(30.7) for C ∞,ω ) the respective classes of curves which are described in (30.8) and

(30.9). The derivative can be computed pointwise on M .

30.11. Relation between spaces of real analytic and holomorphic sections

in ¬nite dimensions. Now let us assume that p : F ’ N is a ¬nite dimensional

holomorphic vector bundle over a ¬nite dimensional complex manifold N . For a

subset A ⊆ N let H(M ⊇ A ← F |A) be the space of germs along A of holomorphic

sections W ’ F |W for open sets W in N containing A. We equip H(M ⊇ A ←

F |A) with the locally convex topology induced by the inductive cone H(M ⊇

W ← F |W ) ’ H(M ⊇ A ← F |A) for all such W . This is Hausdor¬ since jet

prolongations at points in A separate germs.

For a real analytic ¬nite dimensional vector bundle p : E ’ M let C ω (M ← E) be

the space of real analytic sections s : M ’ E. Furthermore, let C ω (M ⊇ A ← E|A)

30.11

30.11 30. Spaces of sections of vector bundles 301

denote the space of germs at a subset A ⊆ M of real analytic sections de¬ned

near A. The complexi¬cation of this real vector space is the complex vector space

H(M ⊇ A ← EC |A), because germs of real analytic sections s : A ’ E extend

uniquely to germs along A of holomorphic sections W ’ EC for open sets W in

MC containing A, compare (11.2).

We topologize C ω (M ⊇ A ← E|A) as subspace of H(M ⊇ A ← EC |A).

Theorem. Structure on spaces of germs of sections. If p : E ’ M is a

real analytic ¬nite dimensional vector bundle and A a closed subset of M , then the

space C ω (M ⊇ A ← E|A) is convenient. Its bornology is generated by the cone

(ψ± )—

C ω (M ⊇ A ← E|A) ’ ’ ’ C ω (U± ⊇ U± © A, R)k ,

’’

where (U± , ψ± )± is an arbitrary real analytic vector bundle atlas of E. If A is

compact, the space C ω (M ⊇ A ← E|A) is nuclear.

The uniform boundedness principle for all point-evaluations holds if these separate

points. This follows from (11.6).

Proof. We show the corresponding result for holomorphic germs. By taking real

parts the theorem then follows. So let q : F ’ N be a holomorphic ¬nite dimen-

sional vector bundle, and let A be a closed subset of N . Then H(M ⊇ A ← F |A)

is a bornological locally convex space, since it is an inductive limit of the spaces

H(W ← F |W ) for open sets W containing A, which are nuclear and Fr´chet by e

(30.5). If A is compact, H(M ⊇ A ← F |A) is nuclear as countable inductive limit.

Let (U± , ψ± )± be a holomorphic vector bundle atlas for F . Then we consider the

cone

(ψ± )—

H(M ⊇ A ← F |A) ’ ’ ’ H(U± ⊇ U± © A, Ck ) = H(U± ⊇ U± © A, C)k .

’’

Obviously, each mapping is continuous, so the cone induces a bornology which is

coarser than the given one, and which is complete by (11.4).

It remains to show that every subset B ⊆ H(M ⊇ A ← F |A), such that (ps± )— (B)

is bounded in every H(U± ⊇ U± © A, C)k , is bounded in H(F |W ) for some open

neighborhood W of A in N .

Since all restriction mappings to smaller subsets are continuous, it su¬ces to show

the assertions of the theorem for some re¬nement of the atlas (U± ). Let us pass

¬rst to a relatively compact re¬nement. By topological dimension theory, there is a

further re¬nement such that any dimR N + 2 di¬erent sets have empty intersection.

We call the resulting atlas again (U± ). Let (K± ) be a cover of N consisting of

compact subsets K± ⊆ U± for all ±.

