Let a, b ∈ G such that Ua © Ub = …. Then

¯ ¯

ua —¦ u’1 = g ’1 —¦ »a’1 —¦ »b —¦ g : ub (Ua © Ub ) ’ ua (Ua © Ub )

¯ ¯ ¯ ¯

b

¯

= g ’1 —¦ »a’1 b —¦ p —¦ (exp |W )

= g ’1 —¦ p —¦ »a’1 b —¦ (exp |W )

= pr1 —¦ f ’1 —¦ »a’1 b —¦ (exp |W ) is smooth.

5.12. Let : G — M ’ M be a left action. Then we have partial mappings

x

: G ’ M , given by a (x) = x (a) = (a, x) = a.x.

a : M ’ M and

M

For any X ∈ g we de¬ne the fundamental vector ¬eld ζX = ζX ∈ X(M ) by

ζX (x) = Te ( x ).X = T(e,x) .(X, 0x ).

Lemma. In this situation the following assertions hold:

(1) ζ : g ’ X(M ) is a linear mapping.

(2) Tx ( a ).ζX (x) = ζAd(a)X (a.x).

(3) RX — 0M ∈ X(G — M ) is -related to ζX ∈ X(M ).

(4) [ζX , ζY ] = ’ζ[X,Y ] .

Proof. (1) is clear.

(2) a x (b) = abx = aba’1 ax = ax conja (b), so we get Tx ( a ).ζX (x) =

Tx ( a ).Te ( x ).X = Te ( a —¦ x ).X = Te ( ax ). Ad(a).X = ζAd(a)X (ax).

(3) —¦ (Id — a ) = —¦ (ρa — Id) : G — M ’ M , so we get ζX ( (a, x)) =

T(e,ax) .(X, 0ax ) = T .(Id —T ( a )).(X, 0x ) = T .(T (ρa )—Id).(X, 0x ) = T .(RX —

0M )(a, x).

(4) [RX — 0M , RY — 0M ] = [RX , RY ] — 0M = ’R[X,Y ] — 0M is -related to

[ζX , ζY ] by (3) and by 3.10. On the other hand ’R[X,Y ] — 0M is -related to

’ζ[X,Y ] by (3) again. Since is surjective we get [ζX , ζY ] = ’ζ[X,Y ] .

5.13. Let r : M — G ’ M be a right action, so r : G ’ Di¬(M ) is a group

ˇ

anti homomorphism. We will use the following notation: ra : M ’ M and

rx : G ’ M , given by rx (a) = ra (x) = r(x, a) = x.a.

M

For any X ∈ g we de¬ne the fundamental vector ¬eld ζX = ζX ∈ X(M ) by

ζX (x) = Te (rx ).X = T(x,e) r.(0x , X).

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

5. Lie subgroups and homogeneous Spaces 47

Lemma. In this situation the following assertions hold:

(1) ζ : g ’ X(M ) is a linear mapping.

(2) Tx (ra ).ζX (x) = ζAd(a’1 )X (x.a).

(3) 0M — LX ∈ X(M — G) is r-related to ζX ∈ X(M ).

(4) [ζX , ζY ] = ζ[X,Y ] .

5.14. Theorem. Let : G — M ’ M be a smooth left action. For x ∈ M let

Gx = {a ∈ G : ax = x} be the isotropy subgroup of x in G, a closed subgroup

of G. Then x : G ’ M factors over p : G ’ G/Gx to an injective immersion

¯

ix : G/Gx ’ M , which is G-equivariant, i.e. a —¦ ix = ix —¦ »a for all a ∈ G. The

image of ix is the orbit through x.

The fundamental vector ¬elds span an integrable distribution on M in the

sense of 3.20. Its leaves are the connected components of the orbits, and each

orbit is an initial submanifold.

Proof. Clearly x factors over p to an injective mapping ix : G/Gx ’ M ; by

the universal property of surjective submersions ix is smooth, and obviously

¯ ¯

it is equivariant. Thus Tp(a) (ix ).Tp(e) (»a ) = Tp(e) (ix —¦ »a ) = Tp(e) ( a —¦ ix ) =

Tx ( a ).Tp(e) (ix ) for all a ∈ G and it su¬ces to show that Tp(e) (ix ) is injective.

Let X ∈ g and consider its fundamental vector ¬eld ζX ∈ X(M ). By 3.14 and

5.12.3 we have

(exp(tX), x) = (FlRX —0M (e, x)) = FlζX ( (e, x)) = FlζX (x).

(1) t t

t

So exp(tX) ∈ Gx , i.e. X ∈ gx , if and only if ζX (x) = 0x . In other words,

0x = ζX (x) = Te ( x ).X = Tp(e) (ix ).Te p.X if and only if Te p.X = 0p(e) . Thus ix

is an immersion.

Since the connected components of the orbits are integral manifolds, the fun-

damental vector ¬elds span an integrable distribution in the sense of 3.20; but

also the condition 3.25.2 is satis¬ed. So by theorem 3.22 each orbit is an initial

submanifold in the sense of 2.14.

¯

5.15. A mapping f : M ’ M between two manifolds with left (or right) actions

and ¯ of a Lie group G is called G-equivariant if f —¦ a = ¯a —¦f ( or f —¦ra = ra —¦f )

¯

for all a ∈ G. Sometimes we say in short that f is a G-mapping. From formula

5.14.(1) we get

Lemma. If G is connected, then f is G-equivariant if and only if the funda-

¯

mental ¬eld mappings are f related, i.e. T f —¦ ζX = ζX —¦ f for all X ∈ g.

Proof. The image of the exponential mapping generates the connected compo-

nent of the unit.

5.16. Semidirect products of Lie groups. Let H and K be two Lie groups

and let : H — K ’ K be a left action of H in K such that each h : K ’ K

is a group homomorphism. So the associated mapping ˇ : H ’ Aut(K) is a

homomorphism into the automorphism group of K. Then we can introduce the

following multiplication on K — H

(1) (k, h)(k , h ) := (k h (k ), hh ).

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

48 Chapter I. Manifolds and Lie groups

It is easy to see that this de¬nes a Lie group G = K H called the semidirect

product of H and K with respect to . If the action is clear from the context we

write G = K H only. The second projection pr2 : K H ’ H is a surjective

smooth homomorphism with kernel K —{e}, and the insertion inse : H ’ K H,

inse (h) = (e, h) is a smooth group homomorphism with pr2 —¦ inse = IdH .

