

original one. Since A is _nite dimensional this proceedure stabilizes. This yields

1 = e1 + _ _ _ + ek with minimal idempotents. Multiplying this relation by a

minimal idempotent e, we _nd that e appears exactly once in the right hand

side. Then we may decompose A as A = A1 _ _ _ _ _ Ak, where Ai := eiA.

Now each Ai has only one nonzero idempotent, namely ei, and it su_ces to

investigate each Ai separately. To simplify the notation we suppose that A = Ai,

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298 Chapter VIII. Product preserving functors

so that now 1 is the only nonzero idempotent of A. Let N := fn 2 A : nk =

0 for some kg be the ideal of all nilpotent elements in A.

We claim that any x 2 A n N is invertible. If not then xA _ A is a proper

ideal, and since A is _nite dimensional the decreasing sequence

A _ xA _ x2A _ _ _ _

of ideals must become stationary. If xkA = 0 then x 2 N, thus there is a k such

that xk+`A = xkA 6= 0 for all ` > 0. Then x2kA = xkA and there is some y 2 A

with xk = x2ky. So we have (xky)2 = xky 6= 0, and since 1 is the only nontrivial

idempotent of A we have xky = 1. So xk􀀀1y is an inverse of x as required.

So the quotient algebra A=N is a _nite dimensional _eld, so A=N equals R

or C. If A=N = C, let x 2 A be such that x + N =

􀀀1 2 C = A=N. Then

1 + x2 + N = N = 0 in C, so 1 + x2 is nilpotent and A cannot be formally real.

Thus A=N = R and A = R _ 1 _ N as required. _

35.2. De_nition. A Weil algebra A is a real commutative algebra with unit

which is of the form A = R _ 1 _ N, where N is a _nite dimensional ideal of

nilpotent elements.

So by lemma 35.1 a formally real and _nite dimensional unital commutative

algebra is the direct sum of _nitely many Weil algebras.

35.3. Some algebraic preliminaries. Let A be a commutative algebra with

unit and let M be a module over A. The semidirect product A[M] of A and M

or the idealisator of M is the algebra (A _M;+; _), where (a1;m1) _ (a2;m2) =

(a1a2; a1m2 + a2m1). Then M is a (nilpotent) ideal of A[M].

Let Mm_n = f(tij) : tij 2 M; 1 _ i _ m; 1 _ j _ ng be the space of all

(m _ n)-matrices with entries in the module M. If S 2 Ar_m and T 2 Mm_n

then the product of matrices ST 2 Mr_n is de_ned by the usual formula.

For a matrix U = (uij) 2 An_n the determinant is given by the usual formula

det(U) =

_2Sn

sign _

i=1 ui;_(i). It is n-linear and alternating in the columns

of U.

Lemma. If m = (mi) 2 Mn_1 is a column vector of elements in the A-module

M and if U = (uij) 2 An_n is a matrix with Um = 0 2 Mn_1 then we have

det(U)mi = 0 for each i.

Proof. We may compute in the idealisator A[M], or assume without loss of generality

that all mi 2 A. Let u_j denote the j-th column of U. Then

uijmj = 0

for all i means that m1u_1 = 􀀀

j>1mju_j , thus

det(U)m1 = det(m1u_1; u_2; : : : ; u_n)

= det(􀀀

j>1mju_j ; u_2; : : : ; u_n) = 0 _

Lemma. Let I be an ideal in an algebra A and let M be a _nitely generated

A-module. If IM = M then there is an element a 2 I with (1 􀀀 a)M = 0.

Proof. Let M =

i=1 Ami for generators mi 2 M. Since IM = M we have

mi =

j=1 tijmj for some T = (tij) 2 In_n. This means (1n 􀀀 T)m = 0 for

m = (mj) 2 Mn_1. By the _rst lemma we get det(1n 􀀀T)mj = 0 for all j. But

det(1n 􀀀 T) = 1 􀀀 a for some a 2 I. _

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35. Weil algebras and Weil functors 299

Lemma of Nakayama. Let (A; I) be a local algebra (i.e. an algebra with a

unique maximal ideal I) and let M be an A-module. Let N1;N2 _ M be

submodules with N1 _nitely generated. If N1 _ N2+IN1 then we have N1 _ N2.

In particular IN1 = N1 implies N1 = 0.

Proof. Let IN1 = N1. By the lemma above there is some a 2 I with (1􀀀a)N1 =

0. Since I is a maximal ideal (so A=I is a _eld), 1 􀀀 a is invertible. Thus

N1 = 0. If N1 _ N2 + IN1 we have I((N1 + N2)=N2) = (N1 + N2)=N2 thus

(N1 + N2)=N2 = 0 or N1 _ N2. _

35.4. Lemma. Any ideal I of _nite codimension in the algebra of germs

En := C1

0 (Rn;R) contains some power Mk

n of the maximal ideal Mn of germs

vanishing at 0.

Proof. Consider the chain of ideals En _ I +Mn _ I +M2

_ _ _ _ . Since I has

_nite codimension we have I+Mk

n = I+Mk+1

n for some k. SoMk

_ I+MnMk

which implies Mk

_ I by the lemma of Nakayama 35.3 since Mk

n is _nitely

generated by the monomials of order k in n variables. _

35.5. Theorem. Let A be a unital real commutative algebra. Then the following

assertions are equivalent.

(1) A is a Weil algebra.

(2) A is a _nite dimensional quotient of an algebra of germs En = C1

0 (Rn;R)

for some n.

(3) A is a _nite dimensional quotient of an algebra R[X1; : : : ;Xn] of polynomials.

(4) A is a _nite dimensional quotient of an algebra R[[X1; : : : ;Xn]] of formal

power series.

(5) A is a quotient of an algebra Jk

0 (Rn;R) of jets.

Proof. Let A = R _ 1 _ N, where N is the maximal ideal of nilpotent elements,

which is generated by _nitely many elements, say X1; : : : ;Xn. Since

R[X1; : : : ;Xn] is the free real unital commutative algebra generated by these

elements, A is a quotient of this polynomial algebra. There is some k such that

xk+1 = 0 for all x 2 N, so A is even a quotient of the jet algebra Jk

0 (Rn;R).

Since the jet algebra is itself a quotient of the algebra of germs and the algebra

of formal power series, the same is true for A. That all these _nite dimensional

quotients are Weil algebras is clear, since they all are formally real and have

only one nonzero idempotent. _

If A is a quotient of the jet algebra Jr

0 (Rn;R), we say that the order of A is

at most r.

35.6. The width of a Weil algebra. Consider the square N2 of the nilpotent

ideal N of a Weil algebra A. The dimension of the real vector space N=N2 is

called the width of A.

Let M _ R[x1; : : : ; xn] denote the ideal of all polynomials without constant

term and let I _ R[x1; : : : ; xn] be an ideal of _nite codimension which

is contained in M2. Then the width of the factor algebra A = R[x1; : : : ; xn]=I

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300 Chapter VIII. Product preserving functors

is n. Indeed the nilpotent ideal of A is M=I and (M=I)2 = M2=I, hence

(M=I)=(M=I)2 _= M=M2 is of dimension n.

35.7. Proposition. If M is a smooth manifold and I is an ideal of _nite

codimension in the algebra C1(M;R), then C1(M;R)=I is a direct sum of

_nitely many Weil algebras.

If A is a _nite dimensional commutative real algebra with unit, then we have

Hom(C1(M;R);A) = Hom(C1(M;R);W(A)), where W(A) is the subalgebra

of A generated by all idempotent and nilpotent elements of A (the so-called Weil

part of A). In particular W(A) is formally real.

Proof. The algebra C1(M;R) is formally real, so the _rst assertion follows from

lemma 35.1. If ' : C1(M;R) ! A is an algebra homomorphism, then the kernel

of ' is an ideal of _nite codimension in C1(M;R), so the image of ' is a direct

sum of Weil algebras and is thus generated by its idempotent and nilpotent

elements. _

35.8. Lemma. Let M be a smooth manifold and let ' : C1(M;R) ! A be an

algebra homomorphism into a Weil algebra A.

Then there is a point x 2 M and some k _ 0 such that ker ' contains the

ideal of all functions which vanish at x up to order k.

Proof. Since '(1) = 1 the kernel of ' is a nontrivial ideal in C1(M;R) of _nite

codimension.

If _ is a closed subset of M we let C1(_;R) denote the algebra of all real

valued functions on _ which are restrictions of smooth functions on M. For a

smooth function f let Zf := f􀀀1(0) be its zero set. For a subset S _ C1(_;R)

we put ZS :=

fZf : f 2 Sg.

Claim 1. Let I be an ideal of _nite codimension in C1(_;R). Then ZI is a

_nite subset of _ and ZI = ; if and only if I = C1(_;R).

ZI is _nite since C1(_;R)=I is _nite dimensional. Zf = ; implies that f is

invertible. So if I 6= C1(_;R) then fZf : f 2 Ig is a _lter of nonempty closed

sets, since Zf \ Zg = Zf2+g2 . Let h 2 C1(M;R) be a positive proper function,

i.e. inverse images under h of compact sets are compact. The square of the

geodesic distance with respect to a complete Riemannian metric on a connected

manifold M is such a function. Then we put f = hj_ 2 C1(_;R). The sequence

f; f2; f3; : : : is linearly dependent mod I, since I has _nite codimension, so

g =

i=1 _ifi 2 I for some (_i) 6= 0 in Rn. Then clearly Zg is compact. So this

_lter of closed nonempty sets contains a compact set and has therefore nonempty

intersection ZI =

f2I Zf .

