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 suppose now that the same vector bundle (E, p, 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 cover (U± ) (where we identify cohomologous ones) form a set

ˇ ˇ

H 1 ((U± ), GL(V )), 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 that re¬nes (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 induces

ˇ ˇ

therefore a mapping 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),

lim ˇ ˇ

so we may form the inductive limit ’ H 1 (U, GL(V )) =: H 1 (M, GL(V )) over

’

all open covers of M directed by re¬nement.

ˇ

Theorem. There is a bijective correspondence between H 1 (M, GL(V )) and the

set of all isomorphism classes of vector 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

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± ’

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52 Chapter II. Di¬erential forms

’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. So (E, p, 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, so „± ·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 was already shown in the discussion before the

theorem, and the argument can be easily re¬ned to show also that isomorphic

bundles give cohomologous cocycles.

Remark. If GL(V ) is an abelian group, i.e. if V is of real or complex dimension

ˇ

1, 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 clear 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 isomorphism) leads to the concept

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

of the whole situation.

6.5. Let (U± , ψ± ) be a vector bundle atlas on a vector bundle (E, p, M ). Let

(ej )k be a basis of the standard ¬ber V . We consider the section sj (x) :=

j=1

’1

ψ± (x, ej ) for x ∈ U± . Then the sj : U± ’ E are local sections of E such that

(sj (x))k is a basis of Ex for each x ∈ U± : we say that s = (s1 , . . . , sk ) is a

j=1

local frame ¬eld for E over U± .

Now let conversely U ‚ M be an open set and let sj : U ’ E be local

sections of E such that s = (s1 , . . . , sk ) is a local frame ¬eld of E over U . Then s

determines a unique vector bundle chart (U, ψ) of E such that sj (x) = ψ ’1 (x, ej ),

in the following way. We de¬ne f : U — Rk ’ E|U by f (x, v 1 , . . . , v k ) :=

k j

j=1 v sj (x). Then f is smooth, invertible, and a ¬ber linear isomorphism, so

(U, ψ = f ’1 ) is the vector bundle chart promised above.

6.6. A vector sub bundle (F, p, M ) of a vector bundle (E, p, M ) is a vector bundle

and a vector bundle homomorphism „ : F ’ E, which covers IdM , such that

„x : Ex ’ Fx is a linear embedding for each x ∈ M .

Lemma. Let • : (E, p, M ) ’ (E , q, N ) be a vector bundle homomorphism

such that rank(•x : Ex ’ E•(x) ) is constant in x ∈ M . Then ker •, given by

(ker •)x = ker(•x ), is a vector sub bundle of (E, p, M ).

Proof. This is a local question, so we may assume that both bundles are trivial:

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6. Vector bundles 53

let E = M — Rp and let F = N — Rq , then •(x, v) = (•(x), •(x).v), where • :

M ’ L(Rp , Rq ). The matrix •(x) has rank k, so by the elimination procedure

we can ¬nd p’k linearly independent solutions vi (x) of the equation •(x).v = 0.

The elimination procedure (with the same lines) gives solutions vi (y) for y near

x, so near x we get a local frame ¬eld v = (v1 , . . . , vp’k ) for ker •. By 6.5 ker •

is then a vector sub bundle.

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

the category of ¬nite dimensional vector spaces and linear mappings into itself,

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

k

F(V ) = Λk (V ) (the k-th exterior power), or F(V ) = V , and the like.

If (E, p, 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 smooth functions F(•±β ) : x ’ F(•±β (x)), U±β ’

GL(F(V )). Since F is a covariant functor, F(•±β ) satis¬es again the cocycle

condition 6.4.1, and cohomology of cocycles 6.4.2 is respected, so there exists

a unique vector bundle (F(E) := V B(F(•±β )), p, M ), the value at the vector

bundle (E, p, 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 duality functor F(V ) = V — , then

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

±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, and similarly in

other constructions.

So for vector bundles (E, p, M ) and (F, q, M ) we have the following vector

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

—

E — F , and so on.

6.8. Pullbacks of vector bundles. Let (E, p, M ) be a vector bundle and let

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

same typical ¬ber and a vector bundle homomorphism

wE

p— f

f —E

p

f —p

u u

wM

f

N

are 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)

in the sense of 2.19.

