derivative of a product h of two functions f and g is given by

d2 h(0, 0)(w1 , w2 ) = d2 f (0, 0)(w1 , w2 ) g(0, 0)

+ df (0, 0)(w1 ) dg(0, 0)(w2 )

+ df (0, 0)(w2 ) dg(0, 0)(w1 )

+ f (0, 0) d2 g(0, 0)(w1 , w2 ).

Thus ‚ is a derivation since the middle terms give ¬nite dimensional operators in

L2 ( 2 , 2 ; R). It is not a direct sum of two operational tangent vectors on 2 since

functions f depending only on the j-th factor have as second derivative forms with

nonzero (j,j) entry only. Hence D0 (prj )(‚)(f ) = ‚(f —¦ prj ) = ((f —¦ prj ) (0)) = 0,

but ‚ = 0.

28.16

287

29. Vector Bundles

29.1. Vector bundles. Let p : E ’ M be a smooth mapping between manifolds.

By a vector bundle chart on (E, p, M ) we mean a pair (U, ψ), where U is an open

subset in M , and where ψ is a ¬ber respecting di¬eomorphism as in the following

diagram:

w

‘ ψ

E|U := p’1 (U )

U —V

‘“

‘ pr

£

p 1

U.

Here V is a ¬xed convenient vector space, called the standard ¬ber or the typical

¬ber, real for the moment.

’1

Two vector bundle charts (U1 , ψ1 ) and (U2 , ψ2 ) are called compatible, if ψ1 —¦ ψ2 is

’1

a ¬ber linear isomorphism, i.e., (ψ1 —¦ ψ2 )(x, v) = (x, ψ1,2 (x)v) for some mapping

ψ1,2 : U1,2 := U1 © U2 ’ GL(V ). The mapping ψ1,2 is then unique and smooth

into L(V, V ), and it is called the transition function between the two vector bundle

charts.

A vector bundle atlas (U± , ψ± )±∈A for p : E ’ M is a set of pairwise compatible

vector bundle charts (U± , ψ± ) such that (U± )±∈A is an open cover of M . Two vector

bundle atlas are called equivalent, if their union is again a vector bundle atlas.

A (smooth) vector bundle p : E ’ M consists of manifolds E (the total space), M

(the base), and a smooth mapping p : E ’ M (the projection) together with an

equivalence class of vector bundle atlas: We must know at least one vector bundle

atlas. The projection p turns out to be a surjective smooth mapping which has the

0-section as global smooth right inverse. Hence it is a ¬nal smooth mapping, see

(27.15).

If all mappings mentioned above are real analytic we call p : E ’ M a real analytic

vector bundle. If all mappings are holomorphic and V is a complex vector space we

speak of a holomorphic vector bundle.

29.2. Remark. Let p : E ’ M be a ¬nite dimensional real analytic vector bundle.

If we extend the transition functions ψ±β to ψ±β : U±β ’ GL(VC ) = GL(V )C , we

see that there is a holomorphic vector bundle (EC , pC , MC ) over a complex (even

Stein) manifold MC such that E is isomorphic to a real part of EC |M , compare

(11.1). The germ of it along M is unique. Real analytic sections s : M ’ E

coincide with certain germs along M of holomorphic sections W ’ EC for open

neighborhoods W of M in MC .

Note that every smooth ¬nite dimensional vector bundle admits a compatible real

analytic structure, see [Hirsch, 1976, p. 101].

29.3. We will now give a formal description of the set vector bundles with ¬xed

base M and ¬xed standard ¬ber V , up to equivalence. We only treat smooth vector

29.3

288 Chapter VI. In¬nite dimensional manifolds 29.3

bundles; similar descriptions are possible for real analytic and holomorphic vector

bundles.

Let us ¬rst ¬x an open cover (U± )±∈A of M . If p : E ’ M is a vector bundle

which admits a vector bundle atlas (U± , ψ± ) with the given open cover, then we

’1

have ψ± —¦ ψβ (x, v) = (x, ψ±β (x)v) for transition functions ψ±β : U±β = U± © Uβ ’

GL(V ) ‚ L(V, V ), which are smooth. This family of transition functions satis¬es

ψ±β (x) · ψβγ (x) = ψ±γ (x) for each x ∈ U±βγ = U± © Uβ © Uγ ,

(1)

for all x ∈ U± .

ψ±± (x) = e

Condition (1) is called a cocycle condition, and thus we call the family (ψ±β ) the

cocycle of transition functions for the vector bundle atlas (U± , ψ± ).

Let us now suppose that the same vector bundle p : E ’ M is described by an

equivalent vector bundle atlas (U± , •± ) with the same open cover (U± ). Then the

vector bundle charts (U± , ψ± ) and (U± , •± ) are compatible for each ±, so •± —¦

’1

ψ± (x, v) = (x, „± (x)v) for some „± : U± ’ GL(V ). But then we have

’1

(x, „± (x)ψ±β (x)v) = (•± —¦ ψ± )(x, ψ±β (x)v)

’1 ’1

’1

= (•± —¦ ψ± —¦ ψ± —¦ ψβ )(x, v) = (•± —¦ ψβ )(x, v)

= (•± —¦ •’1 —¦ •β —¦ ψβ )(x, v) = (x, •±β (x)„β (x)v).

’1

β

So we get

for all x ∈ U±β .

(2) „± (x)ψ±β (x) = •±β (x)„β (x)

We say that the two cocycles (ψ±β ) and (•±β ) of transition functions over the cover

(U± ) are cohomologous. The cohomology classes of cocycles (ψ±β ) over the open

ˇ

cover (U± ) (where we identify cohomologous ones) form a set H 1 ((U± ), GL(V ))

ˇ

the ¬rst Cech cohomology set of the open cover (U± ) with values in the sheaf

C ∞ ( , GL(V )) =: GL(V ).

Now let (Wi )i∈I be an open cover of M re¬ning (U± ) with Wi ‚ Uµ(i) , where

µ : I ’ A is some re¬nement mapping. Then for any cocycle (ψ±β ) over (U± )

we de¬ne the cocycle µ— (ψ±β ) =: (•ij ) by the prescription •ij := ψµ(i),µ(j) |Wij .