Claim 2. If I is an ideal of _nite codimension in C1(M;R) and if a function

f 2 C1(M;R) vanishes near ZI , then f 2 I.

Let ZI _ U1 _ U1 _ U2 where U1 and U2 are open in M such that fjU2 = 0.

The restriction mapping C1(M;R) ! C1(M n U1;R) is a surjective algebra

homomorphism, so the image I0 of I is again an ideal of _nite codimension in

C1(MnU1;R). But clearly ZI0 = ;, so by claim 1 we have I0 = C1(MnU1;R).

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

35. Weil algebras and Weil functors 301

Thus there is some g 2 I such that gj(M n U1) = fj(M n U1). Now choose

h 2 C1(M;R) such that h = 0 on U1 and h = 1 o_ U2. Then f = fh = gh 2 I.

Claim 3. For the ideal ker ' in C1(M;R) the zero set ZI consists of one point

x only.

Since ker ' is a nontrivial ideal of _nite codimension, Zker ' is not empty and

_nite by claim 1. For any function f 2 C1(M;R) which is 1 or 0 near the points

in Zker ' the element '(f) is an idempotent of the Weil algebra A. Since 1 is

the only nonzero idempotent of A, the zero set ZI consists of one point.

Now by claims 2 and 3 the ideal ker ' contains the ideal of all functions which

vanish near x. So ' factors to the algebra C1

x (M;R) of germs at x, compare

35.5.(2). Now ker ' _ C1

x (M;R) is an ideal of _nite codimension, so by lemma

35.4 the result follows. _

35.9. Corollary. The evaluation mapping ev : M ! Hom(C1(M;R);R),

given by ev(x)(f) := f(x), is bijective.

This result is sometimes called the exercise of Milnor, see [Milnor-Stashe_,

74, p. 11]. Another (similar) proof of it can be found in the mathematical short

story in the introduction to chapter VIII.

Proof. By lemma 35.8, for every ' 2 Hom(C1(M;R);R) there is an x 2 M

and a k _ 0 such that ker ' contains the ideal of all functions vanishing at

x up to order k. Since the codimension of ker ' is 1, we have ker ' = ff 2

C1(M;R) : f(x) = 0g. Then for any f 2 C1(M;R) we have f 􀀀f(x)1 2 ker ',

so '(f) = f(x). _

35.10. Corollary. For two manifolds M1 and M2 the mapping

C1(M1;M2) ! Hom(C1(M2;R);C1(M1;R))

f 7! (f_ : g 7! g _ f)

is bijective.

Proof. Let x1 2 M1 and ' 2 Hom(C1(M2;R);C1(M1;R)). Then evx1

_ '

is in Hom(C1(M2;R);R), so by 35.9 there is a unique x2 2 M2 such that

evx1

_ ' = evx2 . If we write x2 = f(x1), then f : M1 ! M2 and '(g) = g _ f for

all g 2 C1(M2;R). This also implies that f is smooth. _

35.11. Chart description of Weil functors. Let A = R _ 1 _ N be a Weil

algebra. We want to associate to it a functor TA :Mf !Mf from the category

Mf of all _nite dimensional second countable manifolds into itself. We will give

several descriptions of this functor, and we begin with the most elementary and

basic construction, the idea of which goes back to [Weil, 53].

Step 1. If p(t) is a real polynomial, then for any a 2 A the element p(a) 2 A is

uniquely de_ned; so we have a (polynomial) mapping TA(p) : A ! A.

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

302 Chapter VIII. Product preserving functors

Step 2. If f 2 C1(R;R) and _1 + n 2 R _ 1 _ N = A, we consider the Taylor

expansion j1f(_)(t) =

j=0

f(j)(_)

j! tj of f at _ and we put

TA(f)(_1 + n) := f(_)1 +

j=1

f(j)(_)

j! nj ;

which is _nite sum, since n is nilpotent. Then TA(f) : A ! A is smooth and we

get TA(f _ g) = TA(f) _ TA(g) and TA(IdR) = IdA.

Step 3. For f 2 C1(Rm;R) we want to de_ne the value of TA(f) at the vector

(_11 + n1; : : : ; _m1 + nm) 2 Am = A _ : : : _ A. Let again j1 P f(_)(t) =

_2Nm

_!d_f(_)t_ be the Taylor expansion of f at _ 2 Rm for t 2 Rm. Then

we put

TA(f)(_11 + n1; : : : ; _m1 + nm) := f(_)1 +

j_j_1

_!d_f(_)n_1

1 : : : n_m

m ;

which is again a _nite sum.

Step 4. For f 2 C1(Rm;Rk) we apply the construction of step 3 to each component

fj : Rm ! R of f to de_ne TA(f) : Am ! Ak.

Since the Taylor expansion of a composition is the composition of the Taylor

expansions we have TA(f _ g) = TA(f) _ TA(g) and TA(IdRm) = IdAm.

If ' : A ! B is a homomorphism between two Weil algebras we have 'k _

TAf = TBf _ 'm for f 2 C1(Rm;Rk).

Step 5. Let _ = _A : A ! A=N = R be the projection onto the quotient _eld

of the Weil algebra A. This is a surjective algebra homomorphism, so by step 4

the following diagram commutes for f 2 C1(Rm;Rk):

Am w

TAf

_kA

Rm w

f Rk

If U _ Rm is an open subset we put TA(U) := (_m

A )􀀀1(U) = U _ Nm, which is

an open subset in TA(Rm) := Am. If f : U ! V is a smooth mapping between

open subsets U and V of Rm and Rk, respectively, then the construction of steps

3 and 4, applied to the Taylor expansion of f at points in U, produces a smooth

mapping TAf : TAU ! TAV , which _ts into the following commutative diagram:

U _ Nm[[[]pr1

TAU w

TAf

TAV

_kA

V _ Nk

pr1

U w

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

35. Weil algebras and Weil functors 303

We have TA(f _ g) = TAf _ TAg and TA(IdU) = IdTAU, so TA is now a covariant

functor on the category of open subsets of Rm's and smooth mappings between

them.

Step 6. In 1.14 we have proved that the separable connected smooth manifolds

are exactly the smooth retracts of open subsets in Rm's. If M is a smooth

manifold, let i : M ! Rm be an embedding, let i(M) _ U _ Rm be a tubular

neighborhood and let q : U ! U be the projection of U with image i(M). Then

q is smooth and q _ q = q. We de_ne now TA(M) to be the image of the smooth

retraction TAq : TAU ! TAU, which by 1.13 is a smooth submanifold.

If f : M ! M0 is a smooth mapping between manifolds, we de_ne TAf :

TAM ! TAM0 as

TAM _ TAU

TA(i0_f_q)

􀀀􀀀􀀀􀀀􀀀􀀀􀀀! TAU0 TAq0

􀀀􀀀􀀀! TAU0;

which takes values in TAM0.

It remains to show, that another choice of the data (i; U; q;Rm) for the manifold

M leads to a di_eomorphic submanifold TAM, and that TAf is uniquely

de_ned up to conjugation with these di_eomorphisms for M and M0. Since this

is a purely formal manipulation with arrows we leave it to the reader and give

instead the following:

Step 6'. Direct construction of TAM for a manifold M using atlases.

Let M be a smooth manifold of dimension m, let (U_; u_) be a smooth atlas

of M with chart changings u__ := u_ _ u􀀀1

_ : u_(U__) ! u_(U__). Then the

smooth mappings

TA(u_(U__)) w

TA(u__)

TA(u_(U__))

u_(U__) w

u__

u_(U__)

form again a cocycle of chart changings and we may use them to glue the open

sets TA(u_(U_)) = u_(U_) _ Nm _ TA(Rm) = Am in order to obtain a smooth

manifold which we denote by TAM. By the diagram above we see that TAM

will be the total space of a _ber bundle T(_A;M) = _A;M : TAM ! M, since

the atlas (TA(U_); TA(u_)) constructed just now is already a _ber bundle atlas.

Thus TAM is Hausdor_, since two points xi can be separated in one chart if

they are in the same _ber, or they can be separated by inverse images under

_A;M of open sets in M separating their projections.

This construction does not depend on the choice of the atlas. For two atlases

have a common re_nement and one may pass to this.

If f 2 C1(M;M0) for two manifolds M, M0, we apply the functor TA to

the local representatives of f with respect to suitable atlases. This gives local

representatives which _t together to form a smooth mapping TAf : TAM !

TAM0. Clearly we again have TA(f _ g) = TAf _ TAg and TA(IdM) = IdTAM, so

that TA :Mf !Mf is a covariant functor.

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304 Chapter VIII. Product preserving functors

35.12. Remark. If we apply the construction of 35.11, step 6' to the algebra

A = 0, which we did not allow (1 6= 0 2 A), then T0M depends on the choice

of the atlas. If each chart is connected, then T0M = _0(M), computing the

connected components of M. If each chart meets each connected component of

M, then T0M is one point.

35.13. Theorem. Main properties of Weil functors. Let A = R _ 1 _ N

be a Weil algebra, where N is the maximal ideal of nilpotents. Then we have:

1. The construction of 35.11 de_nes a covariant functor TA : Mf ! Mf

such that (TAM; _A;M;M;NdimM) is a smooth _ber bundle with standard _ber

NdimM. For any f 2 C1(M;M0) we have a commutative diagram

TAM w

TAf

_A;M

TAM0

_A;M0

M w

M0.