For any ¬nite set A of indices let us now consider all non empty intersections

UA := ±∈A U± and KA := ±∈A K± . Since by (8.4) (or (8.6)) the space H(UA ⊇

A © KA , C) is a regular inductive limit, there are open sets WA ⊆ UA containing

30.11

302 Chapter VI. In¬nite dimensional manifolds 30.12

A © KA , such that B|(A © KA ) (more precisely (ψA )— (B|(A © KA )) for some suitable

vector bundle chart mapping ψA ) is contained and bounded in H(WA , C)k . By

passing to smaller open sets, we may assume that WA1 ⊆ WA2 for A1 ⊇ A2 . Now

we de¬ne the subset

WA , where WA := WA \

W := K± .

A ±∈A

/

The set W is open since (K± ) is a locally ¬nite cover. For x ∈ A let A := {± : x ∈

K± }, then x ∈ WA .

Now we show that every germ s ∈ B has a unique extension to W . For every A

the germ of s along A © KA has a unique extension sA to a section over WA and

for A1 ⊆ A2 we have sA1 |WA2 = sA2 . We de¬ne the extension sW by sW |WA =

sA |WA . This is well de¬ned since one may check that WA1 © WA2 ⊆ WA1 ©A2 .

B is bounded in H(M ⊇ W ← F |W ) if it is uniformly bounded on each compact

subset K of W . This is true since each K is covered by ¬nitely many W± and

B|A © K± is bounded in H(W± , C).

30.12. Real analytic sections are dense. Let p : E ’ M be a real analytic

¬nite dimensional vector bundle. Then there is another real analytic vector bundle

p : E ’ M such that the Whitney sum E •E ’ M is real analytically isomorphic

to a trivial bundle M —Rk ’ M . This is seen as follows: By [Grauert, 1958, theorem

3] there is a closed real analytic embedding i : E ’ Rk for some k. Now the ¬ber

derivative along the zero section gives a ¬berwise linear and injective real analytic

mapping E ’ Rk , which induces a real analytic embedding j of the vector bundle

p : E ’ M into the trivial bundle M —Rk ’ M . The standard inner product on Rk

gives rise to the real analytic orthogonal complementary vector bundle E := E ⊥

and a real analytic Riemannian metric on the vector bundle E.

Now we can easily show that the space C ω (M ← E) of real analytic sections of the

vector bundle E ’ M is dense in the space of smooth sections, in the Whitney

C ∞ -topology: A smooth section corresponds to a smooth function M ’ Rk , which

we may approximate by a real analytic function in the Whitney C ∞ -topology, using

[Grauert, 1958, Proposition 8]. The latter one can be projected to a real analytic

approximating section of E.

Clearly, an embedding of the real analytic vector bundle into another one induces

a linear embedding of the spaces of real analytic sections onto a direct summand.

In this situation the orthogonal projection onto the vertical bundle V E within

T (M — Rk ) gives rise to a real analytic linear connection (covariant derivative)

: C ω (M ← T M ) — C ω (M ← E) ’ C ω (M ← E). If c : R ’ M is a smooth

or real analytic curve in M then the parallel transport Pt(c, t)v ∈ Ec(t) along c

is smooth or real analytic, respectively, in (t, v) ∈ R — Ec(0) . It is given by the

di¬erential equation ‚t Pt(c, t)v = 0.

More generally, for ¬ber bundles we get a similar result.

30.12

30.13 30. Spaces of sections of vector bundles 303

Lemma. Let p : E ’ M be a locally trivial real analytic ¬nite dimensional ¬ber

bundle. Then the set C ω (M ← E) of real analytic sections is dense in the space

C ∞ (M ← E) of smooth sections, in the Whitney C ∞ -topology.

The Whitney topology, even in in¬nite dimensions, will be explained in (41.10).

Proof. By the results of Grauert cited above, we choose a real analytic embedding

i : E ’ Rk onto a closed submanifold. Let ix : Ex ’ Rk be the restriction to

the ¬ber over x ∈ M . Using the standard inner product on Rk and the a¬ne

structure, we consider the orthogonal tubular neighborhood T (ix (Ex ))⊥ ⊃ Vx ∼=

k

Ux ‚ R , with projection qx : Ux ’ ix (Ex ), where we choose Vx so small that

U := x∈M {x} — Ux is open in M — Rk . Then q : U ’ (p, i)(E) ‚ M — Rk is real

analytic.