The mapping µ— respects the cohomology relations and thus induces a mapping

ˇ ˇ

µ : H 1 ((U± ), GL(V )) ’ H 1 ((Wi ), GL(V )). One can show that the mapping µ—

depends on the choice of the re¬nement mapping µ only up to cohomology (use

„i = ψµ(i),·(i) |Wi if µ and · are two re¬nement mappings), so we may form the

ˇ ˇ

inductive limit lim H 1 (U, GL(V )) =: H 1 (M, GL(V )) over all open covers of M

’’

directed by re¬nement.

ˇ

Theorem. H 1 (M, GL(V )) is bijective to the set of all isomorphism classes of vec-

tor bundles over M with typical ¬ber V .

Proof. Let (ψ±β ) be a cocycle of transition functions ψ±β : U±β ’ GL(V ) over

some open cover (U± ) of M . We consider the disjoint union ±∈A {±} — U± — V

29.3

29.4 29. Vector bundles 289

and the following relation on it: (±, x, v) ∼ (β, y, w) if and only if x = y and

ψβ± (x)v = w.

By the cocycle property (1) of (ψ±β ), this is an equivalence relation. The space

of all equivalence classes is denoted by E = V B(ψ±β ), and it is equipped with

the quotient topology. We put p : E ’ M , p[(±, x, v)] = x, and we de¬ne the

vector bundle charts (U± , ψ± ) by ψ± [(±, x, v)] = (x, v), ψ± : p’1 (U± ) =: E|U± ’

’1

U± — V . Then the mapping ψ± —¦ ψβ (x, v) = ψ± [(β, x, v)] = ψ± [(±, x, ψ±β (x)v)] =

(x, ψ±β (x)v) is smooth, so E becomes a smooth manifold. E is Hausdor¬: let u = v

in E; if p(u) = p(v) we can separate them in M and take the inverse image under

p; if p(u) = p(v), we can separate them in one chart. Hence p : E ’ M is a vector

bundle.

Now suppose that we have two cocycles (ψ±β ) over (U± ), and (•ij ) over (Vi ).

Then there is a common re¬nement (Wγ ) for the two covers (U± ) and (Vi ). The

construction described a moment ago gives isomorphic vector bundles if we restrict

the cocycle to a ¬ner open cover. So we may assume that (ψ±β ) and (•±β ) are

cocycles over the same open cover (U± ). If the two cocycles are cohomologous, i.e.,

„± · ψ±β = •±β · „β on U±β , then a ¬ber linear di¬eomorphism „ : V B(ψ±β ) ’

V B(•±β ) is given by „ [(±, x, v)] = [(±, x, „± (x)v)]. By relation (2), this is well

de¬ned, so the vector bundles V B(ψ±β ) and V B(•±β ) are isomorphic.

Most of the converse direction has already been shown above, and the argument

given can easily be re¬ned to show that isomorphic vector bundles give cohomolo-

gous cocycles.

Remark. If GL(V ) is an abelian group (if V is real or complex 1-dimensional),

ˇ

then H 1 (M, GL(V )) is a usual cohomology group with coe¬cients in the sheaf

GL(V ), and it can be computed with the methods of algebraic topology. If GL(V ) is

not abelian, then the situation is rather mysterious: there is no accepted de¬nition

ˇ ˇ

for H 2 (M, GL(V )) for example. So H 1 (M, GL(V )) is more a notation than a

mathematical concept.

A coarser relation on vector bundles (stable equivalence) leads to the concept of

topological K-theory, which can be handled much better, but is only a quotient of

the true situation.

29.4. Let p : E ’ M and q : F ’ N be vector bundles. A vector bundle

homomorphism • : E ’ F is a ¬ber respecting, ¬ber linear smooth mapping

wF

•

E

u u

p q

wN

•

M

i.e., we require that •x : Ex ’ F•(x) is linear. We say that • covers •, which turns

out to be smooth. If • is invertible, it is called a vector bundle isomorphism.

29.4

290 Chapter VI. In¬nite dimensional manifolds 29.6

29.5. Constructions with vector bundles. Let F be a covariant functor from

the category of convenient vector spaces and bounded linear mappings into it-

self, such that F : L(V, W ) ’ L(F(V ), F(W )) is smooth. Then F will be called

a smooth functor for shortness™ sake. Well known examples of smooth functors

are F (V ) = ˜ β k V , the k-th iterated convenient tensor product, F(V ) = Λk (V )

(the k-th exterior product, the skew symmetric elements in ˜ k V ), or F(V ) =

β

Lk (V ; R), in particular F(V ) = V , also F(V ) = D0 V (see the proof of lemma

sym

(28.9)), and similar ones.

If p : E ’ M is a vector bundle, described by a vector bundle atlas with cocycle

of transition functions •±β : U±β ’ GL(V ), where (U± ) is an open cover of M ,

then we may consider the functions F(•±β ) : x ’ F(•±β (x)), U±β ’ GL(F(V )),

which are smooth into L(F(V ), F(V )). Since F is a covariant functor, F(•±β )

satis¬es again the cocycle condition (29.3.1), and cohomology of cocycles (29.3.2)

p

is respected, so there exists a unique vector bundle F(E) := V B(F(•±β ) ’ M ,

’

the value at the vector bundle p : E ’ M of the canonical extension of the functor

F to the category of vector bundles and their homomorphisms.

If F is a contravariant smooth functor like the duality functor F(V ) = V , then we

have to consider the new cocycle F(•’1 ) = F(•β± ) instead.

±β

If F is a contra-covariant smooth bifunctor like L(V, W ), then the rule

’1

F(V B(ψ±β ), V B(•±β )) := V B(F(ψ±β , •±β ))

describes the induced canonical vector bundle construction.

So for vector bundles p : E ’ M and q : F ’ M we have the following vector

bundles with base M : Λk E, E • F , E — , ΛE := k≥0 Λk E, E —β F , L(E, F ), and

˜

so on.

29.6. Pullback of vector bundles. Let p : E ’ M be a vector bundle, and let

f : N ’ M be smooth. Then the pullback vector bundle f — p : f — E ’ N with the

same typical ¬ber and a vector bundle homomorphism

wE

p— f

—

fE

p

f —p

u u

wM

f

N

is de¬ned as follows. Let E be described by a cocycle (ψ±β ) of transition functions

over an open cover (U± ) of M , E = V B(ψ±β ). Then (ψ±β —¦ f ) is a cocycle of

transition functions over the open cover (f ’1 (U± )) of N , and the bundle is given

by f — E := V B(ψ±β —¦ f ). As a manifold we have f — E = N — E.