So (TA; _A) is a bundle functor on Mf, which gives a vector bundle on Mf if

and only if N is nilpotent of order 2.

2. The functor TA : Mf ! Mf is multiplicative: it respects products.

It maps the following classes of mappings into itself: immersions, initial immersions,

embeddings, closed embeddings, submersions, surjective submersions,

_ber bundle projections. It also respects transversal pullbacks, see 2.19. For

_xed manifolds M and M0 the mapping TA : C1(M;M0) ! C1(TAM; TAM0) is

smooth, i.e. it maps smoothly parametrized families into smoothly parametrized

families.

3. If (U_) is an open cover of M then TA(U_) is also an open cover of TAM.

4. Any algebra homomorphism ' : A ! B between Weil algebras induces

a natural transformation T('; ) = T' : TA ! TB. If ' is injective, then

T(';M) : TAM ! TBM is a closed embedding for each manifold M. If ' is

surjective, then T(';M) is a _ber bundle projection for each M. So we may

view T as a co-covariant bifunctor from the category of Weil algebras timesMf

to Mf.

Proof. 1. The main assertion is clear from 35.11. The _ber bundle _A;M :

TAM ! M is a vector bundle if and only if the transition functions TA(u__) are

_ber linear NdimM ! NdimM. So only the _rst derivatives of u__ should act on

N, so any product of two elements in N must be 0, thus N has to be nilpotent

of order 2.

2. The functor TA respects products in the category of open subsets of Rm's

by 35.11, step 4 and 5. All the other assertions follow by looking again at the

chart structure of TAM and by taking into account that f is part of TAf (as the

base mapping).

3. This is obvious from the chart structure.

4. We de_ne T(';Rm) := 'm : Am ! Bm. By 35.11, step 4, this restricts to

a natural transformation TA ! TB on the category of open subsets of Rm's and

by gluing also on the category Mf. Obviously T is a co-covariant bifunctor on

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

35. Weil algebras and Weil functors 305

the indicated categories. Since _B _ ' = _A (' respects the identity), we have

T(_B;M) _T(';M) = T(_A;M), so T(';M) : TAM ! TBM is _ber respecting

for each manifold M. In each _ber chart it is a linear mapping on the typical

_ber NdimM

! NdimM

B .

So if ' is injective, T(';M) is _berwise injective and linear in each canonical

_ber chart, so it is a closed embedding.

If ' is surjective, let N1 := ker ' _ NA, and let V _ NA be a linear complement

to N1. Then for m = dimM and for the canonical charts we have the

commutative diagram:

TAM w

T(';M)

TBM

TA(U_) w

T(';U_)

TA(u_)

TB(U_)

TB(u_)

u_(U_) _ Nm

A w

Id_('jNA)m

u_(U_) _ Nm

_ V m w

Id_0 _ Iso

u_(U_) _ 0 _ Nm

So T(';M) is a _ber bundle projection with standard _ber (ker ')m. _

35.14. Theorem. Algebraic description of Weil functors. There are

bijective mappings _M;A : Hom(C1(M;R);A) ! TA(M) for all smooth manifolds

M and all Weil algebras A, which are natural in M and A. Via _ the

set Hom(C1(M;R);A) becomes a smooth manifold and Hom(C1( ;R);A) is

a global expression for the functor TA.

Proof. Step 1. Let (xi) be coordinate functions on Rn. By lemma 35.8 for

' 2 Hom(C1(Rn;R);A) there is a point x(') = (x1('); : : : ; xn(')) 2 Rn such

that ker ' contains the ideal of all f 2 C1(Rn;R) vanishing at x(') up to some

order k, so that '(xi) = xi(') _ 1 + '(xi 􀀀 xi(')), the latter summand being

nilpotent in A of order _ k. Applying ' to the Taylor expansion of f at x(')

up to order k with remainder gives

'(f) =

j_j_k

@j_jf

@x_ (x(')) '(x1 􀀀 x1('))_1 : : : '(xn 􀀀 xn('))_n

= TA(f)('(x1); : : : ; '(xn)):

So ' is uniquely determined by the elements '(xi) in A and the mapping

_Rn;A : Hom(C1(Rn;R);A) ! An;

_(') := ('(x1); : : : ; '(xn))

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306 Chapter VIII. Product preserving functors

is injective. Furthermore for g = (g1; : : : ; gm) 2 C1(Rn;Rm) and coordinate

functions (y1; : : : ; ym) on Rm we have

(_Rm;A _ (g_)_)(') = ('(y1 _ g); : : : ; '(ym _ g))

= ('(g1); : : : ; '(gm))

􀀀

TA(g1)('(x1); : : : ; '(xn)); : : : ; TA(gm)('(x1); : : : ; '(xn))

;

so _Rn;A is natural in Rn. It is also bijective since any (a1; : : : ; an) 2 An

de_nes a homomorphism ' : C1(Rn;R) ! A by the prescription '(f) :=

TAf(a1; : : : ; an).

Step 2. Let i : U ! Rn be the embedding of an open subset. Then the image of

the mapping

Hom(C1(U;R);A) (i_)_

􀀀􀀀􀀀! Hom(C1(Rn;R);A) _Rn;A 􀀀􀀀􀀀􀀀! An

is the set _􀀀1

A;Rn(U) = TA(U) _ An, and (i_)_ is injective.

To see this let ' 2 Hom(C1(U;R);A). By lemma 35.8 ker ' contains the

ideal of all f vanishing up to some order k at a point x(') 2 U _ Rn, and since

'(xi) = xi(') _ 1 + '(xi 􀀀 xi(')) we have

_A;Rn(_Rn;A(' _ i_)) = _nA

('(x1); : : : ; '(xn)) = x(') 2 U:

As in step 1 we see that the mapping

_􀀀1

A;Rn(U) 3 (a1; : : : ; an) 7! (C1(U;R) 3 f 7! TA(f)(a1; : : : ; an))

is the inverse to _Rn;A _ (i_)_.

Step 3. The two functors Hom(C1( ;R);A) and TA : Mf ! Set coincide

on all open subsets of Rn's, so they have to coincide on all manifolds, since

smooth manifolds are exactly the retracts of open subsets of Rn's by 1.14.1.

Alternatively one may check that the gluing process described in 35.11, step

6, works also for the functor Hom(C1( ;R);A) and gives a unique manifold

structure on it which is compatible to TAM. _

35.15. Covariant description of Weil functors. Let A be a Weil algebra,

which by 35.5.(2) can be viewed as En=I, a _nite dimensional quotient of the

algebra En = C1

0 (Rn;R) of germs at 0 of smooth functions on Rn.

De_nition. Let M be a manifold. Two mappings f; g : Rn ! M with f(0) =

g(0) = x are said to be I-equivalent, if for all germs h 2 C1

x (M;R) we have

h _ f 􀀀 h _ g 2 I.

The equivalence class of a mapping f : Rn ! M will be denoted by jA(f)

and will be called the A-velocity at 0 of f. Let us denote by JA(M) the set of

all A-velocities on M.

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

35. Weil algebras and Weil functors 307

There is a natural way to extend JA to a functor Mf ! Set. For every

smooth mapping f : M ! N between manifolds we put JA(f)(jA(g)) := jA(f_g)

for g 2 C1(Rn;M).

Now one can repeat the development of the theory of (n; r)-velocities for the

more general space JA(M) instead of Jk

0 (Rn;M) and show that JA(M) is a

smooth _ber bundle over M, associated to a higher order frame bundle. This

development is very similar to the computations done in 35.11 and we will in

fact reduce the whole situation to 35.11 and 35.14 by the following

35.16. Lemma. There is a canonical equivalence

JA(M) ! Hom(C1(M;R);A);

jA(f) 7! (C1(M;R) 3 g 7! jA(g _ f) 2 A);

which is natural in A and M and a di_eomorphism, so the functor JA :Mf !

FM is equivalent to TA.

Proof. We just have to note that JA(R) = En=I = A. _

Let us state explicitly that a trivial consequence of this lemma is that theWeil

functor determined by the Weil algebra En=Mk+1

n = Jk

0 (Rn;R) is the functor

n of (n; r)-velocities from 12.8.

35.17. Theorem. Let A and B be Weil algebras. Then we have:

(1) We get the algebra A back from the Weil functor TA by TA(R) = A

with addition +A = TA(+R), multiplication mA = TA(mR) and scalar

multiplication mt = TA(mt) : A ! A.

(2) The natural transformations TA ! TB correspond exactly to the algebra

homomorphisms A ! B

Proof. (1) This is obvious. (2) For a natural transformation ' : TA ! TB its

value 'R : TA(R) = A ! TB(R) = B is an algebra homomorphisms. The inverse

of this mapping is already described in theorem 35.13.4. _

35.18. The basic facts from the theory of Weil functors are completed by the

following assertion, which will be proved in more general context in 36.13.

Proposition. Given two Weil algebras A and B, the composed functor TA _TB

is a Weil functor generated by the tensor product A B.

Corollary. (See also 37.3.) There is a canonical natural equivalence TA _ TB

TB _ TA generated by the exchange algebra isomorphism A B _= B A.