Conversely we consider an exact sequence of Lie groups and homomorphisms

j p

{e} ’ K ’ G ’ H ’ {e}.

’’

(2)

So j is injective, p is surjective, and the kernel of p equals the image of j.

We suppose furthermore that the sequence splits, so that there is a smooth

homomorphism i : H ’ G with p —¦ i = IdH . Then the rule h (k) = i(h)ki(h’1 )

(where we suppress j) de¬nes a left action of H on K by automorphisms. It

H ’ G given by (k, h) ’ ki(h) is an

is easily seen that the mapping K

isomorphism of Lie groups. So we see that semidirect products of Lie groups

correspond exactly to splitting short exact sequences.

Semidirect products will appear naturally also in another form, starting from

right actions: Let H and K be two Lie groups and let r : K — H ’ K be a right

action of H in K such that each rh : K ’ K is a group homomorphism. Then

the multiplication on H — K is given by

¯¯ ¯

¯¯

(h, k)(h, k) := (hh, rh (k)k).

(3)

This de¬nes a Lie group G = H r K, also called the semidirect product of H

and K with respect to r. If the action r is clear from the context we write

G = H K only. The ¬rst projection pr1 : H K ’ H is a surjective smooth

homomorphism with kernel {e} — K, and the insertion inse : H ’ H K,

inse (h) = (h, e) is a smooth group homomorphism with pr1 —¦ inse = IdH .

Conversely we consider again a splitting exact sequence of Lie groups and

homomorphisms

j p

{e} ’ K ’ G ’ H ’ {e}.

’’

The splitting is given by a homomorphism i : H ’ G with p —¦ i = IdH . Then

the rule rh (k) = i(h’1 )ki(h) (where we suppress j) de¬nes now a right action

of H on K by automorphisms. It is easily seen that the mapping H r K ’ G

given by (h, k) ’ i(h)k is an isomorphism of Lie groups.

Remarks

The material in this chapter is standard. The concept of initial submani-

folds in 2.14“2.17 is due to Pradines, the treatment given here follows [Albert,

Molino]. The proof of theorem 3.16 is due to [Mauhart, 90]. The main re-

sults on distributions of non constant rank (3.18“3.25) are due to [Sussman, 73]

and [Stefan, 74], the treatment here follows [Lecomte]. The proof of the Baker-

Campbell-Hausdor¬ formula 4.29 is adapted from [Sattinger, Weaver, 86], see

also [Hilgert, Neeb, 91]. ¦

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

49

CHAPTER II.

DIFFERENTIAL FORMS

This chapter is still devoted to the fundamentals of di¬erential geometry,

but here the deviation from the standard presentations is already large. In

the section on vector bundles we treat the Lie derivative for natural vector

bundles, i.e. functors which associate vector bundles to manifolds and vector

bundle homomorphisms to local di¬eomorphisms. We give a formula for the Lie

derivative of the form of a commutator, but it involves the tangent bundle of the

vector bundle in question. So we also give a careful treatment to this situation.

The Lie derivative will be discussed in detail in chapter XI; here it is presented

in a somewhat special situation as an illustration of the categorical methods we

are going to apply later on. It follows a standard presentation of di¬erential

forms and a thorough treatment of the Fr¨licher-Nijenhuis bracket via the study

o

of all graded derivations of the algebra of di¬erential forms. This bracket is a

natural extension of the Lie bracket from vector ¬elds to tangent bundle valued

di¬erential forms. We believe that this bracket is one of the basic structures of

di¬erential geometry (see also section 30), and in chapter III we will base nearly

all treatment of curvature and the Bianchi identity on it.

6. Vector bundles

6.1. Vector bundles. Let p : E ’ M be a smooth mapping between mani-

folds. 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 U —V

ee ψ

E|U := p’1 (U )

eg pr

e£

p

1

U.

Here V is a ¬xed ¬nite dimensional vector space, called the standard ¬ber or the

typical ¬ber, real as a rule, unless otherwise speci¬ed.

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

’1 ’1

ψ2 is 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, and it is called the transition function between the two vector bundle

charts.

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

50 Chapter II. Di¬erential forms

A vector bundle atlas (U± , ψ± )±∈A for (E, p, 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 atlases are called equivalent, if their union is again a vector bundle

atlas.

A vector bundle (E, p, 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 atlases; so we must know at least one vector

bundle atlas. The projection p turns out to be a surjective submersion.

The tangent bundle (T M, πM , M ) of a manifold M is the ¬rst example of a

vector bundle.

6.2. Let us ¬x a vector bundle (E, p, M ) for the moment. On each ¬ber Ex :=

p’1 (x) (for x ∈ M ) there is a unique structure of a real vector space, induced

from 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.

A section u of (E, p, 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 (E, p, M ) will be denoted by

either C ∞ (E) = C ∞ (E, p, M ) = C ∞ (E ’ M ). Clearly it is a vector space with

¬ber wise addition and scalar multiplication.

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

’1

ping 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

’1

section can be formed by x ’ ± g± (x) · ψ± (x, f± (x)). So a smooth vector

bundle has ˜many™ smooth sections.

6.3. Let (E, p, M ) and (F, q, N ) be vector bundles. A vector bundle homomor-

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

wF

•

E

u u

p q

w N.

•

M

So we require that •x : Ex ’ F•(x) is linear. We say that • covers •. If • is

invertible, it is called a vector bundle isomorphism.

The smooth vector bundles together with their homomorphisms form a cate-

gory VB.

6.4. We will now give a formal description of the amount of vector bundles with

¬xed base M and ¬xed standard ¬ber V , up to isomorphisms which cover the

identity on M .

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

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

’1

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

U± © Uβ ’ GL(V ), which are smooth. This family of transition functions

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

6. Vector bundles 51

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 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± ’

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

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:

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

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.

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

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

F (M )x the expression dt |0 ((FlX )— s)(x) makes sense and therefore the section

d

t

X—

d

LX s := dt |0 (Flt ) s

is globally de¬ned and is an element of C ∞ (F (M )). It is called the Lie derivative

of s along X.