(f,M,p)

The vector bundle f — E has the following universal property: For any vector bundle

q : F ’ P , vector bundle homomorphism • : F ’ E, and smooth g : P ’ N such

29.6

29.8 29. Vector bundles 291

that f —¦ g = •, there is a unique vector bundle homomorphism ψ : F ’ f — E with

ψ = g and p— f —¦ ψ = •.

wf E wE

p— f

ψ —

F

q p

f —p

u u u

wN wM

g f

P

29.7. Proposition. Let p : E ’ M be a smooth vector bundle with standard

¬ber V , and suppose that M and the product of the model space of M and V are

smoothly paracompact. In particular this holds if M and V are metrizable and

smoothly paracompact.

Then the total space E is smoothly paracompact.

Proof. If M and V are metrizable and smoothly paracompact then by (27.9) the

product M — V is smoothly paracompact. Let M be modeled on the convenient

vector space F . Let (U± ) be an open cover of E. We choose Wβ ‚ Wβ ‚ Wβ in

M such that the (Wβ ) are an open cover of M and the Wβ are open, trivializing

for the vector bundle E, and domains of charts for M . We choose a partition

of unity (fβ ) on M which is subordinated to (Wβ ). Then E|Wβ ∼ Wβ — V is

=

di¬eomorphic to an open subset of the smoothly paracompact convenient vector

space F — V . We consider the open cover of F — V consisting of (U± © E|Wβ )±

and (F \ supp(fβ )) — V and choose a subordinated partition of unity consisting

of (g±β )± and one irrelevant function. Since the g±β have support with respect to

E|Wβ in U± © E|Wβ they extend to smooth functions on the whole of E. Then

( β g±β (fβ —¦ p))± is a partition of unity which is subordinated to U± .

29.8. Theorem. For any vector bundle p : E ’ M with M smoothly regular

there is a smooth vector bundle embedding into a trivial vector bundle over M with

locally (over M ) splitting image. If the ¬bers are Banach spaces, and M is smoothly

paracompact then the ¬ber of the trivial bundle can be chosen as Banach space as

well.

A ¬berwise short exact sequence of vector bundles over a smoothly paracompact

manifold M which is locally splitting is even globally splitting.

Proof. We choose ¬rst a vector bundle atlas, then smooth bump functions with

supports in the base sets of the atlas such that the carriers still cover M , then we

re¬ne the atlas such that in the end we have an atlas (U± , ψ± : E|U± ’ U± — E± )

and functions f± ∈ C ∞ (M, R) with U± ⊃ supp(f± ) such that (carr(f± )) is an open

cover of M .

Then we de¬ne a smooth vector bundle homomorphism

¦:E’M— E±

±

¦(u) = (p(u), (f± (p(u)) · ψ± (u))± ).

29.8

292 Chapter VI. In¬nite dimensional manifolds 29.9

This gives a locally splitting embedding with the following inverse

1 ’1

(x, (vβ )β ) ’ ψ± (x, v± )

f± (x)

over carr(f± ).

If the ¬bers are Banach spaces and M is smoothly paracompact, we may assume

that the family (ψ± )± is a smooth partition of unity. Then we may take as ¬ber

of the trivial bundle the space {(x± )± ∈ ± E± : ( x± )± ∈ c0 } supplied with the

supremum norm of the norms of the coordinates.

The second assertion follows since we may glue the local splittings with the help of

a partition of unity.

29.9. The kinematic tangent bundle of a vector bundle. Let p : E ’ M be

a vector bundle with ¬ber addition +E : E—M E ’ E and ¬ber scalar multiplication

mE : E ’ E. Then πE : T E ’ E, the tangent bundle of the manifold E, is itself

t

a vector bundle, with ¬ber addition +T E and scalar multiplication mT E

t

If (U± , ψ± : E|U± ’ U± — V )±∈A is a vector bundle atlas for E, and if (u± : U± ’

u± (U± ) ‚ F ) is a manifold atlas for M , then (E|U± , ψ± )±∈A is an atlas for the

manifold E, where

ψ± := (u± — IdV ) —¦ ψ± : E|U± ’ U± — V ’ u± (U± ) — V ‚ F — V.

Hence, the family (T (E|U± ), T ψ± : T (E|U± ) ’ T (u± (U± )—V ) = (u± (U± )—V —F —

V )±∈A is the atlas describing the canonical vector bundle structure of πE : T E ’ E.

The transition functions are:

’1

(ψ± —¦ ψβ )(x, v) = (x, ψ±β (x)v)

(u± —¦ u’1 )(x) = u±β (x)

β

(ψ± —¦ (ψβ )’1 )(x, v) = (u±β (x), ψ±β (u’1 (x))v)

β

(T ψ± —¦ T (ψβ )’1 )(x, v; ξ, w) =

= u±β (x), ψ±β (u’1 (x))v; d(u±β )(x)ξ, (d(ψ±β —¦ u’1 )(x)ξ)v + ψ±β (u’1 (x))w .

β β β

So we see that for ¬xed (x, v) the transition functions are linear in (ξ, w) ∈ F — V .

This describes the vector bundle structure of the tangent bundle πE : T E ’ E.

For ¬xed (x, ξ) the transition functions of T E are also linear in (v, w) ∈ V — V .

This gives a vector bundle structure on T p : T E ’ T M . Its ¬ber addition will

be denoted by T (+E ) : T (E —M E) = T E —T M T E ’ T E, since it is the tangent

mapping of +E . Likewise, its scalar multiplication will be denoted by T (mE ). One

t

might say that the vector bundle structure on T p : T E ’ T M is the derivative of

the original one on E.

The subbundle {Ξ ∈ T E : T p.Ξ = 0 in T M } = (T p)’1 (0) ⊆ T E is denoted by V E

and is called the vertical bundle over E. The local form of a vertical vector Ξ is

T ψ± .Ξ = (x, v; 0, w), so the transition functions look like

(T ψ± —¦ T (ψβ )’1 )(x, v; 0, w) = (u±β (x), ψ±β (u’1 (x)v; 0, ψ±β (u’1 (x)w).