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308 Chapter VIII. Product preserving functors

36. Product preserving functors

36.1. A covariant functor F : Mf ! Mf is said to be product preserving, if

the diagram

F(M1) F(pr1)

􀀀􀀀􀀀􀀀 F(M1 _M2) F(pr2)

􀀀􀀀􀀀􀀀! F(M2)

is always a product diagram. Then F(point) = point, by the following argument:

F(point) u F(point _ point) F(pr1)

F(pr2)

F(point)

point

􀀀􀀀􀀀 􀀀􀀀􀀀􀀀_

44444446

Each of f1, f, and f2 determines each other uniquely, thus there is only one

mapping f1 : point ! F(point), so the space F(point) is single pointed.

The basic purpose of this section is to prove the following

Theorem. Let F be a product preserving functor together with a natural transformation

_F : F ! Id such that (F; _F ) satis_es the locality condition 18.3.(i).

Then F = TA for some Weil algebra A.

This will be a special case of much more general results below. The _nal proof

will be given in 36.12. We will _rst extract uniquely a sum of Weil algebras from

a product preserving functor, then we will reconstruct the functor from this

algebra under mild conditions.

36.2. We denote the addition and the multiplication on the reals by +;m :

R2 ! R, and for _ 2 R we let m_ : R ! R be the scalar multiplication by _ and

we also consider the mapping _ : point ! R onto the value _.

Theorem. Let F : Mf ! Mf be a product preserving functor. Then either

F(R) is a point or F(R) is a _nite dimensional real commutative and formally real

algebra with operations F(+), F(m), scalar multiplication F(m_), zero F(0),

and unit F(1), which is called Al(F). If ' : F1 ! F2 is a natural transformation

between two such functors, then Al(') := 'R : Al(F1) ! Al(F2) is an algebra

homomorphism.

Proof. Since F is product preserving, we have F(point) = point. All the laws

for a commutative ring with unit can be formulated by commutative diagrams

of mappings between products of the ring and the point. We do this for the ring

R and apply the product preserving functor F to all these diagrams, so we get

the laws for the commutative ring F(R) with unit F(1) with the exception of

F(0) 6= F(1) which we will check later for the case F(R) 6= point. Addition F(+)

and multiplication F(m) are morphisms in Mf, thus smooth and continuous.

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

36. Product preserving functors 309

For _ 2 R the mapping F(m_) : F(R) ! F(R) equals multiplication with the

element F(_) 2 F(R), since the following diagram commutes:

F(R)

u _=

AAAAAAAAAAAC

F(m_)

F(R) _ point w

Id_F(_)

u _=

F(R) _ F(R) wF(R)

F(R _ point) w

F(Id__)

F(R _ R)

''''')

F(m)

We may investigate now the di_erence between F(R) = point and F(R) 6= point.

In the latter case for _ 6= 0 we have F(_) 6= F(0) since multiplication by F(_)

equals F(m_) which is a di_eomorphism for _ 6= 0 and factors over a one pointed

space for _ = 0. So for F(R) 6= point which we assume from now on, the group

homomorphism _ 7! F(_) from R into F(R) is actually injective.

In order to show that the scalar multiplication _ 7! F(m_) induces a continuous

mapping R _ F(R) ! F(R) it su_ces to show that R ! F(R), _ 7! F(_),

is continuous.

(F(R); F(+); F(m􀀀1); F(0)) is a commutative Lie group and is second countable

as a manifold since F(R) 2 Mf. We consider the exponential mapping

exp : L ! F(R) from the Lie algebra L into this group. Then exp(L) is

an open subgroup of F(R), the connected component of the identity. Since

fF(_) : _ 2 Rg is a subgroup of F(R), if F(_) =2 exp(L) for all _ 6= 0, then

F(R)= exp(L) is a discrete uncountable subgroup, so F(R) has uncountably many

connected components, in contradiction to F(R) 2 Mf. So there is _0 6= 0 in

R and v0 6= 0 in L such that F(_0) = exp(v0). For each v 2 L and r 2 N,

hence r 2 Q, we have F(mr) exp(v) = exp(rv). Now we claim that for any

sequence _n ! _ in R we have F(_n) ! F(_) in F(R). If not then there is a

sequence _n ! _ in R such that F(_n) 2 F(R) n U for some neighborhood U of

F(_) in F(R), and by considering a suitable subsequence we may also assume

that 2n2 (_n+1 􀀀 _) is bounded. By lemma 36.3 below there is a C1-function

f : R ! R with f( _0

2n ) = _n and f(0) = _. Then we have

F(_n) = F(f)F(m2􀀀n)F(_0) = F(f)F(m2􀀀n) exp(v0) =

= F(f) exp(2􀀀nv0) ! F(f) exp(0) = F(f(0)) = F(_);

contrary to the assumption that F(_n) =2 U for all n. So _ 7! F(_) is a continuous

mapping R ! F(R), and F(R) with its manifold topology is a real _nite

dimensional commutative algebra, which we will denote by Al(F) from now on.

The evaluation mapping evIdR : Hom(C1(R;R); Al(F)) ! Al(F) is bijective

since it has the right inverse x 7! (C1(R;R) 3 f 7! F(f)x). But by 35.7 the

evaluation map has values in the Weil part W(Al(F)) of Al(F), so the algebra

Al(F) is generated by its idempotent and nilpotent elements and has to be

formally real, a direct sum of Weil algebras by 35.1. _

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310 Chapter VIII. Product preserving functors

Remark. In the case of product preserving bundle functors the smoothness of

_ 7! F(_) is a special case of the regularity proved in 20.7. In fact one may also

conclude that F(R) is a smooth algebra by the results from [Montgomery-Zippin,

55], cited in 5.10.

36.3. Lemma. [Kriegl, 82] Let _n ! _ in R, let tn 2 R, tn > 0, tn ! 0 strictly

monotone, such that _

_n 􀀀 _n+1

(tn 􀀀 tn+1)k ; n 2 N

is bounded for all k. Then there is a C1-function f : R ! R with f(tn) = _n

and f(0) = _ such that f is at at each tn.

Proof. Let ' 2 C1(R;R), ' = 0 near 0, ' = 1 near 1, and 0 _ ' _ 1 elsewhere.

Then we put

f(t) =

8>>><

>>>:

_ for t _ 0;

t 􀀀 tn+1

tn 􀀀 tn+1

(_n 􀀀 _n+1) + _n+1 for tn+1 _ t _ tn;

_1 for t1 _ t;

and one may check by estimating the left and right derivatives at all tn that f

is smooth. _

36.4. Product preserving functors without Weil algebras. Let F :

Mf ! Mf be a functor with preserves products and assume that it has

the property that F(R) = point. Then clearly F(Rn) = F(R)n = point and

F(M) = point for each smoothly contractible manifold M. Moreover we have:

Lemma. Let f0; f1 : M ! N be homotopic smooth mappings, let F be as

above. Then F(f0) = F(f1) : F(M) ! F(N).

Proof. A continuous homotopy h : M_[0; 1] ! N between f0 and f1 may _rst be

reparameterized in such a way that h(x; t) = f0(x) for t < " and h(x; t) = f1(x)

for 1 􀀀 " < t, for some " > 0. Then we may approximate h by a smooth

mapping without changing the endpoints f0 and f1. So _nally we may assume

that there is a smooth h : M _ R ! N such that h _ insi = fi for i = 0; 1 where

inst : M ! M _ R is given by inst(x) = (x; t). Since

F(M) u F(M _ R) F(pr1)

F(pr2)

F(R)

F(M) _ point point

is a product diagram we see that F(pr1) = IdF(M). Since pr1 _ inst = IdM we

get also F(inst) = IdF(M) and thus F(f0) = F(h) _ F(ins0) = F(h) _ F(ins1) =

F(f1). _

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

36. Product preserving functors 311

Examples. For a manifold M let M =

M_ be the disjoint union of its connected

components and put ~H1(M) :=

_ H1(M_;R), using singular homology

with real coe_cients, for example. If M is compact, ~H1(M) 2Mf and ~H1 becomes

a product preserving functor from the category of all compact manifolds

into Mf without a Weil algebra.

For a connected manifold M the singular homology group H1(M;Z) with

integer coe_cients is a countable discrete set, since it is the abelization of the

fundamental group _1(M), which is a countable group for a separable connected

manifold. Then again by the Kunneth theorem H1( ;Z) is a product preserving

functor from the category of connected manifolds into Mf without a Weil

algebra.

More generally let K be a _nite CW-complex and let [K;M] denote the

discrete set of all (free) homotopy classes of continuous mappings K ! M,

where M is a manifold. Algebraic topology tells us that this is a countable set.

Clearly [K; ] then de_nes a product preserving functor without a Weil algebra.

Since we may take the product of such functors with other product preserving

functors we see, that the Weil algebra does not determine the functor at all. For

conditions which exclude such behaviour see theorem 36.8 below.

36.5. Convention. Let A = A1__ _ __Ak be a formally real _nite dimensional

commutative algebra with its decomposition into Weil algebras. In this section

we will need the product preserving functor TA := TA1

_ : : : _ TAk : Mf !

Mf which is given by TA(M) := TA1 (M) _ : : : _ TAk (M). Then 35.13.1 for

TA has to be modi_ed as follows: _A;M : TAM ! Mk is a _ber bundle. All

other conclusions of theorem 35.13 remain valid for this functor, since they are

preserved by the product, with exception of 35.13.3, which holds for connected

manifolds only now. Theorem 35.14 remains true, but the covariant description

(we will not use it in this section) 35.15 and 35.16 needs some modi_cation.