Lemma. In this situation we have

(1) (FlX )— (FlX )— s = (FlX )— s, whenever de¬ned.

t r t+r

(2) dt (Flt ) s = (Flt ) LX s = LX (FlX )— s, so

X— X—

d

t

[LX , (Flt ) ] := LX —¦ (Flt ) ’ (FlX )— —¦ LX = 0, whenever de¬ned.

X— X—

t

(3) (FlX )— s = s for all relevant t if and only if LX s = 0.

t

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58 Chapter II. Di¬erential forms

Proof. (1) is clear. (2) is seen by the following computations.

X— X— X— X—

d d

dr |0 (Flr ) (Flt ) s = LX (Flt ) s.

dt (Flt ) s =

X— X— X—

d d

dr |0 ((Flt ) (Flr ) s)(x)

dt ((Flt ) s)(x) =

X X X X

d

dr |0 F (Fl’t )(F (Fl’r ) —¦ s —¦ Flr )(Flt (x))

=

F (FlX ) dr |0 (F (FlX ) —¦ s —¦ FlX )(FlX (x))

d

= ’t ’r r t

((FlX )— LX s)(x),

= t

since F (FlX ) : F (M )FlX (x) ’ F (M )x is linear.

’t t

(3) follows from (2).

6.16. Let F1 , F2 be two vector bundle functors on Mfm . Then the tensor

product (F1 — F2 )(M ) := F1 (M ) — F2 (M ) is again a vector bundle functor and

for si ∈ C ∞ (Fi (M )) there is a section s1 — s2 ∈ C ∞ ((F1 — F2 )(M )), given by

the pointwise tensor product.

Lemma. In this situation, for X ∈ X(M ) we have

LX (s1 — s2 ) = LX s1 — s2 + s1 — LX s2 .

In particular, for f ∈ C ∞ (M, R) we have LX (f s) = df (X) s + f LX s.

Proof. Using the bilinearity of the tensor product we have

X—

d

LX (s1 — s2 ) = dt |0 (Flt ) (s1 — s2 )

X—

— (FlX )— s2 )

d

dt |0 ((Flt ) s1

= t

X— X—

d d

dt |0 (Flt ) s1 — s2 + s1 — dt |0 (Flt ) s2

=

= LX s1 — s2 + s1 — LX s2 .

6.17. Let • : F1 ’ F2 be a linear natural transformation between vector bun-

dle functors on Mfm , i.e. for each M ∈ Mfm we have a vector bundle ho-

momorphism •M : F1 (M ) ’ F2 (M ) covering the identity on M , such that

F2 (f ) —¦ •M = •N —¦ F1 (f ) holds for any f : M ’ N in Mfm (we shall see in

14.11 that for every natural transformation • : F1 ’ F2 in the purely categorical

sense each morphism •M : F1 (M ) ’ F2 (M ) covers IdM ).

Lemma. In this situation, for s ∈ C ∞ (F1 (M )) and X ∈ X(M ), we have

LX (•M s) = •M (LX s).

Proof. Since •M is ¬ber linear and natural we can compute as follows.

X— X

FlX )(x)

d d

LX (•M s)(x) = dt |0 ((Flt ) (•M s))(x) dt |0 (F2 (Fl’t ) —¦ •M —¦ s —¦

= t

•M —¦ dt |0 (F1 (FlX ) —¦ s FlX )(x) = (•M LX s)(x).

d

—¦

= ’t t

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6. Vector bundles 59

6.18. A tensor ¬eld of type p is a smooth section of the natural bundle

q

q— p

T M— T M . For such tensor ¬elds, by 6.15 the Lie derivative along

any vector ¬eld is de¬ned, by 6.16 it is a derivation with respect to the tensor

product, and by 6.17 it commutes with any kind of contraction or ˜permutation

of the indices™. For functions and vector ¬elds the Lie derivative was already

de¬ned in section 3.

6.19. Let F be a vector bundle functor on Mfm and let X ∈ X(M ) be a

vector ¬eld. We consider the local vector bundle homomorphism F (FlX ) ont

X X X X

F (M ). Since F (Flt ) —¦ F (Fls ) = F (Flt+s ) and F (Fl0 ) = IdF (M ) we have

F

X

= ds |0 F (FlX ) —¦ F (FlX ) = X F —¦ F (FlX ), so we get F (FlX ) = FlX ,

d d

dt F (Flt ) s t t t t

X

d

F

where X = ds |0 F (Fls ) ∈ X(F (M )) is a vector ¬eld on F (M ), which is called

the ¬‚ow prolongation or the canonical lift of X to F (M ). If it is desirable for

technical reasons we shall also write X F = FX.

Lemma.

(1) X T = κM —¦ T X.

(2) [X, Y ]F = [X F , Y F ].

(3) X F : (F (M ), pM , M ) ’ (T F (M ), T (pM ), T M ) is a vector bundle homo-

morphism for the T (+)-structure.

(4) For s ∈ C ∞ (F (M )) and X ∈ X(M ) we have

LX s = vprF (M ) (T s —¦ X ’ X F —¦ s).

(5) LX s is linear in X and s.

Proof. (1) is an easy computation. F (FlX ) is ¬ber linear and this implies (3).

t

(4) is seen as follows:

X X

d

dt |0 (F (Fl’t ) —¦ s —¦ Flt )(x) in F (M )x

(LX s)(x) =

vprF (M ) ( dt |0 (F (FlX ) —¦ s —¦ FlX )(x) in V F (M ))

d

= ’t t

vprF (M ) (’X F —¦ s —¦ FlX (x) + T (F (FlX )) —¦ T s —¦ X(x))

= 0 0

vprF (M ) (T s —¦ X ’ X F —¦ s)(x).

=

(5) LX s is homogeneous of degree 1 in X by formula (4), and it is smooth as a

mapping X(M ) ’ C ∞ (F (M )), so it is linear. See [Fr¨licher, Kriegl, 88] for the

o

convenient calculus in in¬nite dimensions.

(2) Note ¬rst that F induces a smooth mapping between appropriate spaces

of local di¬eomorphisms which are in¬nite dimensional manifolds (see [Kriegl,

Michor, 91]). By 3.16 we have

Y X Y X

‚

‚t 0 (Fl’t —¦ Fl’t —¦ Flt —¦ Flt ),

0=

1 ‚2 Y X Y X

2 ‚t2 |0 (Fl’t —¦ Fl’t —¦ Flt —¦ Flt )

[X, Y ] =

[X,Y ]

‚

= ‚t 0 Flt .