β β

29.9

30.1 30. Spaces of sections of vector bundles 293

They are linear in (v, w) ∈ V — V for ¬xed x, so V E is a vector bundle over M . It

coincides with 0— (T E, T p, T M ), the pullback of the bundle T E ’ T M over the

M

zero section. We have a canonical isomorphism vlE : E —M E ’ V E, called the

d

vertical lift, given by vlE (ux , vx ) := dt |0 (ux +tvx ), which is ¬ber linear over M . The

local representation of the vertical lift is (T ψ± —¦ vlE —¦ (ψ± — ψ± )’1 )((x, u), (x, v)) =

(x, u; 0, v).

If (and only if) • : (E, p, M ) ’ (F, q, N ) is a vector bundle homomorphism, then

we have vlF —¦ (• —M •) = T • —¦ vlE : E —M E ’ V F ‚ T F . So vl is a natural

transformation between certain functors on the category of vector bundles and their

homomorphisms.

The mapping vprE := pr2 —¦ vl’1 : V E ’ E is called the vertical projection. Note

E

’1

also the relation pr1 —¦vlE = πE |V E.

29.10. The second kinematic tangent bundle of a manifold. All of (29.9)

is valid for the second tangent bundle T 2 M = T T M of a manifold, but here we

have one more natural structure at our disposal. The canonical ¬‚ip or involution

κM : T 2 M ’ T 2 M is de¬ned locally by

(T 2 u —¦ κM —¦ T 2 u’1 )(x, ξ; ·, ζ) = (x, ·; ξ, ζ),

where (U, u) is a chart on M . Clearly, this de¬nition is invariant under changes of

charts.

The ¬‚ip κM has the following properties:

κN —¦ T 2 f = T 2 f —¦ κM for each f ∈ C ∞ (M, N ).

(1)

T (πM ) —¦ κM = πT M .

(2)

πT M —¦ κM = T (πM ).

(3)

κ’1 = κM .

(4) M

κM is a linear isomorphism from T (πM ) : T T M ’ T M to πT M : T T M ’

(5)

T M , so it interchanges the two vector bundle structures on T T M .

‚‚

(6) κM is the unique smooth mapping T T M ’ T T M satisfying ‚t ‚s c(t, s) =

‚‚

κM ‚s ‚t c(t, s) for each c : R2 ’ M .

All this follows from the local formula given above.

29.11. Remark. In (28.16) we saw that in general D0 (E — F ) = D0 E — D0 F . So

the constructions of (29.9) and (29.10) do not carry over to the operational tangent

bundles.

30. Spaces of Sections of Vector Bundles

30.1. Let us ¬x a vector bundle p : E ’ M for the moment. On each ¬ber

Ex := p’1 (x) (for x ∈ M ) there is a unique structure of a convenient vector space,

induced by any vector bundle chart (U± , ψ± ) with x ∈ U± . So 0x ∈ Ex is a special

element, and 0 : M ’ E, 0(x) = 0x , is a smooth mapping, the zero section.

30.1

294 Chapter VI. In¬nite dimensional manifolds 30.2

A section u of p : E ’ M is a smooth mapping u : M ’ E with p —¦ u = IdM . The

support of the section u is the closure of the set {x ∈ M : u(x) = 0x } in M . The

space of all smooth sections of the bundle p : E ’ M will be denoted by either

C ∞ (M ← E) = C ∞ (E, p, M ) = C ∞ (E). Also the notation “(E ’ M ) = “(p) =

“(E) is used in the literature. Clearly, it is a vector space with ¬ber wise addition

and scalar multiplication.

If (U± , ψ± )±∈A is a vector bundle atlas for p : E ’ M , then any smooth mapping

’1

f± : U± ’ V (the standard ¬ber) de¬nes a local section x ’ ψ± (x, f± (x)) on U± .

If (g± )±∈A is a partition of unity subordinated to (U± ), then a global section can be

’1

formed by x ’ ± g± (x) · ψ± (x, f± (x)). So a smooth vector bundle has ”many”

smooth sections if M admits enough smooth partitions of unity.

We equip the space C ∞ (M ← E) with the structure of a convenient vector space

given by the closed embedding

C ∞ (M ← E) ’ C ∞ (U± , V )

±

s ’ pr2 —¦ψ± —¦ (s|U± ),

where C ∞ (U± , V ) carries the natural structure described in (27.17), see also (3.11).

This structure is independent of the choice of the vector bundle atlas, because

C ∞ (U± , V ) ’ β C ∞ (U±β , V ) is a closed linear embedding for any other atlas

(Uβ )β .

Proposition. The space C ∞ (M ← E) of sections of the vector bundle (E, p, M )

with this structure satis¬es the uniform boundedness principle with respect to the

point evaluations evx : C ∞ (M ← E) ’ Ex for all x ∈ M .

If M is a separable manifold modeled on duals of nuclear Fr´chet spaces, and if

e

each ¬ber Ex is a nuclear Fr´chet space then C ∞ (M ← E) is a nuclear Fr´chet

e e

space and thus smoothly paracompact.

Proof. By de¬nition of the structure on C ∞ (M ← E) the uniform boundedness

principle follows from (5.26) via (5.25).

For the statement about nuclearity note that by (6.1) the spaces C ∞ (U± , V ) are

nuclear since we may assume that the U± form a countable cover of M by charts

which are di¬eomorphic to c∞ -open subsets of duals of nuclear Fr´chet spaces, and

e

closed subspaces of countable products of nuclear Fr´chet spaces are again nuclear

e

Fr´chet. By (16.10) nuclear Fr´chet spaces are smoothly paracompact.

e e

30.2. Lemma. Let M be a smooth manifold and let f : M ’ L(E, F ) be smooth,

where E and F are convenient vector spaces.

Then f— (h)(x) := f (x)(h(x)) is a linear bounded C ∞ (M, E) ’ C ∞ (M, F ) with the

natural structure of convenient vector spaces described in (27.17). The correspond-

ing statements in the real analytic and holomorphic cases are also true.

Proof. This follows from the uniform boundedness principles and the exponential

laws of (27.17).