36.6. Lemma. Let F :Mf !Mf be a product preserving functor. Then the

mapping

_F;M : F(M) ! Hom(C1(M;R); Al(F)) = TAl(F)M

_F;M (x)(f) := F(f)(x);

is smooth and natural in F and M.

Proof. Naturality in F and M is obvious. To show that _ is smooth is more

di_cult. To simplify the notation we let Al(F) =: A = A1 _ _ _ _ _ Ak be the

decomposition of the formally real algebra Al(F) into Weil algebras.

Let h = (h1; : : : ; hn) : M ! Rn be a closed embedding into some high

dimensional Rn. By theorem 35.13.2 the mapping TA(h) : TAM ! TARn is also

a closed embedding. By theorem 35.14, step 1 of the proof (and by reordering the

product), the mapping _Rn;A : Hom(C1(Rn;R);A) ! An is given by _Rn;A(') =

('(xi))ni

=1, where (xi) are the standard coordinate functions on Rn. We have

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

312 Chapter VIII. Product preserving functors

F(Rn) _= F(R)n _= An _= TA(Rn). Now we consider the commuting diagram

F(M)

_F;M

Hom(C1(M;R);A) w

_M;A

(h_)_

TA(M)

TA(h)

Hom(C1(Rn;R);A) w

_Rn;A

TA(Rn) F(Rn)

For z 2 F(M) we have

(_Rn;A _ (h_)_ _ _F;M )(z) = _Rn;A(_F;M (z) _ h_)

􀀀

_F;M (z)(x1 _ h); : : : ; _F;M (z)(xn _ h)

􀀀

_F;M (z)(h1); : : : ; _F;M (z)(hn)

􀀀

F(h1)(z); : : : ; F(hn)(z)

= F(h)(z):

This is smooth in z 2 F(M). Since _M;A is a di_eomorphism and TA(h) is a

closed embedding, _F;M is smooth as required. _

36.7. The universal covering of a product preserving functor. Let

F : Mf ! Mf be a product preserving functor. We will construct another

product preserving functor as follows. For any manifold M we choose a universal

cover qM : ~M ! M (over each connected component ofM separately), and we let

_1(M) denote the group of deck transformations of ~M ! M, which is isomorphic

to the product of all fundamental groups of the connected components of M. It

is easy to see that _1(M) acts strictly discontinuously on TA( ~M ), and by lemma

36.6 therefore also on F( ~M ). So the orbit space

~ F(M) := F( ~M )=_1(M)

is a smooth manifold. For f : M1 ! M2 we choose any smooth lift ~ f : ~M1 ! ~M2,

which is unique up to composition with elements of _1(Mi). Then F ~ f factors

as follows:

F( ~M1) w

F( ~ f)

F( ~M2)

u ~ F(M) w

~ F(f) ~ F(M2):

The resulting smooth mapping ~ F(f) does not depend on the choice of the lift

~ f. So we get a functor ~ F : Mf ! Mf and a natural transformation q = qF :

~ F ! F, induced by F(qM) : F( ~M ) ! F(M), which is a covering mapping. This

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

36. Product preserving functors 313

functor ~ F is again product preserving, because we may choose (M1 _ M2)_ =

1 _ ~M2 and _1(M1 _M2) = _1(M1) _ _1(M2), thus

~ F(M1 _M2) = F((M1 _M2)_)=_1(M1 _M2) =

= F( ~M1)=_1(M1) _ F( ~M2)=_1(M2) = ~ F(M1) _ ~ F(M2):

Note _nally that ~ TA 6= TA if A is sum of at least two Weil algebras. As an example

consider A = R _ R, then TA(M) = M _M, but ~ TA(S1) = R2=Z(2_; 2_) _=

S1 _ R.

36.8. Theorem. Let F be a product preserving functor.

(1) If M is connected, then there exists a unique smooth mapping F;M :

^TAl(F)(M) ! F(M) which is natural in F and M and satis_es _F;M _

F;M = qTAl(F);M :

^TAl(F)(M) w

F;M

hhhjq

F(M)

_F;M

TAl(F)(M).

(2) If F maps embeddings to injective mappings, then _F;M : F(M) !

TAl(F)(M) is injective for all manifolds M, and it is a di_eomorphism for

connected M.

(3) If M is connected and F;M is surjective, then _F;M and F;M are covering

mappings.

Remarks. Condition (2) singles out the functors of the form TA among all

product preserving functors. Condition (3) singles the coverings of the TA's. A

product preserving functor satisfying condition (3) will be called weakly local .

Proof. We let Al(F) =: A = A1 __ _ __Ak be the decomposition of the formally

real algebra Al(F) into Weil algebras. We start with a

Sublemma. If M is connected then _F;M is surjective and near each ' 2

Hom(C1(M;R);A) = TA(M) there is a smooth local section of _F;M .

Let ' = '1 +_ _ _+'k for 'i 2 Hom(C1(M;R);Ai). Then by lemma 35.8 for

each i there is exactly one point xi 2 M such that 'i(f) depends only on a _nite

jet of f at xi. Since M is connected there is a smoothly contractible open set

U in M containing all xi. Let g : Rm ! M be a di_eomorphism onto U. Then

(g_)_ : Hom(C1(Rm;R);A) ! Hom(C1(M;R);A) is an embedding of an open

neighborhood of ', so there is _' 2 Hom(C1(Rm;R);A) depending smoothly on

' such that (g_)_( _') = '. Now we consider the mapping

Hom(C1(Rm;R);A) _Rm

􀀀􀀀􀀀! TA(Rm) _= F(Rm) F(g)

􀀀􀀀􀀀!

F(g)

􀀀􀀀􀀀! F(M) _M 􀀀􀀀! Hom(C1(M;R);A):

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

314 Chapter VIII. Product preserving functors

We have (_M _ F(g) _ _Rm)( _') = ((g_)_ _ _Rm _ _Rm)( _') = (g_)_( _') = ',

since it follows from lemma 36.6 that _Rm _ _Rm = Id. So the mapping sU :=

F(g) _ _Rm _ (g__)􀀀1 : TAU ! F(M) is a smooth local section of _M de_ned

near '. We may also write sU = F(iU) _ (_F;U )􀀀1 : TAU ! F(M), since for

contractible U the mapping _F;U is clearly a di_eomorphism. So the sublemma

is proved.

(1) Now we start with the construction of F;M . We note _rst that it su_ces

to construct F;M for simply connected M because then we may induce it for

not simply connected M using the following diagram and naturality.

fTA( ~M ) TA ~M w

F; ~M

F( ~M )

u fTA(M) w

F;M

F(M):

Furthermore it su_ces to construct F;M for high dimensional M since then we

have

fTA(M _ R) w

F;M_R

F(M _ R)

u fTA(M) _ F(R) w

F;M _ IdF(R)

F(M) _ F(R):

So we may assume that M is connected, simply connected and of high dimension.

For any contractible subset U of M we consider the local section sU of _F;M

constructed in the sublemma and we just put F;M (') := sU(') for ' 2 TAU _

TAM. We have to show that F;M is well de_ned. So we consider contractible

U and U0 in M with ' 2 TA(U \ U0). If _(') = (x1; : : : ; xk) 2 Mk as in

the sublemma, this means that x1; : : : ; xk 2 U \ U0. We claim that there are

contractible open subsets V , V 0, and W of M such that x1; : : : ; xk 2 V \ V 0 \

W and that V _ U \ W and V 0 _ U0 \ W. Then by the naturality of _

we have sU(') = sV (') = sW(') = sV 0 (') = sU0 (') as required. For the

existence of these sets we choose an embedding H : R2 ! M such that c(t) =

H(t; sin t) 2 U, c0(t) = H(t;􀀀sin t) 2 U0 and H(2_j; 0) = xj for j = 1; : : : ; k.

This embedding exists by the following argument. We connect the points by

a smooth curve in U and a smooth curve in U0, then we choose a homotopy

between these two curves _xing the xj 's, and we approximate the homotopy by

an embedding, using transversality, again _xing the xj 's. For this approximation

we need dimM _ 5, see [Hirsch, 76, chapter 3]. Then V , V 0, and W are just

small tubular neighborhoods of c, c0, and H.

(2) Since a manifold M has at most countably many connected components,

there is an embedding I : M ! Rn for some n. Then from

F(M) v w

F(i)

_F;M

F(Rn)

_F;Rn

TA(M) w

TA(i) TA(Rn),

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36. Product preserving functors 315

lemma 36.6, and the assumption it follows that _F;M is injective. If M is furthermore

connected then the sublemma implies furthermore that _F;M is a diffeomorphism.

(3) Since __ = q, and since q is a covering map and is surjective, it follows

that both _ and are covering maps. _

In the example F = TR_R considered at the end of 36.7 we get that F;S1 :

~ F(S1) = R2=Z(2_; 2_) ! F(S1) = S1 _ S1 = R2=(Z(2_; 0) _ Z(0; 2_)) is the

covering mapping induced from the injection Z(2_; 2_) ! Z(2_; 0) _ Z(0; 2_).

36.9. Now we will determine all weakly local product preserving functors F on

the category conMf of all connected manifolds with Al(F) equal to some given

formally real _nite dimensional algebra A with k Weil components. Let F be

such a functor.

For a connected manifold M we de_ne C(M) by the following transversal

pullback:

C(M) w

F(M)

TRk (M) Mk w 0 TAM;

where 0 is the natural transformation induced by the inclusion of the subalgebra

Rk generated by all idempotents into A.