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60 Chapter II. Di¬erential forms

Applying F to these curves (of local di¬eomorphisms) we get

F F F F

(FlY —¦ FlX —¦ FlY —¦ FlX ),

‚

0= ’t ’t t t

‚t 0

YF XF YF F

1 ‚2

—¦ FlX )

[X F , Y F ] = 2 |0 (Fl’t —¦ Fl’t —¦ Flt t

2 ‚t

2 Y X Y X

1‚

2 ‚t2 |0 F (Fl’t —¦ Fl’t —¦ Flt —¦ Flt )

=

[X,Y ]

‚

) = [X, Y ]F .

= F (Flt

‚t 0

See also section 50 for a purely ¬nite dimensional proof of a much more general

result.

6.20. Proposition. For any vector bundle functor F on Mfm and X, Y ∈

X(M ) we have

[LX , LY ] := LX —¦ LY ’ LY —¦ LX = L[X,Y ] : C ∞ (F (M )) ’ C ∞ (F (M )).

So L : X(M ) ’ End C ∞ (F (M )) is a Lie algebra homomorphism.

Proof. See section 50 for a proof of a much more general formula.

6.21. Theorem. Let M be a manifold, let •i : R — M ⊃ U•i ’ M be smooth

mappings for i = 1, . . . , k where each U•i is an open neighborhood of {0} — M

in R — M , such that each •i is a di¬eomorphism on its domain, •i = IdM , and

t 0

j j ’1

—¦ (•t ) —¦ •j —¦ •i .

i ’1

‚ i i j i

‚t 0 •t = Xi ∈ X(M ). We put [• , • ]t = [•t , •t ] := (•t ) t t

∞

Let F be a vector bundle functor, let s ∈ C (F (M )) be a section. Then for

each formal bracket expression P of lenght k we have

|0 P (•1 , . . . , •k )— s

‚

for 1 ¤ < k,

0= t t

‚t

1 ‚k k—

∈ C ∞ (F (M )).

1

LP (X1 ,...,Xk ) s = k! ‚tk |0 P (•t , . . . , •t ) s

Proof. This can be proved with similar methods as in the proof of 3.16. A

concise proof can be found in [Mauhart, Michor, 92]

6.22. A¬ne bundles. Given a ¬nite dimensional a¬ne space A modelled on

a vector space V = A, we denote by + the canonical mapping A — A ’ A,

(p, v) ’ p + v for p ∈ A and v ∈ A. If A1 and A2 are two a¬ne spaces and

f : A1 ’ A2 is an a¬ne mapping, then we denote by f : A1 ’ A2 the linear

mapping given by f (p + v) = f (p) + f (v).

Let p : E ’ M be a vector bundle and q : Z ’ M be a smooth mapping

such that each ¬ber Zx = q ’1 (x) is an a¬ne space modelled on the vector space

Ex = p’1 (x). Let A be an a¬ne space modelled on the standard ¬ber V of E.

We say that Z is an a¬ne bundle with standard ¬ber A modelled on the vector

bundle E, if for each vector bundle chart ψ : E|U = p’1 (U ) ’ U — V on E

there exists a ¬ber respecting di¬eomorphism • : Z|U = q ’1 (U ) ’ U — A such

that •x : Zx ’ A is an a¬ne morphism satisfying •x = ψx : Ex ’ V for each

x ∈ U . We also write E = Z to have a functorial assignment of the modelling

vector bundle.

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7. Di¬erential forms 61

Let Z ’ M and Y ’ N be two a¬ne bundles. An a¬ne bundle morphism

f : Z ’ Y is a ¬ber respecting mapping such that each fx : Zx ’ Yf (x) is an

a¬ne mapping, where f : M ’ N is the underlying base mapping of f . Clearly

the rule x ’ fx : Zx ’ Yf (x) induces a vector bundle homomorphism f : Z ’ Y

over the same base mapping f .

7. Di¬erential forms

7.1. The cotangent bundle of a manifold M is the vector bundle T — M := (T M )— ,

the (real) dual of the tangent bundle.

‚ ‚

If (U, u) is a chart on M , then ( ‚u1 , . . . , ‚um ) is the associated frame ¬eld

j

‚ ‚

over U of T M . Since ‚ui |x (uj ) = duj ( ‚ui |x ) = δi we see that (du1 , . . . , dum ) is

the dual frame ¬eld on T — M over U . It is also called a holonomous frame ¬eld.

A section of T — M is also called a 1-form.

p

7.2. According to 6.18 a tensor ¬eld of type on a manifold M is a smooth

q

section of the vector bundle

p times q times

p q

T — M = T M — · · · — T M — T — M — · · · — T — M.

TM —

The position of p (up) and q (down) can be explained as follows: If (U, u) is a

chart on M , we have the holonomous frame ¬eld

— duj1 — · · · — dujq

‚ ‚ ‚

— — ··· —

‚ui1 ‚ui2 ‚uip i∈{1,... ,m}p ,j∈{1,... ,m}q

p

over U of this tensor bundle, and for any -tensor ¬eld A we have

q

i ...i

— duj1 — · · · — dujq .

‚ ‚

Aj1 ...jp ‚ui1 — · · · —

A|U = ‚uip

1 q

i,j

The coe¬cients have p indices up and q indices down, they are smooth functions

on U . From a strictly categorical point of view the position of the indices should

be exchanged, but this convention has a long tradition.

7.3 Lemma. Let ¦ : X(M ) — · · · — X(M ) = X(M )k ’ C ∞ ( T M ) be a

∞

mapping which is k-linear over C (M, R) then ¦ is given by a k -tensor ¬eld.

Proof. For simplicity™s sake we put k = 1, = 0, so ¦ : X(M ) ’ C ∞ (M, R) is a

C ∞ (M, R)-linear mapping: ¦(f.X) = f.¦(X).

Claim 1. If X | U = 0 for some open subset U ‚ M , then we have ¦(X) |

U = 0.

Let x ∈ U . We choose f ∈ C ∞ (M, R) with f (x) = 0 and f | M \ U = 1. Then

f.X = X, so ¦(X)(x) = ¦(f.X)(x) = f (x).¦(X)(x) = 0.