30.2

30.3 30. Spaces of sections of vector bundles 295

30.3. Lemma. Under additional assumptions we have alternative descriptions of

the convenient structure on the vector space of sections C ∞ (M ← E):

(1) If M is smoothly regular, choose a smooth closed embedding E ’ M —F into

a trivial vector bundle with ¬ber a convenient vector space F by (29.8). Then

C ∞ (M ← E) can be considered as a closed linear subspace of C ∞ (M, F ),

with the natural structure from (27.17).

(2) If there exists a smooth linear covariant derivative with unique parallel

transport on p : E ’ M , see , then we equip C ∞ (M ← E) with the initial

structure with respect to the cone:

)—

Pt(c,

C (M ← E) ’ ’ ’ ’ C ∞ (R, Ec(0) ),

∞

’’’

s ’ (t ’ Pt(c, t)’1 s(c(t))),

where c ∈ C ∞ (R, M ) and Pt denotes the parallel transport.

The space C ∞ (M ← E) of sections of the vector bundle p : E ’ M with this struc-

ture satis¬es the uniform boundedness principle with respect to the point evaluations

evx : C ∞ (M ← E) ’ Ex for all x ∈ M .

If M is a separable manifold modeled on duals of nuclear Fr´chet spaces, and if

e

each ¬ber Ex is a nuclear Fr´chet space then C ∞ (M ← E) is a nuclear Fr´chet

e e

space and thus smoothly paracompact.

If in (1) M is even smoothly paracompact we may choose a ˜complementary™ smooth

vector bundle p : E ’ M such that the Whitney sum is trivial E •M E ∼ M — F ,

=

see also (29.8).

: X(M ) — C ∞ (M ← E) ’ C ∞ (M ← E) with

For a linear covariant derivative

unique parallel transport we require that the parallel transport Pt(c, t)v ∈ Ec(t)

along each smooth curve c : R ’ M for all v ∈ Ec(0) and t ∈ R is the unique

solution of the di¬erential equation ‚t Pt(c, t)v = 0. See (32.12) till (32.16).

Proof. This structure is independent of the choice of the vector bundle atlas,

because C ∞ (U± , V ) ’ β C ∞ (U±β , V ) is a closed linear embedding for any other

atlas (Uβ ).

The structures from (30.1) and (1) give even the same locally convex topology if

we equip C ∞ (M, F ) with the initial topology given by the following diagram.

wC

inj—

C ∞ (M ← E) ∞

(M, F )

u u

(pr2 —¦ψ± )—

wC

C ∞ (U± , V ) ∞

(U± , F )

where the bottom arrow is a push forward with the vector bundle embedding h :

U± ’ L(V, F ) of trivial bundles, given by h§ := pr2 —¦ inj —¦ψ± : U± — V ’ F , which

’1

is bounded by (30.2).

30.3

296 Chapter VI. In¬nite dimensional manifolds 30.4

We now show that the identity from description (2) to description (30.1) is bounded.

The restriction mapping C ∞ (M ← E) ’ C ∞ (U± ← E|U± ) is obviously bounded

for description (2) on both sides. Hence, it su¬ces to check for a trivial bundle E =

M — V , that the identity from description (2) to description (30.1) is bounded. For

the constant parallel transport Ptconst the result follows from proposition (27.17).

The change to an arbitrary parallel transport is done as follows: For each C ∞ -curve

c : R ’ M the diagram

wC

)—

Pt (c,

∞ ∞

C (M, V ) (R, V )

¢

u

h—

Ptconst (c, )— = c—

C ∞ (R, V )

commutes, where h : R ’ GL(V ) is given by h(t)(v) = Pt(c, t)v with inverse

h’1 (t)(w) = Pt(c, t)’1 w = Pt(c( +t), ’t)w, and its push forward is bibounded

by (30.2).

Finally, we show that the identity from description (30.1) to description (2) is

bounded. The structure on C ∞ (R, Ec(0) ) is initial with respect to the restriction

maps to a covering by intervals I which is subordinated to the cover c’1 (U± ) of

)—

Pt(c,

R. Thus, it su¬ces to show that the map C (M ← E) ’ ’ ’ ’ C ∞ (I, Ec(0) ) is

∞

’’’

bounded for the structure (30.1) on C ∞ (M ← E). This map factors as

wC

(pr2 —¦ψ± )—

C ∞ (M ← E) ∞

(U± , V )

)—

u

Pt(c,

(c|I)—

C ∞ (R, Ec(0) )

u u

u h—

C ∞ (I, Ec(0) ) C ∞ (I, V )

where

h(t)(v) := Pt(c, t)’1 (ψ± (c(t), v)) = Pt(c(

’1 ’1

+t), ’t)(ψ± (c(t), v))

is again a smooth map I ’ L(V, Ec(0) )).

30.4. Spaces of smooth sections with compact supports. For a smooth

vector bundle p : E ’ M with ¬nite dimensional second countable base M and

∞

standard ¬ber V we denote by Cc (M ← E) the vector space of all smooth sections

with compact supports in M .

∞

Lemma. The following structures of a convenient vector space on Cc (M ← E)

are all equivalent:

∞

(1) Let CK (M ← E) be the space of all smooth sections of E ’ M with supports

contained in the ¬xed compact subset K ‚ M , a closed linear subspace of

30.4

30.4 30. Spaces of sections of vector bundles 297

C ∞ (M ← E). Consider the ¬nal convenient vector space structure on

∞

Cc (M ← E) induced by the cone

∞ ∞

CK (M ← E) ’ Cc (M ← E)

∞

where K runs through a basis for the compact subsets of M . Then Cc (M ←

∞

E) is even the strict and regular inductive limit of spaces CK (M ← E)

where K runs through a countable base of compact sets.

(2) Choose a second smooth vector bundle q : E ’ M such that the Whitney

sum is trivial (29.8): E•E ∼ M —F . Then Cc (M ← E) can be considered

∞

=

∞

as a closed direct summand of Cc (M, F ).

∞

The space Cc (M ← E) satis¬es the uniform boundedness principle with respect to

the point evaluations. Moreover, if the standard ¬ber V is a nuclear Fr´chet space

e

∞

and the base M is in addition separable then Cc (M ← E) is smoothly paracompact.