Now we consider the following diagram: In it every square is a pullback, and

each vertical mapping is a covering mapping, if F is weakly local, by theorem

36.8.

k w 0

TA ~M

u ~M k=_1(M) w

fTA(M)

C(M) w

F(M)

Mk wTA(M):

Thus F(M) = TA( ~M )=G, where G is the group of deck transformations of

the covering C(M) ! ~M k, a subgroup of _1(M)k containing _1(M) (with its

diagonal action on ~M k). Here g = (g1; : : : ; gk) 2 _1(M)k acts on TA( ~M ) =

TA1 ( ~M ) _ : : : _ TAk ( ~M ) via TA1 (g1) _ : : : _ TAk (gk). So we have proved

36.10. Theorem. A weakly local product preserving functor F on the category

conMf of all connected manifolds is uniquely determined by specifying

a formally real _nite dimensional algebra A = Al(F) and a product preserving

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316 Chapter VIII. Product preserving functors

functor G : conMf ! Groups satisfying _1 _ G _ _k

1 , where _1 is the fundamental

group functor, sitting as diagonal in _k

1 , and where k is the number of

Weil components of A.

The statement of this theorem is not completely rigorous, since _1 depends

on the choice of a base point.

36.11. Corollary. On the category of simply connected manifolds a weakly

local product preserving functor is completely determined by its algebra A =

Al(F) and coincides with TA.

If the algebra Al(F) = A of a weakly local functor F is a Weil algebra (the

unit is the only idempotent), then F = TA on the category conMf of connected

manifolds. In particular F is a bundle functor and is local in the sense of 18.3.(i).

36.12. Proof of theorem 36.1. Using the assumptions we may conclude that

_F;M : F(M) ! M is a _ber bundle for each M 2 Mf, using 20.3, 20.7, and

20.8. Moreover for an embedding iU : U ! M of an open subset F(iU) : F(U) !

F(M) is the embedding onto F(M)jU = _􀀀1

F;M (U). Let A = Al(F). Then A can

have only one idempotent, for even the bundle functor pr1 : M _M ! M is not

local. So A is a Weil algebra.

By corollary 36.11 we have F = TA on connected manifolds. Since F is local,

it is fully determined by its values on smoothly contractible manifolds, i.e. all

Rm's. _

36.13. Lemma. For product preserving functors F1 and F2 on Mf we have

Al(F2 _ F1) = Al(F1) Al(F2) naturally in F1 and F2.

Proof. Let B be a real basis for Al(F1). Then

Al(F2 _ F1) = F2(F1(R)) = F2(

b2B

R _ b) _=

b2B

F2(R) _ b;

so the formula holds for the underlying vector spaces. Now we express the

multiplication F1(m) : Al(F1) P _ Al(F1) ! Al(F1) in terms of the basis: bibj =

k ck

ijbk, and we use

F2(F1(m)) = (F1(m)_)_ : Hom(C1(Al(F1) _ Al(F1);R); Al(F2)) !

! Hom(C1(Al(F1);R); Al(F2))

to see that the formula holds also for the multiplication. _

Remark. We chose the order Al(F1) Al(F2) so that the elements of Al(F2)

stand on the right hand side. This coincides with the usual convention for writing

an atlas for the second tangent bundle and will be essential for the formalism

developed in section 37 below.

36.14. Product preserving functors on not connected manifolds. Let

F be a product preserving functor Mf !Mf. For simplicity's sake we assume

that F maps embeddings to injective mappings, so that on connected manifolds

it coincides with TA where A = Al(F). For a general manifold we have TA(M) _=

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36. Product preserving functors 317

Hom(C1(M;R);A), but this is not the unique extension of FjconMf to Mf,

as the following example shows: Consider Pk(M) = M _ : : : _ M (k times),

given by the product of Weil algebras Rk. Now let Pc

k (M) =

_ Pk(M_) be the

disjoint union of all Pk(M_) where M_ runs though all connected components

of M. Then Pc

k is a di_erent extension of PkjconMf to Mf.

Let us assume now that A = Al(F) is a direct sum on k Weil algebras,

A = A1 _ _ _ _ _ Ak and let _ : TA ! Pk be the natural transformation induced

by the projection on the subalgebra Rk generated by all idempotents. Then also

Fc(M) = _􀀀1(Pc

k (M)) _ TA(M) is an extension of FjconMf to Mf which

di_ers from TA. Clearly we have Fc(M) =

_ F(M_) where the disjoint union

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

n for product

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 _F;M : F(M) ! TA(M) is injective for

each manifold M. We have to show that the set f1; : : : ; kg can be divided into

equivalence classes I1; : : : ; In such that F(M) _ TA(M) is the inverse image

under _ : 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 2 Ir for some r

implies that Ni = Nj . Then each Ir gives rise to Gc

r = TcL

i2Ir

To _nd the equivalence classes we consider X = f1; : : : ; kg as a discrete manifold

and consider F(X) _ TA(X) = Xk. Choose an element i = (i1; : : : ; ik) 2

F(X) with maximal number of distinct members. The classes Ir will then be

the non-empty sets of the form fs : is = jg 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) = f(d1; : : : ; dk) 2 Dk : s; t 2 Ir implies ds = dt for all rg:

Suppose not, then there exist d = (d1; : : : ; dk) 2 F(D) and r; s; t with s; t 2 Ir

and ds 6= 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)) 2 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 ! _0(M) be the projection of M

onto the (discrete) set of its connected components. For a 2 F(M) the value

F(p)(a) 2 F(_0(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 2 M such s; t 2 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 _􀀀1(x1; : : : ; xk) _ TA(M) is contained in F(M) _ TA(M). Let m = dimM

(or the maximum of dimMi 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 _ f1; : : : ; ng. We choose y1; : : : ; yk 2 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)(_􀀀1(y1; : : : ; yk)) =

_􀀀1(x1; : : : ; xk), and from the last step of the proof we know that F(N) contains

_􀀀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 ! _0(M) onto the space of connected components of M.

Since _0(M) 2 Mf, the functor F acts on this discrete set. Since F is weakly

local and maps points to points, F(_0(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

_1 _ G _ _k

1 , 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

TR = R2;

T(+)(a; a0)(b; b0) = (a + b; a0 + b0);

T(m)(a; a0)(b; b0) = (ab; ab0 + a0b);

T(m_)(a; a0) = (_a; _a0):

So the Weil algebra TR = Al(T) =: D is the algebra generated by 1 and _ with

_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 + a0_)(b + b0_) = ab + (ab0 + a0b)_ for the multiplication in TR.

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 multiplications

m_ for _ 2 R, which act on TR by m_(1) = 1 and m_(_) = _:_.

(2) The projection _ : 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 TM

for a _nite dimensional vector space V with _berwise tensor product. Moreover

for the space of natural transformations between two such functors we have

Nat(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 TM. Finally by 35.17.(2) we have

Nat(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 _1 : F1 ! Id, _2 : F2 ! Id induced by (_:1 + n) 7!

_ 2 R. In general we will write _F : F ! Id. Thus we have also

F2_1 : F2 _ F1 ! F2 and _2F1 : F2 _ F1 ! F1.

(2) The zero sections 01 : Id ! F1 and 02 : Id ! F2 induced by R ! A1,

_ 7! _:1. Then we have F201 : F2 ! F2 _ F1 and 02F1 : F1 ! F2 _ F1.

(3) The isomorphism A1A2

A2A1, given by a1a2 7! a2a1 induces

the canonical ip mapping _F1;F2 = _ : F2 _ F1 ! F1 _ F2. We have

_F1;F2 = _􀀀1

F2;F1 .

(4) The multiplication m in A is a homomorphism AA ! A which induces

a natural transformation _ = _F : 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

(_2F1; F2_1); 0F1_IdF2

_ _F2_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 _ factors to a natural transformation _F2_F1 : F2_F1 !

F1 _ F2 with vl _ _F2_F1 = _F2;F1

_ vl.

(7) The multiplication _ induces a natural transformation __vl : F _F ! F.

It is clear that _ 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 T2 = T _ T be the second tangent

bundle. Its Weil algebra is D2 := Al(T2) = D D = R4 with generators

1, _1, and _2 and with relations _2

1 = _2

2 = 0. Then (1; _1; _2; _1_2) is the

standard basis of R4 = T2R in the usual description, which we also used in 6.12.