Now a smooth section of E corresponds to a smooth function f : M ’ Rk with

f (x) ∈ ix (Ex ). We may approximate f by a real analytic function g : M ’ Rk such

that g(x) ∈ Ux for each x. Then h(x) = qx (g(x)) corresponds to a real analytic

approximating section.

By looking at the trivial ¬ber bundle pr1 : N — N ’ M this lemma says that for

¬nite dimensional real analytic manifolds M and N the space C ω (M, N ) of real

analytic mappings is dense in C ∞ (M, N ), in the Whitney C ∞ -topology. Moreover,

for a smooth ¬nite dimensional vector bundle p : E ’ M there is a smoothly

isomorphic structure of a real analytic vector bundle. Namely, as smooth vector

bundle E is the pullback f — E(k, n) of the universal bundle E(k, n) ’ G(k, n) over

the Grassmann manifold G(k, n) for n high enough via a suitable smooth mapping

f : M ’ G(k, n). Choose a smoothly compatible real analytic structure on M and

choose a real analytic mapping g : M ’ G(k, n) which is near enough to f in the

Whitney C ∞ -topology to be smoothly homotopic to it. Then g — E(k, n) is a real

analytic vector bundle and is smoothly isomorphic to E = f — E(k, n).

30.13. Corollary. Let be a real analytic linear connection on a ¬nite dimen-

sional vector bundle p : E ’ M , which exists by (30.12). Then the following cone

generates the bornology on C ω (M ← E).

)—

Pt(c,

C (M ← E) ’ ’ ’ ’ C ± (R, Ec(0) ),

ω

’’’

s ’ (t ’ Pt(c, t)’1 s(c(t))),

for all c ∈ C ± (R, M ) and ± = ω, ∞.

Proof. The bornology induced by the cone is coarser that the given one by (30.6).

A still coarser bornology is induced by all curves subordinated to some vector

bundle atlas. Hence, by theorem (30.6) it su¬ces to check for a trivial bundle that

this bornology coincides with the given one. So we assume that E is trivial. For

the constant parallel transport the result follows from lemma (11.9).

30.13

304 Chapter VI. In¬nite dimensional manifolds 30.14

The change to an arbitrary real analytic parallel transport is done as follows: For

each C ± -curve c : R ’ M the diagram

wC

)—

Pt (c,

u

ω ±

C (M ← E) (R, Ec(0) )

∼

c—

u =

wC

C (c— E)

± ±

(R — Ec(0) )

—

c —

Pt (Id, )

commutes and the the bottom arrow is an isomorphism by (30.10), so the structure

induced by the cone does not depend on the choice of the connection.

30.14. Lemma. Curves in spaces of sections.

(1) For a real analytic ¬nite dimensional vector bundle p : E ’ M a curve

c : R ’ C ω (M ← E) is smooth if and only if c§ : R — M ’ E satis¬es the

following condition:

For each n there is an open neighborhood Un of R — M in R —

MC and a (unique) C n -extension c : Un ’ EC (29.2) such that

˜

c(t, ) is holomorphic for all t ∈ R.

˜

(2) For a smooth ¬nite dimensional vector bundle p : E ’ M a curve c :

R ’ C ∞ (M ← E) is real analytic if and only if c§ satis¬es the following

condition:

For each n there is an open neighborhood Un of R — M in C — M

and a (unique) C n -extension c : Un ’ E — C such that c( , x) :

˜ ˜

Un © (C — {x}) ’ Ex — C is holomorphic for all x ∈ M .