Proof. Since CK (M ← E) is closed in C ∞ (M ← E) the inductive limit CK (M ←

∞ ∞

∞ ∞

E) ’ Cc (M ← E) is strict. So the limit is regular (52.8) and hence Cc (M ← E)

∞

is convenient with the structure in (1). The direct sum property CK (M ← E) ‚

∞

CK (M, F ) from (30.3.1) passes through the direct limits, so the equivalence of

statements (1) and (2) follows.

∞

We now show that Cc (M ← E) satis¬es the uniform boundedness principle for the

point evaluations. Using description (2) and (5.25) for a direct sum we may assume

∞

that the bundle is trivial, hence we only have to consider Cc (M, V ) for a convenient

∞

vector space V . Now let F be a Banach space, and let f : F ’ Cc (M, V ) be a

linear mapping, such that evx —¦f : F ’ V is bounded for each x ∈ M . Then by

the uniform boundedness principle (27.17) it is bounded into C ∞ (M, V ). We claim

∞

that f has values even in CK (M, V ) for some K, so it is bounded therein, and

∞

hence in Cc (M, V ), as required.

If not we can recursively construct the following data: a discrete sequence (xn ) in

M , a bounded sequence (yn ) in the Banach space F , and linear functionals n ∈ V

such that ±

= 0 if n < k,

| k (f (yn )(xk ))| = 1 if n = k,

< 1 if n > k.

Namely, we choose yn ∈ F and xn ∈ M such that f (yn )(xn ) = 0 in V , and xn has

distance 1 to m<n supp(f (ym )) (in a complete Riemannian metric, where closed

bounded subsets are compact). By shrinking yn we may get | m (f (yn )(xm ))| < 1

for m < n. Then we choose n ∈ V such that n (f (yn )(xn )) = 1.

Then y := n 21 yn ∈ F , and f (y)(xk ) = 0 for all k since | k (f (y)(xk ))| > 0. So

n

∞

f (y) ∈ Cc (M, V ).

/

For the last assertion, if the standard ¬ber V is a nuclear Fr´chet space and the

e

base M is separable then C ∞ (M ← E) is a nuclear Fr´chet space by the propo-

e

∞

sition in (30.1), so each closed linear subspace CK (M ← E) is a nuclear Fr´chet

e

∞

space, and by (16.10) the countable strict inductive limit Cc (M ← E) is smoothly

paracompact.

30.4

298 Chapter VI. In¬nite dimensional manifolds 30.6

30.5. Spaces of holomorphic sections. Let q : F ’ N be a holomorphic

vector bundle over a complex (i.e., holomorphic) manifold N with standard ¬ber

V , a complex convenient vector space. We denote by H(N ← F ) the vector space

of all holomorphic sections s : N ’ F , equipped with the topology which is initial

with respect to the cone

(pr —¦ψ± )—

’ 2 ’’

H(N ← F ) ’ H(U± ← F |U± ) ’ ’ ’ ’ H(U± , V )

’

where the convenient structure on the right hand side is described in (27.17), see

also (7.21).

By (5.25) and (8.10) the space H(N ← F ) of sections satis¬es the uniform bound-

edness principle for the point evaluations.

For a ¬nite dimensional holomorphic vector bundle the topology on H(N ← F )

turns out to be nuclear and Fr´chet by (8.2), so by (16.10) H(N ← F ) is smoothly

e

paracompact.

30.6. Spaces of real analytic sections. Let p : E ’ M be a real analytic

vector bundle with standard ¬ber V . We denote by C ω (M ← E) the vector space

of all real analytic sections. We will equip it with one of the equivalent structures

of a convenient vector space described in the next lemma.

Lemma. The following structures of a convenient vector space on the space of

sections C ω (M ← E) are all equivalent:

(1) Choose a vector bundle atlas (U± , ψ± ), and consider the initial structure

with respect to the cone

(pr —¦ψ± )—

C ω (M ← E) ’ C ω (U± ← E|U± ) ’ ’ ’ ’ C ω (U± , V ),

’ 2 ’’

’

where the spaces C ω (U± , V ) are equipped with the structure of (27.17).

(2) If M is smoothly regular, choose a smooth closed embedding E ’ M — F

into a trivial vector bundle with ¬ber a convenient vector space F . Then

C ω (M ← E) can be considered as a closed linear subspace of C ω (M, F ).

The space C ω (M ← E) satis¬es the uniform boundedness principle for the point

evaluations evx : C ω (M ← E) ’ Ex .

If the base manifold is compact ¬nite dimensional real analytic, and if the standard

¬ber is a ¬nite dimensional vector space, then C ω (M ← E) is smoothly paracom-

pact.

Proof. We use the following diagram

wC

inj—

C ω (M ← E) ω

(M, F )

u u

(pr2 —¦ψ± )—

wC

C ω (U± , V ) ω

(U± , F ),

30.6

30.9 30. Spaces of sections of vector bundles 299

where the bottom arrow is a push forward with the vector bundle embedding h :

U± ’ L(V, F ) of trivial bundles, given by h§ := pr2 —¦ inj —¦ψ± : U± — V ’ F , which

’1

is bounded by (30.2). The uniform boundedness principle follows from (11.12).

For proving that C ω (M ← E) is smoothly paracompact we use the second descrip-

tion. Then C ω (M ← E) is a direct summand in a space C ω (M, V ), where M is

a compact real analytic manifold and V is a ¬nite dimensional real vector space.

The function space C ω (M, V ) is smoothly paracompact by (11.4).

30.7. C ∞,ω -mappings. Let M and N be real analytic manifolds. A mapping

f : R — M ’ N is said to be of class C ∞,ω if for each (t, x) ∈ R — M and each

real analytic chart (V, v) of N with f (t, x) ∈ V there are a real analytic chart

(U, u) of M with x ∈ U , an open interval t ∈ I ‚ R such that f (I — U ) ‚ V , and

v —¦ f —¦ (I — u’1 ) : I — u(U ) ’ v(V ) is of class C ∞,ω in the sense of (11.20), i.e., the

canonical associate is a smooth mapping (v —¦ f —¦ (I — u’1 ))∨ : I ’ C ω (u(U ), v(V )).

The mapping is said to be C ω,∞ if the canonical associate is a real analytic mapping

(v —¦ f —¦ (I — u’1 ))∨ : I ’ C ∞ (u(U ), v(V )), see (11.20.2).

These notions are well de¬ned by the composition theorem for C ∞,ω -mappings

(11.22), and the obvious generalization of (11.21) is true.