From the list of natural transformations in 37.1 we get _T : (_1; _2) 7! (_; 0),

T_ : (_1; _2) 7! (0; _), and _ = +_ (_T; T_) : T2 ! T; (_1; _2) 7! (_; _). Then we

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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:

_ : T2 ! T2;

_(a1 + x1_1 + x2_2 + x3_1_2) = a1 + x2_1 + x1_2 + x3_1_2:

vl : T ! T2; vl(a1 + x_) = a1 + x_1_2:

m_T : T2 ! T2;

m_T(a1 + x1_1 + x2_2 + x3_1_2) = a1 + x1_1 + _x2_2 + _x3_1_2:

Tm_ : T2 ! T2;

Tm_(a1 + x1_1 + x2_2 + x3_1_2) = a1 + _x1_1 + x2_2 + _x3_1_2:

(+T) : T2 _T T2 ! T2;

(+T)((a1 + x1_1 + x2_2 + x3_1_2); (a1 + x1_1 + y2_2 + y3_1_2)) =

= a1 + x1_1 + (x2 + y2)_2 + (x3 + y3)_1_2:

(T+)((a1 + x1_1 + x2_2 + x3_1_2); (a1 + y1_1 + x2_2 + y3_1_2)) =

= a1 + (x1 + y1)_1 + x2_2 + (x3 + y3)_1_2:

The space of all natural transformations Nat(T; T2) _= Hom(D;D2) turns out to

be the real algebraic variety R2[

RR2 consisting of all homomorphisms _ 7! x1_1+

x2_2 +x3_1_2 with x1x2 = 0, since _2 = 0. The homomorphism _ 7! x_1 +y_1_2

corresponds to the natural transformation (+T) _ (vl _ my; 0T _ mx), and the

homomorphism _ 7! x_2+y_1_2 corresponds to (T+)_(vl _my; T0_mx). So any

element in Nat(T; T2) can be expressed in terms of the natural transformations

f0T; T0; (T+); (+T); T_; _T; vl;m_ for _ 2 Rg:

Similarly Nat(T2; T2) _= Hom(D2;D2) turns out to be the real algebraic variety

(R2 [

R R2) _ (R2 [

R R2) consisting of all

x1_1 + x2_2 + x3_1_2

y1_1 + y2_2 + y3_1_2

with x1x2 = y1y2 = 0. Again any element of Nat(T2; T2) can be written in

terms of f0T; T0; (T+); (+T); T_; _T; _;m_T; Tm_ for _ 2 Rg. If for example

x2 = y1 = 0 then the corresponding transformation is

(+T) _ (my2T _ Tmx1 ; (T+) _ (vl _ + _ (mx3

_ _T;my3

_ T_); 0T _ mx1

_ _T)):

Note also the relations T_ _ _ = _T, _ _ (T+) = (+T) _ (_ _ _), _ _ vl = vl,

__Tm_ = m_T; so _ interchanges the two vector bundle structures on T2 ! T,

namely ((+T);m_T; _T) and ((T+); Tm_; T_), and vl : T ! T2 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 _F;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 _=

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:_ = _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 _F;M : F(M) ! M. Note that this space is in_nite dimensional in

general. Recall from theorem 35.14 that

F(M) = TA(M) _M;A 􀀀􀀀􀀀 Hom(C1(M;R);A)

is an isomorphism. For f 2 C1(M;R) we can decompose F(f) = TA(f) :

F(M) ! F(R) = A = R:1 _ N into

F(f) = TA(f) = (f _ _) _ N(f);

N(f) : F(M) ! N:

Lemma.

(1) Each Xx 2 F(M)x = _􀀀1(x) for x 2 M de_nes an R-linear mapping

DXx : C1(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 2 M.

(2) Each R-linear mapping _ : C1(M;R) ! N which satis_es the expansion

property at x 2 M is of the form _ = DXx for a unique Xx 2 F(M)x.

(3) The R-linear mappings _ : C1(M;R) ! C1(M;N) = N C1(M;R)

which have the expansion property

(a) _(f:g) = _(f):g + f:_(g) + _(f):_(g); f; g 2 C1(M;R);

are exactly those induced (via 1 and 2) by the smooth sections of _ :

F(M) ! M.

Linear mappings satisfying the expansion property 1 will be called expansions:

if N is generated by _ with _k+1 = 0, so that F(M) = Jk

0 (R;M), then these

are parametrized Taylor expansions of f to order k (applied to a k-jet of a

curve through each point). For X 2 XF (M) we will write DX : C1(M;R) !

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322 Chapter VIII. Product preserving functors

C1(M;N) = N C1(M;R) for the expansion induced by X. Note the de_ning

equation

F(f) _ X = f:(b) 1 + DX(f) = (f:1;DX(f)) or

f(x):1 + DX(f)(x) = F(f)(X(x)) = _􀀀1

M;A(X(x))(f):

Proof. (1) and (2). For ' 2 Hom(C1(M;R);A) = F(M) we consider the foot

point _(_M;A(')) = _M;R(_(')) = x 2 M and _M;A(') = Xx 2 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 2 M the mapping f 7! _(f)(x) 2 N is of the form DX(x) for a

unique X(x) 2 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 = _F _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 [aX; bY ]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 2 N and

X 2 X(M) is a smooth vector _eld on M, then by lemma 37.5 we have aX 2

XT_F (M) and for f 2 C1(M;R) we use Tf(X) = f:1 + df(X) to get

(T _ F)(f)(a X) = (IdN Tf)(a X)

= f:1 + a:df(X) = f:1 + a:X(f)

= f:1 + DaX(f) by 37.6.(b). Thus

DT_F

aX(a) (f) = DaX(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 _ : C1(M;R) ! N C1(M;R) satisfying

_(f:g) = _(f):g + f:_(g):

These mappings are called derivations.

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37. Examples and applications 323

Now we denote L := LR(C1(M;R);N C1(M;R)) for short, and for _,

_ 2 L we de_ne

(b) _ _ _ := (m IdC1(M;R)) _ (IdN _) _ _ : C1(M;R) !

! N C1(M;R) ! N N C1(M;R) ! N C1(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.

(1) For X, Y 2 XF (M) we have DFX

_Y = DFX

_ DF

Y + DFX

+ DF

Y .

(2) For X, Y 2 XT_F (M) we have DF

[X;Y ]T_F

= DFX

_ DF

􀀀 DF

_ DFX

(3) For _ 2 L de_ne

exp(_) :=

i=1

i! __i

log(_) :=

i=1

(􀀀1)i􀀀1

__i:

Then exp; log : L ! L are bijective and inverse to each other. exp(_) is

an expansion if and only if _ 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.

_ _ (_ _ _) = (m IdC1(M;R)) _ (IdN _) _ (_ _ _)

= (m Id) _ (IdN _) _ (m Id) _ (IdN _) _ _

= (m Id) _ (m IdN Id) _ (IdNN _) _ (IdN _) _ _

= (m Id) _ (IdN m Id) _ (IdNN _) _ (IdN _) _ _

= (m Id) _ (IdN

􀀀

(m Id) _ (IdN _) _ _

_ _

= (_ _ _) _ _:

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:1 + DFX

_Y (f) = F(f) _ (X _ Y ) by 37.6.(b)

= F(f) _ _F;M _ F(Y ) _ X by 37.7(1)

= _F;R

_ F(F(f)) _ F(Y ) _ X since _ is natural

= m _ F(F(f) _ Y ) _ X

= m _ F(f:1;DF

Y (f)) _ X by 37.6.(b)

= m _

􀀀

(1 F(f) _ X) _ (F(DF

Y (f)) _ X)

= m _

1 (f:1 + DFX

(f)) +

􀀀

Y (f) 1 + (IdN DFX

)(DF

Y (f))

= f:1 + DFX

(f) + DF

Y (f) + (DFX

_ DF

Y )(f):

(3) For vector _elds X, Y 2 X(M) on M and a, b 2 N we have

DT_F

[aX;bY ]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)))

= (m IdC1(M;R)) _ (IdN DT_F

aX) _ DT_F

bY (f) 􀀀 : : :

= (DT_F

_ DT_F

􀀀 DT_F

_ DT_F

aX)(f):

(4) After adjoining a unit to L we see that exp(_) = e_ 􀀀 1 and log(_) =

log(1 + _). So exp and log are inverse to each other in the ring of formal power

series of one variable. The elements 1 and _ generate a quotient of the power

series ring in R:1 _ L, and the formal expressions of exp and log commute with

taking quotients. So exp = log􀀀1. 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 C1(M;R), given by

C_ := m _ (IdA _) : A C1(M;R) !

! A N C1(M;R) _ A A C1(M;R) ! A C1(M;R);

where m : A A ! A is the multiplication. We have C_(a f) = a:_(f).

Lemma.

(1) C___ = C_ _ C_, so C is an algebra homomorphism.

(2) _ 2 L is an expansion if and only if Id+C_ is an automorphism of the

commutative algebra A C1(M;R).

(3) _ 2 L is a derivation if and only if C_ is a derivation of the algebra

A C1(M;R).

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

37. Examples and applications 325

Proof. This is obvious. _

37.11. Proof of theorem 37.7. 1. It is easily checked that L is a group with

multiplication ___ = ___+_+_, with unit 0, and with inverse _􀀀1 =

i=1(􀀀_)_i

(recall that _ is a nilpotent multiplication). As noted already at the of 37.8 the

mapping DF : XF (M) ! L is an isomorphism onto the subgroup of expansions,

because Id+C _ DF : XF (M) ! L ! End(A C1(M;R)) is an isomorphism

onto the subgroup of automorphisms.

2. C _DT_F : XT_F (M) ! End(AC1(M;R)) is a Lie algebra isomorphism

onto the sub Lie algebra of End(A C1(M;R)) of derivations.

3. De_ne exp : XT_F (M) ! XF (M) by DF

exp(X) = exp(DFX

). The Baker-

Campbell-Hausdor_ formula holds for

exp : Der(A C1(M;R)) ! Aut(A C1(M;R));

since the Lie algebra of derivations is nilpotent.

4. This is obvious since we used only natural constructions. _

37.12. The Lie bracket. We come back to the tangent bundle functor T and

its iterates. For T the structures described in theorem 37.7 give just the addition

of vector _elds. In fact we have X _ Y = X + Y , and [X; Y ]T_T = 0.

But we may consider other structures here. We have by 37.1 Al(T) = D =

R:1_R:_ for _2 = 0. So N _= R with the nilpotent multiplication 0, but we still

have the usual multiplication, now called m, on R.

