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 .