We choose one factor to be R because we need the c∞ -topology of the product to

be the product of the c∞ -topologies, see (4.15) and (4.22).

30.8. Lemma. Curves in spaces of sections.

(1) For a smooth vector bundle p : E ’ M a curve c : R ’ C ∞ (M ← E) is

smooth if and only if c§ : R — M ’ E is smooth.

(2) For a holomorphic vector bundle p : E ’ M a curve c : D ’ H(M ← E)

is holomorphic if and only if c§ : D — M ’ E is holomorphic.

(3) For a real analytic vector bundle p : E ’ M a curve c : R ’ C ω (M ← E)

is real analytic if and only if the associated mapping c§ : R — M ’ E is

real analytic.

(4) For a real analytic vector bundle p : E ’ M a curve c : R ’ C ω (M ← E)

is smooth if and only if c§ : R — M ’ E is C ∞,ω , see (30.7). A curve

c : R ’ C ∞ (M ← E) is real analytic if and only if c§ : R — M ’ E is

C ω,∞ , see (11.20).

Proof. By the descriptions of the structures ((30.1) for the smooth case, (30.5) for

the holomorphic case, and (30.6) for the real analytic case) we may assume that M

is open in a convenient vector space F , and we may consider functions with values

in the standard ¬ber instead of sections. The statements then follow from the

respective exponential laws ((3.12) for the smooth case, (7.22) for the holomorphic

case, (11.18) for the real analytic case, and the de¬nition in (11.20) for the C ∞,ω

and C ω,∞ cases).

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

(1) For a smooth vector bundle p : E ’ M with ¬nite dimensional base manifold

M a curve c : R ’ Cc (M ← E) is smooth if and only if c§ : R — M ’ E

∞

30.9

300 Chapter VI. In¬nite dimensional manifolds 30.11

is smooth and satis¬es the following condition:

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 a real analytic ¬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 condition

∞

of (1) above and c§ : R — M ’ E is C ω,∞ , see (30.7).

Compare this with (42.5) and (42.12).

∞

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

∞

locally as a smooth curve into some step CK (M ← E) for some compact K in M ,

and this is by (30.8.1) equivalent to smoothness of c§ and to condition (1). An

analogous proof applies to the real analytic case.

30.10. Corollary. Let p : E ’ M and p : E ’ M be smooth vector bundles

with ¬nite dimensional base manifold. Let W ⊆ E be an open subset, and let

∞

f : W ’ E be a ¬ber respecting smooth (nonlinear) mapping. Then Cc (M ←

∞

W ) := {s ∈ Cc (M ← E) : s(M ) ⊆ W } is open in the convenient vector space

∞ ∞ ∞

Cc (M ← E). The mapping f— : Cc (M ← W ) ’ Cc (M ← E ) is smooth with

∞ ∞ ∞

derivative (dv f )— : Cc (M ← W ) — Cc (M ← E) ’ Cc (M ← E ), where the

d

vertical derivative dv f : W —M E ’ E is given by dv f (u, w) := dt |0 f (u + tw).

∞ ∞

If the vector bundles and f are real analytic then f— : Cc (M ← W ) ’ Cc (M ←

E ) is real analytic with derivative (dv f )— .

If M is compact and the vector bundles and f are real analytic then C ω (M ←

W ) := {s ∈ C ω (M ← E) : s(M ) ⊆ W } is open in the convenient vector space

C ω (M ← E), and the mapping f— : C ω (M ← W ) ’ C ω (M ← E ) is real analytic

with derivative (dv f )— .

∞ ∞

Proof. The set Cc (M ← W ) is open in Cc (M ← E) since its intersection with

∞

each CK (M ← E) is open therein, see corollary (4.16), and the colimit is strict

and regular by (30.4). Then f— has all the stated properties, since it preserves (by

(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 ):

31.5

308 Chapter VI. In¬nite dimensional manifolds 31.5

w A—F

TA f

A—E

u u

π—E π—F

wF

f

E

If U ‚ E is a c∞ -open subset we put TA (U ) := (π — E)’1 (U ) = (1 — U ) — (N — E),

which is a c∞ -open subset in TA (E) := A — E. If f : U ’ V is a smooth mapping

between c∞ -open subsets U and V of E and F , respectively, then the construction

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

mapping TA f : TA U ’ TA V , which ¬ts into the following commutative diagram:

wT

‘ TA f

U — (N — E) V — (N — F )

TA U AV

‘“ &

pr ‘ &pr

u u)

&

π—E π—F

1 1

wV

f

U

We have TA (f —¦ g) = TA f —¦ TA g and TA (IdU ) = IdTA U , so TA is now a covariant

functor on the category of c∞ -open subsets of convenient vector spaces and smooth

mappings between them.

Step 5. Let M be a smooth manifold, let (U± , u± : U± ’ u± (U± ) ‚ E± ) be a

smooth atlas of M with chart changings u±β := u± —¦ u’1 : uβ (U±β ) ’ u± (U±β ).

β

Then the smooth mappings

wT

TA (u±β )

TA (uβ (U±β )) A (u± (U±β ))

u u

π — Eβ π — E±

w u (U

u±β

uβ (U±β ) ±β )

±

form likewise a cocycle of chart changings, and we may use them to glue the c∞ -

open sets TA (u± (U± )) = u± (U± ) — (N — E± ) ‚ TA E± together in order to obtain

a smooth manifold which we denote by TA M . By the diagram above, we see that

TA M will be the total space of a ¬ber bundle T (πA , M ) = πA,M : TA M ’ M , since

the atlas (TA (U± ), TA (u± )) constructed just now is already a ¬ber bundle atlas, see

(37.1) below. So if M is Hausdor¬ then also TA M 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 πA,M of open sets in M separating their projections. This

construction does not depend on the choice of the atlas, because two atlas have a

common re¬nement and one may pass to this.

If f ∈ C ∞ (M, M ) for two manifolds M , M , we apply the functor TA to the local

representatives of f with respect to suitable atlas. This gives local representatives

which ¬t together to form a smooth mapping TA f : TA M ’ TA M . Clearly, we

again have TA (f —¦ g) = TA f —¦ TA g and TA (IdM ) = IdTA M , so that TA : Mf ’ Mf

is a covariant functor.