For X, Y 2 XT (M) we have DX 2 L = LR(C1(M;R);C1(M;R)), a derivation

given by f:1 + DX(f):_ = Tf _ X, see 37.6.(b) | we changed slightly the

notation. So DX(f) = X(f) = df(X) in the usual sense. The space L has one

more structure now, composition, which is determined by specifying a generator

_ of the nilpotent ideal of Al(T). The usual Lie bracket of vector _elds is now

given by D[X;Y ] := DX _ DY 􀀀 DY _ DX.

37.13. Lemma. In the setting of 37.12 we have

(􀀀T) _ (TY _ X; _T _ TX _ Y ) = (T+) _ (vl _ [X; Y ]; 0T _ Y )

in terms of the natural transformations descibed in 37.4

This is a variant of lemma 6.13 and 6.19.(4). The following proof appears

to be more complicated then the earlier ones, but it demonstrates the use of

natural transformations, and we write out carefully the unusual notation.

Proof. For f 2 C1(M;R) and X, Y 2 XT (M) we compute as follows using

repeatedly the de_ning equation for DX from 37.12:

T2f _ TY _ X = T(Tf _ Y ) _ X = T(f:1 _ DY (f):_1) _ X

= (Tf _ X):1 _ (T(DY (f)) _ X):_1; since T preserves products,

= f:1 + DX(f):_2 + (DY (f):1 + DXDY (f):_2):_1

= f:1 + DY (f):_1 + DX(f):_2 + DXDY (f):_1_2:

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

326 Chapter VIII. Product preserving functors

Now we use the natural transformation and their commutation rules from 37.4

to compute:

T2f _ (􀀀T) _ (TY _ X; _T _ TX _ Y ) =

= (􀀀T) _ (T2f _ TY _ X; _T _ T2f _ TX _ Y )

= (􀀀T) _

􀀀

f:1 + DY (f):_1 + DX(f):_2 + DXDY (f):_1_2;

_T (f:1 + DX(f):_1 + DY (f):_2 + DY DX(f):_1_2)

= (􀀀T) _

􀀀

f:1 + DY (f):_1 + DX(f):_2 + DXDY (f):_1_2;

f:1 + DY (f):_1 + DX(f):_2 + DY DX(f):_1_2)

= f:1 + DY (f):_1 + (DXDY 􀀀 DY DX)(f):_1_2

= (T+) _ (0T _ (f:1 + DY (f):_); vl _ (f:1 + D[X;Y ](f):_))

= (T+) _ (0T _ Tf _ Y; vl _ Tf _ [X; Y ])

= (T+) _ (T2f _ 0T _ Y; T2f _ vl _ [X; Y ])

= T2f _ (T+) _ (0T _ Y; vl _ [X; Y ]): _

37.14. Linear connections and their curvatures. Our next application

will be to derive a global formula for the curvature of a linear connection on a

vector bundle which involves the second tangent bundle of the vector bundle.

So let (E; p;M) be a vector bundle. Recall from 11.10 and 11.12 that a linear

connection on the vector bundle E can be described by specifying its connector

K : TE ! E. By lemma 11.10 and by 11.11 any smooth mapping K : TE ! E

which is a (_ber linear) homomorphism for both vector bundle structure on TE,

and which is a left inverse to the vertical lift, K_vlE = pr2 : E_ME ! TE ! E,

speci_es a linear connection.

For any manifold N, smooth mapping s : N ! E, and vector _eld X 2 X(N)

we have then the covariant derivative of s along X which is given by rXs :=

K _ Ts _ X : N ! TN ! TE ! E, see 11.12.

For vector _elds X, Y 2 X(M) and a section s 2 C1(E) the curvature RE

of the connection is given by RE(X; Y )s = ([rX;rY ] 􀀀 r

[X;Y ])s, see 11.12.

37.15. Theorem.

(1) Let K : TE ! E be the connector of a linear connection on a vector

bundle (E; p;M). Then the curvature is given by

RE(X; Y )s = (K _ TK _ _E 􀀀 K _ TK) _ T2s _ TX _ Y

for X, Y 2 X(M) and a section s 2 C1(E).

(2) If s : N ! E is a section along f := p _ s : N ! M then we have for

vector _elds X, Y 2 X(N)

rXrY s 􀀀 rY rXs 􀀀 r

[X;Y ]s =

= (K _ TK _ _E 􀀀 K _ TK) _ T2s _ TX _ Y =

= RE(Tf _ X; Tf _ Y )s:

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

37. Examples and applications 327

(3) Let K : T2M ! M be a linear connection on the tangent bundle. Then

its torsion is given by

Tor(X; Y ) = (K _ _M 􀀀 K) _ TX _ Y:

Proof. (1) Let _rst mEt

: E ! E denote the scalar multiplication. Then we have

0mEt

= vlE where vlE : E ! TE is the vertical lift. We use then lemma

37.13 and the commutation relations from 37.4 and we get in turn:

vlE _ K = @

0mEt

_ K = @

0 K _ mTE

= TK _ @

0mTE

t = TK _ vl(TE;T p;TM):

R(X; Y )s = rXrY s 􀀀 rY rXs 􀀀 r

[X;Y ]s

= K _ T(K _ Ts _ Y ) _ X 􀀀 K _ T(K _ Ts _ X) _ Y 􀀀 K _ Ts _ [X; Y ]

K _ Ts _ [X; Y ] = K _ vlE _ K _ Ts _ [X; Y ]

= K _ TK _ vlTE _ Ts _ [X; Y ]

= K _ TK _ T2s _ vlTM _ [X; Y ]

= K _ TK _ T2s _ ((TY _ X 􀀀 _M _ TX _ Y ) (T􀀀) 0TM _ Y )

= K _ TK _ T2s _ TY _ X 􀀀 K _ TK _ T2s _ _M _ TX _ Y 􀀀 0:

Now we sum up and use T2s _ _M = _E _ T2s to get the result.

(2) The same proof as for (1) applies for the _rst equality, with some obvious

changes. To see that it coincides with RE(Tf _ X; Tf _ Y )s it su_ces to write

out (1) and (T2s _ TX _ Y )(x) 2 T2E in canonical charts induced from vector

bundle charts of E.

(3) We have in turn

Tor(X; Y ) = rXY 􀀀 rY X 􀀀 [X; Y ]

= K _ TY _ X 􀀀 K _ TX _ Y 􀀀 K _ vlTM _ [X; Y ]

K _ vlTM _ [X; Y ] = K _ ((TY _ X 􀀀 _M _ TX _ Y ) (T��) 0TM _ Y )

= K _ TY _ X 􀀀 K _ _M _ TX _ Y 􀀀 0: _

37.16. Weil functors and Lie groups. We have seen in 10.17 that the

tangent bundle TG of a Lie group G is again a Lie group, the semidirect product

g n G of G with its Lie algebra g.

Now let A be a Weil algebra and let TA be its Weil functor. In the notation

of 4.1 the manifold TA(G) is again a Lie group with multiplication TA(_) and

inversion TA(_). By the properties 35.13 of the Weil functor TA we have a surjective

homomorphism _A : TAG ! G of Lie groups. Following the analogy with

the tangent bundle, for a 2 G we will denote its _ber over a by (TA)aG _ TAG,

likewise for mappings. With this notation we have the following commutative

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

328 Chapter VIII. Product preserving functors

diagram:

g N w g A

0 w (TA)0g w

(TA)0 exp

TAg w

TA exp

g w

expG

e w (TA)eG wTAG w

_A G w e

For a Lie group the structural mappings (multiplication, inversion, identity element,

Lie bracket, exponential mapping, Baker-Campbell-Hausdor_ formula,

adjoint action) determine each other mutually. Thus their images under the

Weil functor TA are again the same structural mappings. But note that the

canonical ip mappings have to be inserted like follows. So for example

g A _= TAg = TA(TeG) _ 􀀀! Te(TAG)

is the Lie algebra of TAG and the Lie bracket is just TA([ ; ]). Since the

bracket is bilinear, the description of 35.11 implies that [X a; Y b]TAg =

[X; Y ]g ab. Also TA expG = expTAG. Since expG is a di_eomorphism near

0 and since (TA)0(expG) depends only on the (invertible) jet of expG at 0, the

mapping (TA)0(expG) : (TA)0g ! (TA)eG is a di_eomorphism. Since (TA)0g is

a nilpotent Lie algebra, the multiplication on (TA)eG is globally given by the

Baker-Campbell-Hausdor_ formula. The natural transformation 0G : G ! TAG

is a homomorphism which splits the bottom row of the diagram, so TAG is the

semidirect product (TA)0g n G via the mapping TA_ : (u; g) 7! TA(_g)(u).

Since we will need it later, let us add the following _nal remark: If !G : TG !

TeG is the Maurer Cartan form of G (i.e. the left logarithmic derivative of IdG)

then

_0 _ TA!G _ _ : TTAG _= TATG ! TATeG _= TeTAG

is the Maurer Cartan form of TAG.

Remarks

The material in section 35 is due to [Eck,86], [Luciano, 88] and [Kainz-Michor,

87], the original ideas are from [Weil, 51]. Section 36 is due to [Eck, 86] and

[Kainz-Michor, 87], 36.7 and 36.8 are from [Kainz-Michor, 87], under stronger

locality conditions also to [Eck, 86]. 36.14 is due to [Eck, 86]. The material in

section 37 is from [Kainz-Michor, 87].

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

329

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