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54 Chapter II. Di¬erential forms

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

bundle (F, q, P ), vector bundle homomorphism • : F ’ E and smooth g :

P ’ N such that f —¦ g = •, there is a unique vector bundle homomorphism

ψ : F ’ f — E with ψ = g and p— f —¦ ψ = •.

RR •

F

RT

R u

ψ

wE

p— f

q —

fE

p

f —p

u u u

wN w M.

g f

P

6.9. Theorem. Any vector bundle admits a ¬nite vector bundle atlas.

Proof. Let (E, p, M ) be the vector bundle in question, let dim M = m. Let

(U± , ψ± )±∈A be a vector bundle atlas. Since M is separable, by topological

dimension theory there is a re¬nement of the open cover (U± )±∈A of the form

(Vij )i=1,...,m+1;j∈N , such that Vij © Vik = … for j = k, see the remarks at the end

of 1.1. We de¬ne the set Wi := j∈N Vij (a disjoint union) and ψi |Vij = ψ±(i,j) ,

where ± : {1, . . . , m + 1} — N ’ A is a re¬ning map. Then (Wi , ψi )i=1,...,m+1 is

a ¬nite vector bundle atlas of E.

6.10. Theorem. For any vector bundle (E, p, M ) there is a second vector

bundle (F, p, M ) such that (E •F, p, M ) is a trivial vector bundle, i.e. isomorphic

to M — RN for some N ∈ N.

Proof. Let (Ui , ψi )n be a ¬nite vector bundle atlas for (E, p, M ). Let (gi ) be

i=1

a smooth partition of unity subordinated to the open cover (Ui ). Let i : Rk ’

(Rk )n = Rk — · · · — Rk be the embedding on the i-th factor, where Rk is the

typical ¬ber of E. Let us de¬ne ψ : E ’ M — Rnk by

n

gi (p(u)) ( i —¦ pr2 —¦ ψi )(u) ,

ψ(u) = p(u),

i=1

then ψ is smooth, ¬ber linear, and an embedding on each ¬ber, so E is a vector

⊥

sub bundle of M — Rnk via ψ. Now we de¬ne Fx = Ex in {x} — Rnk with respect

to the standard inner product on Rnk . Then F ’ M is a vector bundle and

E • F ∼ M — Rnk .

=

6.11. The tangent bundle of a vector bundle. Let (E, p, M ) be a vector

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

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

t

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

denoted by mT E .

t

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6. Vector bundles 55

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

(U± , u± ) is also 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 ‚ Rm — V.

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

Rm — V )±∈A is the atlas describing the canonical vector bundle structure of

(T E, πE , E). The transition functions are in turn:

’1

(ψ± —¦ ψβ )(x, v) = (x, ψ±β (x)v) for x ∈ U±β

(u± —¦ u’1 )(y) = u±β (y) for y ∈ uβ (U±β )

β

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

β

(T ψ± —¦ T (ψβ )’1 )(y, v; ξ, w) = u±β (y), ψ±β (u’1 (y))v; d(u±β )(y)ξ,

β

(d(ψ±β —¦ u’1 )(y))ξ)v + ψ±β (u’1 (y))w .

β β

So we see that for ¬xed (y, v) the transition functions are linear in (ξ, w) ∈

Rm — V . This describes the vector bundle structure of the tangent bundle

(T E, πE , E).

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

This gives a vector bundle structure on (T E, T p, 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 ). t

One may say that the second vector bundle structure on T E, that one over T M ,

is the derivative of the original one on E.

The space {Ξ ∈ T E : T p.Ξ = 0 in T M } = (T p)’1 (0) 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 =

(y, v; 0, w), so the transition function looks like (T ψ± —¦ T (ψβ )’1 )(y, v; 0, w) =

(u±β (y), ψ±β (u’1 (y))v; 0, ψ±β (u’1 (y))w). They are linear in (v, w) ∈ V — V for

β β

¬xed y, so V E is a vector bundle over M . It coincides with 0— (T E, T p, T M ),

M

the pullback of the bundle T E ’ T M over the zero section. We have a canonical

isomorphism vlE : E —M E ’ V E, called the vertical lift, given by vlE (ux , vx ) :=

d

dt |0 (ux + tvx ), which is ¬ber linear over M . The local representation of the

vertical lift is (T ψ± —¦ vlE —¦ (ψ± — ψ± )’1 )((y, u), (y, v)) = (y, 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.