Claim 2. If X(x) = 0 then also ¦(X)(x) = 0.

¯

Let (U, u) be a chart centered at x, let V be open with x ∈ V ‚ V ‚ U . Then

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62 Chapter II. Di¬erential forms

X | U = X i ‚ui and X i (x) = 0. We choose g ∈ C ∞ (M, R) with g | V ≡ 1 and

‚

supp g ‚ U . Then (g 2 .X) | V = X | V and by claim 1 ¦(X) | V depends only on

‚

X | V and g 2 .X = i (g.X i )(g. ‚ui ) is a decomposition which is globally de¬ned

‚

on M . Therefore we have ¦(X)(x) = ¦(g 2 .X)(x) = ¦ i

i (g.X )(g. ‚ui ) (x) =

‚

(g.X i )(x).¦(g. ‚ui )(x) = 0.

So we see that for a general vector ¬eld X the value ¦(X)(x) depends only

on the value X(x), for each x ∈ M . So there is a linear map •x : Tx M ’ R for

each x ∈ M with ¦(X)(x) = •x (X(x)). Then • : M ’ T — M is smooth since

‚

• | V = i ¦(g. ‚ui ).dui in the setting of claim 2.

7.4. De¬nition. A di¬erential form or an exterior form of degree k or a k-form

for short is a section of the vector bundle Λk T — M . The space of all k-forms will

be denoted by „¦k (M ). It may also be viewed as the space of all skew symmetric

0

k -tensor ¬elds, i.e. (by 7.3) the space of all mappings

¦ : X(M ) — · · · — X(M ) = X(M )k ’ C ∞ (M, R),

which are k-linear over C ∞ (M, R) and are skew symmetric:

¦(Xσ1 , . . . , Xσk ) = sign σ · ¦(X1 , . . . , Xk )

for each permutation σ ∈ Sk .

We put „¦0 (M ) := C ∞ (M, R). Then the space

dim M

„¦k (M )

„¦(M ) :=

k=0

is an algebra with the following product. For • ∈ „¦k (M ) and ψ ∈ „¦ (M ) and

for Xi in X(M ) (or in Tx M ) we put

(• § ψ)(X1 , . . . , Xk+ ) =

1

sign σ · •(Xσ1 , . . . , Xσk ).ψ(Xσ(k+1) , . . . , Xσ(k+ ) ).

= k! !

σ∈Sk+

This product is de¬ned ¬ber wise, i.e. (• § ψ)x = •x § ψx for each x ∈ M . It

is also associative, i.e. (• § ψ) § „ = • § (ψ § „ ), and graded commutative, i.e.

• § ψ = (’1)k ψ § •. These properties are proved in multilinear algebra.

7.5. If f : N ’ M is a smooth mapping and • ∈ „¦k (M ), then the pullback

f — • ∈ „¦k (N ) is de¬ned for Xi ∈ Tx N by

(f — •)x (X1 , . . . , Xk ) := •f (x) (Tx f.X1 , . . . , Tx f.Xk ).

(1)

Then we have f — (• § ψ) = f — • § f — ψ, so the linear mapping f — : „¦(M ) ’ „¦(N )

is an algebra homomorphism. Moreover we have (g—¦f )— = f — —¦g — : „¦(P ) ’ „¦(N )

if g : M ’ P , and (IdM )— = Id„¦(M ) .

So M ’ „¦(M ) = C ∞ (ΛT — M ) is a contravariant functor from the category

Mf of all manifolds and all smooth mappings into the category of real graded

commutative algebras, whereas M ’ ΛT — M is a covariant vector bundle func-

tor de¬ned only on Mfm , the category of m-dimensional manifolds and local

di¬eomorphisms, for each m separately.

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

7. Di¬erential forms 63

7.6. The Lie derivative of di¬erential forms. Since M ’ Λk T — M is a

vector bundle functor on Mfm , by 6.15 for X ∈ X(M ) the Lie derivative of a

k-form • along X is de¬ned by

X—

d

LX • = dt |0 (Flt ) •.

Lemma. The Lie derivative has the following properties.

(1) LX (• § ψ) = LX • § ψ + • § LX ψ, so LX is a derivation.

(2) For Yi ∈ X(M ) we have

k

(LX •)(Y1 , . . . , Yk ) = X(•(Y1 , . . . , Yk )) ’ •(Y1 , . . . , [X, Yi ], . . . , Yk ).

i=1

(3) [LX , LY ]• = L[X,Y ] •.

k

T — M ’ Λk T — M , given by

Proof. The mapping Alt :

1

(AltA)(Y1 , . . . , Yk ) := sign(σ)A(Yσ1 , . . . , Yσk ),

k!

σ

is a linear natural transformation in the sense of 6.17 and induces an algebra

k—

∞

homomorphism from the tensor algebra k≥0 C ( T M ) onto „¦(M ). So

(1) follows from 6.16.

(2) Again by 6.16 and 6.17 we may compute as follows, where Trace is the

full evaluation of the form on all vector ¬elds:

X(•(Y1 , . . . , Yk )) = LX —¦ Trace(• — Y1 — · · · — Yk )

= Trace —¦LX (• — Y1 — · · · — Yk )

= Trace LX • — (Y1 — · · · — Yk ) + • — ( Y1 — · · · — LX Yi — · · · — Yk ) .

i

Now we use LX Yi = [X, Yi ].

(3) is a special case of 6.20.

7.7. The insertion operator. For a vector ¬eld X ∈ X(M ) we de¬ne the

insertion operator iX = i(X) : „¦k (M ) ’ „¦k’1 (M ) by

(iX •)(Y1 , . . . , Yk’1 ) := •(X, Y1 , . . . , Yk’1 ).

Lemma.

(1) iX is a graded derivation of degree ’1 of the graded algebra „¦(M ), so

we have iX (• § ψ) = iX • § ψ + (’1)deg • • § iX ψ.

(2) [LX , iY ] := LX —¦ iY ’ iY —¦ LX = i[X,Y ] .

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

64 Chapter II. Di¬erential forms

Proof. (1) For • ∈ „¦k (M ) and ψ ∈ „¦ (M ) we have

(iX1 (• § ψ))(X2 , . . . , Xk+ ) = (• § ψ)(X1 , . . . , Xk+ ) =

1

= sign(σ) •(Xσ1 , . . . , Xσk )ψ(Xσ(k+1) , . . . , Xσ(k+ ) ).

k! !