Proof. (1) By theorem (30.6) we may assume that M is open in Rn , and we

consider C ∞ (M, R) instead of C ∞ (M ← E). We note that C ω (M, R) is the real

part of H(Cm ⊇ M, C) by (11.2), which is a regular inductive limit of spaces

H(W, C) for open neighborhoods W of M in Cm by (8.6). By (1.8) the curve c is

smooth if and only if for each n and each bounded interval J ‚ R it factors to a

C n -curve J ’ H(W, C), which sits continuously embedded in C ∞ (W, R2 ). So the

associated mapping R — MC ⊇ J — W ’ C is C n and holomorphic in the second

variable, and conversely.

(2) By (30.1) we may assume that M is open in Rm , and again we consider

C ∞ (M, R) instead of C ∞ (M ← E). We note that C ∞ (M, R) is the projective

limit of the Banach spaces C n (Mi , R), where Mi is a covering of M by compact

cubes. By (9.9) the curve c is real analytic if and only if it is real analytic into

each C n (Mi , R). By (9.6) and (9.5) it extends locally to a holomorphic curve

C ’ C n (Mi , C). Its associated mappings ¬t together to the C n -extension c we

˜

were looking for.

30.14

31.1 31. Product preserving functors on manifolds 305

30.15. Lemma (Curves in spaces of sections with compact support).

For a smooth ¬nite dimensional vector bundle p : E ’ M a curve c : R ’

Cc (M ← E) is real analytic if and only if c§ satis¬es the following two condi-

∞

tions:

(1) For each compact interval [a, b] ‚ R there is a compact subset K ‚ M such

that c§ (t, x) is constant in t ∈ [a, b] for all x ∈ M \ K.

(2) For each n there is an open neighborhood Un of R — M in C — M and a

(unique) C n -extension c : Un ’ E —C such that c( , x) : Un ©(C—{—}) ’

˜ ˜

Ex — C is holomorphic for all x ∈ M .

∞

Proof. By lemma (30.4.1) a curve c : R ’ Cc (M ← E) is real analytic if it factors

∞

locally as a real analytic curve into some step CK (M ← E) for some compact K

in M (this is equivalent to (1)), and real analyticity is equivalent to (2), by lemma

(30.14.2).

31. Product Preserving Functors on Manifolds

In this section, we discuss Weil functors as generalized tangent bundles, namely

those product preserving functors of manifolds which can be described in some

detail. The name Weil functor derives from the fundamental paper [Weil, 1953]

who gave the construction in (31.5) in ¬nite dimensions for the ¬rst time.

31.1. A real commutative algebra A with unit 1 = 0 is called formally real if for

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

n

1

2

A : e = e, e = 0} ‚ A be the set of all nonzero idempotent elements in A. It is not

empty since 1 ∈ E. An idempotent e ∈ E is said to be minimal if for any e ∈ E

we have ee = e or ee = 0.

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

¬nite dimensional as a real vector space.

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

Furthermore, A decomposes as a sum of ideals A = A1 • · · · • Ak where Ai =

ei A = R · ei • Ni , as vector spaces, and Ni is a nilpotent ideal.

Proof. First we remark that every system of nonzero idempotents e1 , . . . , er sat-

isfying ei ej = 0 for i = j is linearly independent over R. Indeed, if we multiply

a linear combination k1 e1 + · · · + kr er = 0 by ei we obtain ki = 0. Consider a

non minimal idempotent e = 0. Then there exists e ∈ E with e = ee =: e = 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 ei ej = 0 for

i = 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 procedure stabilizes. This yields 1 = e1 + · · · + ek

31.1

306 Chapter VI. In¬nite dimensional manifolds 31.4

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 := ei A.

Now each Ai has only one nonzero idempotent, namely ei , and it su¬ces to investi-

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

now 1 is the only nonzero idempotent of A. Let N := {n ∈ A : nk = 0 for some k}

be the ideal of all nilpotent elements in A.