’1

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

’1

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

6.12. The second tangent bundle of a manifold. All of 6.11 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),

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56 Chapter II. Di¬erential forms

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

of charts (T u± equals ψ± from 6.11).

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

(5) κM is a linear isomorphism from the bundle (T T M, T (πM ), T M ) to

(T T M, πT M , T M ), so it interchanges the two vector bundle structures

on T T M .

(6) It is the unique smooth mapping T T M ’ T T M which satis¬es

‚‚ ‚‚ 2

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

All this follows from the local formula given above. We will come back to the

¬‚ip later on in chapter VIII from a more advanced point of view.

6.13. Lemma. For vector ¬elds X, Y ∈ X(M ) we have

[X, Y ] = vprT M —¦ (T Y —¦ X ’ κM —¦ T X —¦ Y ).

We will give global proofs of this result later on: the ¬rst one is 6.19. Another

one is 37.13.

Proof. We prove this locally, so we assume that M is open in Rm , X(x) =

¯ ¯ ¯ ¯

(x, X(x)), and Y (x) = (x, Y (x)). By 3.4 we get [X, Y ](x) = (x, dY (x).X(x) ’

¯ ¯

dX(x).Y (x)), and

vprT M —¦ (T Y —¦ X ’ κM —¦ T X —¦ Y )(x) =

¯ ¯

= vprT M —¦ (T Y.(x, X(x)) ’ κM —¦ T X.(x, Y (x))) =

¯ ¯ ¯ ¯

= vprT M (x, Y (x); X(x), dY (x).X(x))’

¯ ¯ ¯ ¯

’ κM ((x, X(x); Y (x), dX(x).Y (x)) =

¯ ¯ ¯ ¯ ¯

= vprT M (x, Y (x); 0, dY (x).X(x) ’ dX(x).Y (x)) =

¯ ¯ ¯ ¯

= (x, dY (x).X(x) ’ dX(x).Y (x)).

6.14. Natural vector bundles. Let Mfm denote the category of all m-

dimensional smooth manifolds and local di¬eomorphisms (i.e. immersions) be-

tween them. A vector bundle functor or natural vector bundle is a functor F

which associates a vector bundle (F (M ), pM , M ) to each m-manifold M and a

vector bundle homomorphism

w F (N )

F (f )

F (M )

u u

pM pN

wN

f

M

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6. Vector bundles 57

to each f : M ’ N in Mfm , which covers f and is ¬berwise a linear iso-

morphism. We also require that for smooth f : R — M ’ N the mapping

(t, x) ’ F (ft )(x) is also smooth R — F (M ) ’ F (N ). We will say that F maps

smoothly parametrized families to smoothly parametrized families. We shall see

later that this last requirement is automatically satis¬ed. For a characterization

of all vector bundle functors see 14.8.

Examples. 1. T M , the tangent bundle. This is even a functor on the category

Mf .

2. T — M , the cotangent bundle, where by 6.7 the action on morphisms is given

by (T — f )x := ((Tx f )’1 )— : Tx M ’ Tf (x) N . This functor is de¬ned on Mfm

— —

only.

3. Λk T — M , ΛT — M = k≥0 Λk T — M .

k

T —M — T M = T — M — · · · — T — M — T M — · · · — T M , where the

4.

action on morphisms involves T f ’1 in the T — M -parts and T f in the T M -parts.

5. F(T M ), where F is any smooth functor on the category of ¬nite dimen-

sional vector spaces and linear mappings, as in 6.7.

6.15. Lie derivative. Let F be a vector bundle functor on Mfm as described

in 6.14. Let M be a manifold and let X ∈ X(M ) be a vector ¬eld on M . Then

the ¬‚ow FlX , for ¬xed t, is a di¬eomorphism de¬ned on an open subset of M ,

t

which we do not specify. The mapping

w

F (FlX )

t

F (M ) F (M )

u u

pM pM

wM

FlX

t

M

is then a vector bundle isomorphism, de¬ned over an open subset of M .

We consider a section s ∈ C ∞ (F (M )) of the vector bundle (F (M ), pM , M )

and we de¬ne for t ∈ R

(FlX )— s := F (FlX ) —¦ s —¦ FlX .

’t

t t

This is a local section of the vector bundle F (M ). For each x ∈ M the value

((FlX )— s)(x) ∈ F (M )x is de¬ned, if t is small enough. So in the vector space

t