σ

(iX1 • § ψ + (’1)k • § iX1 ψ)(X2 , . . . , Xk+ ) =

1

= sign(σ) •(X1 , Xσ2 , . . . , Xσk )ψ(Xσ(k+1) , . . . , Xσ(k+ ) )+

(k’1)! !

σ

(’1)k

+ sign(σ) •(Xσ2 , . . . , Xσ(k+1) )ψ(X1 , Xσ(k+2) , . . . ).

k! ( ’ 1)! σ

Using the skew symmetry of • and ψ we may distribute X1 to each position by

adding an appropriate sign. These are k+ summands. Since (k’1)! ! + k! ( 1

1

’1)! =

k+

k! ! , and since we can generate each permutation in Sk+ in this way, the result

follows.

(2) By 6.16 and 6.17 we have:

LX iY • = LX Trace1 (Y — •) = Trace1 LX (Y — •)

= Trace1 (LX Y — • + Y — LX •) = i[X,Y ] • + iY LX •.

7.8. The exterior di¬erential. We want to construct a di¬erential operator

„¦k (M ) ’ „¦k+1 (M ) which is natural. We will show that the simplest choice will

work and (later) that it is essentially unique.

So let U be open in Rn , let • ∈ „¦k (Rn ). Then we may view • as an element

of C ∞ (U, Lk (Rn , R)). We consider D• ∈ C ∞ (U, L(Rn , Lk (Rn , R))), and we

alt alt

k+1

∞ n

take its canonical image Alt(D•) in C (U, Lalt (R , R)). Here we write D for

the derivative in order to distinguish it from the exterior di¬erential, which we

de¬ne as d• := (k + 1) Alt(D•), more explicitly as

1

(1) (d•)x (X0 , . . . , Xk ) = sign(σ) D•(x)(Xσ0 )(Xσ1 , . . . , Xσk )

k!

σ

k

(’1)i D•(x)(Xi )(X0 , . . . , Xi , . . . , Xk ),

=

i=0

where the hat over a symbol means that this is to be omitted, and where Xi ∈ Rn .

Now we pass to an arbitrary manifold M . For a k-form • ∈ „¦k (M ) and

vector ¬elds Xi ∈ X(M ) we try to replace D•(x)(Xi )(X0 , . . . ) in formula (1)

by Lie derivatives. We di¬erentiate Xi (•(x)(X0 , . . . )) = D•(x)(Xi )(X0 , . . . ) +

0¤j¤k,j=i •(x)(X0 , . . . , DXj (x)Xi , . . . ), so inserting this expression into for-

mula (1) we get (cf. 3.4) our working de¬nition

k

(’1)i Xi (•(X0 , . . . , Xi , . . . , Xk ))

(2) d•(X0 , . . . , Xk ) :=

i=0

(’1)i+j •([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xk ).

+

i<j

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

7. Di¬erential forms 65

d•, given by this formula, is (k+1)-linear over C ∞ (M, R), as a short computation

involving 3.4 shows. It is obviously skew symmetric, so by 7.3 d• is a (k + 1)-

form, and the operator d : „¦k (M ) ’ „¦k+1 (M ) is called the exterior derivative.

If (U, u) is a chart on M , then we have

•i1 ,... ,ik dui1 § · · · § duik ,

•|U =

i1 <···<ik

‚ ‚

where •i1 ,... ,ik = •( ‚ui1 , . . . , ‚uik ). An easy computation shows that (2) leads

to

d•i1 ,... ,ik § dui1 § · · · § duik ,

(3) d•|U =

i1 <···<ik

so that formulas (1) and (2) really de¬ne the same operator.

7.9. Theorem. The exterior derivative d : „¦k (M ) ’ „¦k+1 (M ) has the follow-

ing properties:

(1) d(• § ψ) = d• § ψ + (’1)deg • • § dψ, so d is a graded derivation of degree

1.

(2) LX = iX —¦ d + d —¦ iX for any vector ¬eld X.

(3) d2 = d —¦ d = 0.

(4) f — —¦ d = d —¦ f — for any smooth f : N ’ M .

(5) LX —¦ d = d —¦ LX for any vector ¬eld X.

Remark. In terms of the graded commutator

[D1 , D2 ] := D1 —¦ D2 ’ (’1)deg(D1 ) deg(D2 ) D2 —¦ D1

for graded homomorphisms and graded derivations (see 8.1) the assertions of

this theorem take the following form:

(2) LX = [iX , d].

(3) 1 [d, d] = 0.

2

(4) [f — , d] = 0.

(5) [LX , d] = 0.

This point of view will be developed in section 8 below.

Proof. (2) For • ∈ „¦k (M ) and Xi ∈ X(M ) we have

(LX0 •)(X1 , . . . , Xk ) = X0 (•(X1 , . . . , Xk ))+

k

(’1)0+j •([X0 , Xj ], X1 , . . . , Xj , . . . , Xk ) by 7.6.2,

+

j=1

(iX0 d•)(X1 , . . . , Xk ) = d•(X0 , . . . , Xk )

k

(’1)i Xi (•(X0 , . . . , Xi , . . . , Xk ))+

=

i=0

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

66 Chapter II. Di¬erential forms

(’1)i+j •([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xk ).

+

0¤i<j

k

(’1)i’1 Xi ((iX0 •)(X1 , . . . , Xi , . . . , Xk ))+

(diX0 •)(X1 , . . . , Xk ) =

i=1

(’1)i+j’1 (iX0 •)([Xi , Xj ], X1 , . . . , Xi , . . . , Xj , . . . , Xk )

+

1¤i<j

k

(’1)i Xi (•(X0 , X1 , . . . , Xi , . . . , Xk ))’

=’

i=1

(’1)i+j •([Xi , Xj ], X0 , X1 , . . . , Xi , . . . , Xj , . . . , Xk ).

’

1¤i<j

By summing up the result follows.

(1) Let • ∈ „¦p (M ) and ψ ∈ „¦q (M ). We prove the result by induction on

p + q.

p + q = 0: d(f · g) = df · g + f · dg.