We claim that any x ∈ A \ N is invertible. Since A is ¬nite dimensional the

decreasing sequence

A ⊃ xA ⊃ x2 A ⊃ · · ·

of ideals must become stationary. If xk A = 0 then x ∈ N , thus there is a k such

that xk+ A = xk A = 0 for all > 0. Then x2k A = xk A, and there is some y ∈ A

with xk = x2k y. So we have (xk y)2 = xk y = 0, and since 1 is the only nontrivial

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

Thus, the quotient algebra A/N is a ¬nite dimensional ¬eld, so A/N equals R or

√

C. If A/N = C, let x ∈ A be such that x + N = ’1 ∈ 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.

31.2. 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 (31.1), a formally real and ¬nite dimensional unital commutative

algebra is the direct sum of ¬nitely many Weil algebras.

31.3. Remark. The evaluation mapping ev : M ’ Hom(C ∞ (M, R), R), given by

ev(x)(f ) := f (x), is bijective if and only if M is smoothly realcompact, see (17.1).

31.4. Corollary. For two manifolds M1 and M2 , with M2 smoothly real compact

and smoothly regular, the mapping

C ∞ (M1 , M2 ) ’ Hom(C ∞ (M2 , R), C ∞ (M1 , R))

f ’ (f — : g ’ g —¦ f )

is bijective.

Proof. Let x1 ∈ M1 and • ∈ Hom(C ∞ (M2 , R), C ∞ (M1 , R)). Then evx1 —¦• is in

Hom(C ∞ (M2 , R), R), so by (17.1) there is a unique x2 ∈ M2 such that evx1 —¦• =

evx2 . If we write x2 = f (x1 ), then f : M1 ’ M2 and •(g) = g —¦ f for all

g ∈ C ∞ (M2 , R). This implies that f is smooth, since M2 is smoothly regular, by

(27.5).

31.4

31.5 31. Product preserving functors on manifolds 307

31.5. 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

smooth manifolds modeled on convenient vector spaces into itself.

Step 1. If f ∈ C ∞ (R, R) and »1 + n ∈ R · 1 • N = A, we consider the Taylor

∞ f (j) (») j

expansion j ∞ f (»)(t) = t of f at », and we put

j=0 j!

∞

f (j) (») j

TA (f )(»1 + n) := f (»)1 + n,

j!

j=1

which is a ¬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 2. If f ∈ C ∞ (R, F ) for a convenient vector space F and »1+n ∈ R·1•N = A,

∞ f (j) (») j

we consider the Taylor expansion j ∞ f (»)(t) = t of f at », and we put

j=0 j!

∞

f (j) (») j

TA (f )(»1 + n) := 1 — f (») + n— ,

j!

j=1

which is a ¬nite sum, since n is nilpotent. Then TA (f ) : A ’ A — F =: TA F is

smooth.

Step 3. For f ∈ C ∞ (E, F ), where E, F are convenient vector spaces, we want to

de¬ne the value of TA (f ) at an element of the convenient vector space TA E = A—E.

Such an element may be uniquely written as 1 — x1 + j nj — xj , where 1 and the

nj ∈ N form a ¬xed ¬nite linear basis of A, and where the xi ∈ E. Let again

j ∞ f (x1 )(y) = 1k k

k≥0 k! d f (x1 )(y ) be the Taylor expansion of f at x1 ∈ E for

y ∈ E. Then we put

TA (f )(1 — x1 + nj — xj ) :=

j

1

nj1 . . . njk — dk f (x1 )(xj1 , . . . , xjk ),

= 1 — f (x1 ) +

k! j

1 ,...,jk

k≥0

which also is a ¬nite sum. A change of basis in N induces the transposed change

in the xi , namely i ( j aj nj ) — xi = j nj — ( i aj xi ), so the value of TA (f )

¯ i¯

i

is independent of the choice of the basis of N . 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 (IdE ) = IdTA E .

If • : A ’ B is a homomorphism between two Weil algebras we have (•—F )—¦TA f =

TB f —¦ (• — E) for f ∈ C ∞ (E, F ).

Step 4. Let π = πA : 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 3 the

following diagram commutes for f ∈ C ∞ (E, F ):