31.5

31.7 31. Product preserving functors on manifolds 309

31.6. Remark. If we apply the construction of (31.5), step 5 to the algebra A = 0,

which we did not allow (1 = 0 ∈ A), then T0 M depends on the choice of the atlas. If

each chart is connected, then T0 M = π0 (M ), computing the connected components

of M . If each chart meets each connected component of M , then T0 M is one point.

31.7. 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 (31.5) de¬nes a covariant functor TA : Mf ’ Mf

such that πA:TA M ’M , M is a smooth ¬ber bundle with standard ¬ber N — E

if M is modeled on the convenient space E. For any f ∈ C ∞ (M, M ) we

have a commutative diagram

wT

TA f

TA M AM

πA,M πA,M

u u

wM.

f

M

So (TA , πA ) is a bundle functor on Mf , which gives a vector bundle functor

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: embeddings of (split-

ting) submanifolds, surjective smooth mappings admitting local smooth sec-

tions, ¬ber bundle projections. For ¬xed manifolds M and M the mapping

TA : C ∞ (M, M ) ’ C ∞ (TA M, TA M ) is smooth, so it maps smoothly pa-

rameterized families to smoothly parameterized families.

(3) If (U± ) is an open cover of M then TA (U± ) is an open cover of TA M .

(4) Any algebra homomorphism • : A ’ B between Weil algebras induces a

natural transformation T (•, ) = T• : TA ’ TB . If • is injective, then

T (•, M ) : TA M ’ TB M 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

times Mf to Mf .

Proof. (1) The main assertion is clear from (31.5). The ¬ber bundle πA,M :

TA M ’ M is a vector bundle if and only if the transition functions TA (u±β ) are

¬ber linear N — E± ’ N — Eβ . So only the ¬rst derivatives of u±β should act on

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

of order 2.

(2) The functor TA respects ¬nite products in the category of c∞ -open subsets of

convenient vector spaces by (31.5), step 3 and 5. All the other assertions follow by

looking again at the chart structure of TA M and by taking into account that f is

part of TA f (as the base mapping).

(3) This is obvious from the chart structure.

(4) We de¬ne T (•, E) := •—E : A—E ’ B—E. By (31.5), step 3, this restricts to a

natural transformation TA ’ TB on the category of c∞ -open subsets of convenient

31.7

310 Chapter VI. In¬nite dimensional manifolds 31.9

vector spaces, and “ by gluing “ also on the category Mf . Obviously, T is a co-

covariant bifunctor on the indicated categories. Since πB —¦ • = πA (• respects the

identity), we have T (πB , M ) —¦ T (•, M ) = T (πA , M ), so T (•, M ) : TA M ’ TB M is

¬ber respecting for each manifold M . In each ¬ber chart it is a linear mapping on

the typical ¬ber NA — E ’ NB — E.

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 if M is modeled on convenient vector spaces E± and for the canonical

charts we have the commutative diagram:

w T uM

u

T (•, M )

TA M B

wT

T (•, U± )

TA (U± ) B (U± )

u u

TA (u± ) TB (u± )

w u (U ) — (N

Id —((•|NA ) — E± )

u± (U± ) — (NA — E± ) — E± )

± ± B

w u (U ) — 0 — (N

Id —0 — iso

u± (U± ) — (N1 — E± ) — (V — E± ) — E± )

± ± B

Hence T (•, M ) is a ¬ber bundle projection with standard ¬ber E± — ker •.

31.8. 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 multipli-

cation mt = TA (mt ) : A ’ A.

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

homomorphisms A ’ B.

Proof. (1) 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 has already been described in theorem (31.7.4).

31.9. Remark. If M is a smoothly real compact and smoothly regular manifold

we consider the set DA (M ) := Hom(C ∞ (M, R), A) of all bounded homomorphisms

from the convenient algebra of smooth functions on M into a Weil algebra A.

Obviously we have a natural mapping TA M ’ DA M which is given by X ’ (f ’

TA (f ).X), using (3.5) and (3.6).

Let D be the algebra of Study numbers R.1 • R.δ with δ 2 = 0. Then TD M = T M ,

the tangent bundle, and DD (M ) is the smooth bundle of all operational tangent

vectors, i.e. bounded derivations at a point x of the algebra of germs C ∞ (M ⊇

{x}, R) see (28.12).

31.9

31.10 31. Product preserving functors on manifolds 311

It would be nice if DA (M ) were a smooth manifold not only for A = D. We do

not know whether this is true. The obvious method of proof hits severe obstacles,

which we now explain.

Let A = R.1 • N be a Weil algebra and let π : A ’ R be the corresponding

projection. Then for • ∈ DA (M ) = Hom(C ∞ (M, R), A) the character π —¦ • equals

evx for a unique x ∈ M , since M is smoothly real compact. Moreover, X :=

• ’ evx .1 : C ∞ (M, R) ’ N satis¬es the expansion property at x:

(1) X(f g) = X(f ).g(x) + f (x).X(g) + X(f ).X(g).

Conversely, a bounded linear mapping X : C ∞ (M, R) ’ N with property (1)

is called an expansion at x. Clearly each expansion at x de¬nes a bounded ho-

momorphism • with π —¦ • = evx . So we view DA (M )x as the set of all ex-

pansions at x. Note ¬rst that for an expansion X ∈ DA (M )x the value X(f )

depends only on the germ of f at x: If f |U = 0 for a neighborhood U of x,

choose a smooth function h with h = 1 o¬ U and h(x) = 0. Then hk f = f and

X(f ) = X(hk f ) = 0 + 0 + X(hk )X(f ) = · · · = X(h)k X(f ), which is 0 for k larger

than the nilpotence index of N .

Suppose now that M = U is a c∞ -open subset of a convenient vector space E. We

can ask whether DA (U )x is a smooth manifold. Let us sketch the di¬culty. A

natural way to proceed would be to apply by induction on the nilpotence index of

N . Let N0 := {n ∈ N : n.N = 0}, which is an ideal in A. Consider the short exact

sequence

p

0 ’ N0 ’ N ’ N/N0 ’ 0

’

¯

and a linear section s : N/N0 ’ N . For X : C ∞ (U, R) ’ N we consider X := p —¦ X

¯

and X0 := X ’ s —¦ X. Then X is an expansion at x ∈ U if and only if

¯