Suppose that (1) is true for p + q < k. Then for X ∈ X(M ) we have by part (2)

and 7.6, 7.7 and by induction

iX d(• § ψ) = LX (• § ψ) ’ d iX (• § ψ)

= LX • § ψ + • § LX ψ ’ d(iX • § ψ + (’1)p • § iX ψ)

= iX d• § ψ + diX • § ψ + • § iX dψ + • § diX ψ ’ diX • § ψ

’ (’1)p’1 iX • § dψ ’ (’1)p d• § iX ψ ’ • § diX ψ

= iX (d• § ψ + (’1)p • § dψ).

Since X is arbitrary, (1) follows.

(3) By (1) d is a graded derivation of degree 1, so d2 = 1 [d, d] is a graded

2

derivation of degree 2 (see 8.1), and is obviously local. Since „¦(M ) is locally

generated as an algebra by C ∞ (M, R) and {df : f ∈ C ∞ (M, R)}, it su¬ces to

show that d2 f = 0 for each f ∈ C ∞ (M, R) (d3 f = 0 is a consequence). But this is

easy: d2 f (X, Y ) = Xdf (Y )’Y df (X)’df ([X, Y ]) = XY f ’Y Xf ’[X, Y ]f = 0.

(4) f — : „¦(M ) ’ „¦(N ) is an algebra homomorphism by 7.6, so f — —¦ d and

d —¦ f — are both graded derivations over f — of degree 1. By the same argument

as in the proof of (3) above it su¬ces to show that they agree on g and dg for

all g ∈ C ∞ (M, R). We have (f — dg)y (Y ) = (dg)f (y) (Ty f.Y ) = (Ty f.Y )(g) =

Y (g —¦ f )(y) = (df — g)y (Y ), thus also df — dg = ddf — g = 0, and f — ddg = 0.

(5) dLX = d iX d + ddiX = diX d + iX dd = LX d.

7.10. A di¬erential form ω ∈ „¦k (M ) is called closed if dω = 0, and it is called

exact if ω = d• for some • ∈ „¦k’1 (M ). Since d2 = 0, any exact form is closed.

The quotient space

ker(d : „¦k (M ) ’ „¦k+1 (M ))

k

H (M ) :=

im(d : „¦k’1 (M ) ’ „¦k (M ))

is called the k-th De Rham cohomology space of M . We will not treat cohomol-

ogy in this book, and we ¬nish with the

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8. Derivations on the algebra of di¬erential forms and the Fr¨licher-Nijenhuis bracket 67

o

Lemma of Poincar´. A closed di¬erential form is locally exact. More pre-

e

k

cisely: let ω ∈ „¦ (M ) with dω = 0. Then for any x ∈ M there is an open

neighborhood U of x in M and a • ∈ „¦k’1 (U ) with d• = ω|U .

Proof. Let (U, u) be chart on M centered at x such that u(U ) = Rm . So we may

just assume that M = Rm .

We consider ± : R—Rm ’ Rm , given by ±(t, x) = ±t (x) = tx. Let I ∈ X(Rm )

be the vector ¬eld I(x) = x, then ±(et , x) = FlI (x). So for t > 0 we have

t

d— — —

I I

d 1

dt (Fllog t ) ω = t (Fllog t ) LI ω

dt ±t ω =

1— —

1

= t ±t (iI dω + diI ω) = t d±t iI ω.

Note that Tx (±t ) = t. Id. Therefore

—

( 1 ±t iI ω)x (X2 , . . . , Xk ) = 1 (iI ω)tx (tX2 , . . . , tXk )

t t

= 1 ωtx (tx, tX2 , . . . , tXk ) = ωtx (x, tX2 , . . . , tXk ).

t

—

So if k ≥ 1, the (k ’ 1)-form 1 ±t iI ω is de¬ned and smooth in (t, x) for all t ∈ R.

t

— —

Clearly ±1 ω = ω and ±0 ω = 0, thus

1

— — d—

’

ω= ±1 ω ±0 ω = dt ±t ωdt

0

1 1

— 1—

d( 1 ±t iI ω)dt

= =d t ±t iI ωdt = d•.

t

0 0

7.11. Vector bundle valued di¬erential forms. Let (E, p, M ) be a vector

bundle. The space of smooth sections of the bundle Λk T — M — E will be denoted

by „¦k (M ; E). Its elements will be called E-valued k-forms.

If V is a ¬nite dimensional or even a suitable in¬nite dimensional vector space,

k

„¦ (M ; V ) will denote the space of all V -valued di¬erential forms of degree k.

The exterior di¬erential extends to this case, if V is complete in some sense.

8. Derivations

on the algebra of di¬erential forms

and the Fr¨licher-Nijenhuis bracket

o

8.1. In this section let M be a smooth manifold. We consider the graded

∞

dim M k k

commutative algebra „¦(M ) = „¦ (M ) = k=’∞ „¦ (M ) of di¬eren-

k=0

tial forms on M , where we put „¦k (M ) = 0 for k < 0 and k > dim M .

We denote by Derk „¦(M ) the space of all (graded) derivations of degree k,

i.e. all linear mappings D : „¦(M ) ’ „¦(M ) with D(„¦ (M )) ‚ „¦k+ (M ) and

D(• § ψ) = D(•) § ψ + (’1)k • § D(ψ) for • ∈ „¦ (M ).

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

68 Chapter II. Di¬erential forms

Lemma. Then the space Der „¦(M ) = k Derk „¦(M ) is a graded Lie alge-

bra with the graded commutator [D1 , D2 ] := D1 —¦ D2 ’ (’1)k1 k2 D2 —¦ D1 as

bracket. This means that the bracket is graded anticommutative, [D1 , D2 ] =

’(’1)k1 k2 [D2 , D1 ], and satis¬es the graded Jacobi identity [D1 , [D2 , D3 ]] =

[[D1 , D2 ], D3 ] + (’1)k1 k2 [D2 , [D1 , D3 ]] (so that ad(D1 ) = [D1 , ] is itself a

derivation of degree k1 ).

Proof. Plug in the de¬nition of the graded commutator and compute.

In section 7 we have already met some graded derivations: for a vector ¬eld

X on M the derivation iX is of degree ’1, LX is of degree 0, and d is of

degree 1. Note also that the important formula LX = d iX + iX d translates to

LX = [iX , d].

8.2. A derivation D ∈ Derk „¦(M ) is called algebraic if D | „¦0 (M ) = 0. Then

D(f.ω) = f.D(ω) for f ∈ C ∞ (M, R), so D is of tensorial character by 7.3. So D

—

induces a derivation Dx ∈ Derk ΛTx M for each x ∈ M . It is uniquely determined

by its restriction to 1-forms Dx |Tx M : Tx M ’ Λk+1 T — M which we may view as

— —

—

an element Kx ∈ Λk+1 Tx M — Tx M depending smoothly on x ∈ M . To express

this dependence we write D = iK = i(K), where K ∈ C ∞ (Λk+1 T — M — T M ) =:

„¦k+1 (M ; T M ). Note the de¬ning equation: iK (ω) = ω —¦ K for ω ∈ „¦1 (M ). We

dim M

call „¦(M, T M ) = k=0 „¦k (M, T M ) the space of all vector valued di¬erential

forms.

Theorem. (1) For K ∈ „¦k+1 (M, T M ) the formula

(iK ω)(X1 , . . . , Xk+ ) =

1

= sign σ .ω(K(Xσ1 , . . . , Xσ(k+1) ), Xσ(k+2) , . . . )

(k+1)! ( ’1)!

σ∈Sk+

for ω ∈ „¦ (M ), Xi ∈ X(M ) (or Tx M ) de¬nes an algebraic graded derivation

iK ∈ Derk „¦(M ) and any algebraic derivation is of this form.

(2) By i([K, L]§ ) := [iK , iL ] we get a bracket [ , ]§ on „¦—+1 (M, T M )

which de¬nes a graded Lie algebra structure with the grading as indicated, and

for K ∈ „¦k+1 (M, T M ), L ∈ „¦ +1 (M, T M ) we have

[K, L]§ = iK L ’ (’1)k iL K,

where iK (ω — X) := iK (ω) — X.

[ , ]§ is called the algebraic bracket or the Nijenhuis-Richardson bracket,

see [Nijenhuis-Richardson, 67].

—

Proof. Since ΛTx M is the free graded commutative algebra generated by the

—

vector space Tx M any K ∈ „¦k+1 (M, T M ) extends to a graded derivation. By

applying it to an exterior product of 1-forms one can derive the formula in (1).

The graded commutator of two algebraic derivations is again algebraic, so the

injection i : „¦—+1 (M, T M ) ’ Der— („¦(M )) induces a graded Lie bracket on

„¦—+1 (M, T M ) whose form can be seen by applying it to a 1-form.

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

8. Derivations on the algebra of di¬erential forms and the Fr¨licher-Nijenhuis bracket 69

o

8.3. The exterior derivative d is an element of Der1 „¦(M ). In view of the formula

LX = [iX , d] = iX d + d iX for vector ¬elds X, we de¬ne for K ∈ „¦k (M ; T M )

the Lie derivation LK = L(K) ∈ Derk „¦(M ) by LK := [iK , d].

Then the mapping L : „¦(M, T M ) ’ Der „¦(M ) is injective, since LK f =

iK df = df —¦ K for f ∈ C ∞ (M, R).

Theorem. For any graded derivation D ∈ Derk „¦(M ) there are unique K ∈

„¦k (M ; T M ) and L ∈ „¦k+1 (M ; T M ) such that

D = LK + iL .

We have L = 0 if and only if [D, d] = 0. D is algebraic if and only if K = 0.

Proof. Let Xi ∈ X(M ) be vector ¬elds. Then f ’ (Df )(X1 , . . . , Xk ) is a

derivation C ∞ (M, R) ’ C ∞ (M, R), so by 3.3 there is a unique vector ¬eld

K(X1 , . . . , Xk ) ∈ X(M ) such that

(Df )(X1 , . . . , Xk ) = K(X1 , . . . , Xk )f = df (K(X1 , . . . , Xk )).

Clearly K(X1 , . . . , Xk ) is C ∞ (M, R)-linear in each Xi and alternating, so K is

tensorial by 7.3, K ∈ „¦k (M ; T M ).

The de¬ning equation for K is Df = df —¦K = iK df = LK f for f ∈ C ∞ (M, R).

Thus D ’ LK is an algebraic derivation, so D ’ LK = iL by 8.2 for unique

L ∈ „¦k+1 (M ; T M ).

Since we have [d, d] = 2d2 = 0, by the graded Jacobi identity we obtain

0 = [iK , [d, d]] = [[iK , d], d] + (’1)k’1 [d, [iK , d]] = 2[LK , d]. The mapping K ’

[iK , d] = LK is injective, so the last assertions follow.

8.4. Applying i(IdT M ) on a k-fold exterior product of 1-forms we see that

i(IdT M )ω = kω for ω ∈ „¦k (M ). Thus we have L(IdT M )ω = i(IdT M )dω ’

d i(IdT M )ω = (k + 1)dω ’ kdω = dω. Thus L(IdT M ) = d.

8.5. Let K ∈ „¦k (M ; T M ) and L ∈ „¦ (M ; T M ). Then obviously [[LK , LL ], d] =

0, so we have

[L(K), L(L)] = L([K, L])

for a uniquely de¬ned [K, L] ∈ „¦k+ (M ; T M ). This vector valued form [K, L] is

called the Fr¨licher-Nijenhuis bracket of K and L.

o

dim M

Theorem. The space „¦(M ; T M ) = k=0 „¦k (M ; T M ) with its usual grading

is a graded Lie algebra for the Fr¨licher-Nijenhuis bracket. So we have

o

[K, L] = ’(’1)k [L, K]

[K1 , [K2 , K3 ]] = [[K1 , K2 ], K3 ] + (’1)k1 k2 [K2 , [K1 , K3 ]]

IdT M ∈ „¦1 (M ; T M ) is in the center, i.e. [K, IdT M ] = 0 for all K.

L : („¦(M ; T M ), [ , ]) ’ Der „¦(M ) is an injective homomorphism of gra-

ded Lie algebras. For vector ¬elds the Fr¨licher-Nijenhuis bracket coincides with

o

the Lie bracket.

Proof. df —¦ [X, Y ] = L([X, Y ])f = [LX , LY ]f . The rest is clear.

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

70 Chapter II. Di¬erential forms

8.6. Lemma. For K ∈ „¦k (M ; T M ) and L ∈ „¦ +1