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CHAPTER II. DIFFERENTIAL FORMS

Back

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 Frolicher-Nijenhuis bracket via the study

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 manifolds.

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

an open subset in M and where is a _ber respecting di_eomorphism as in the

following diagram:

EjU := p􀀀1(U) w

AAAAC

U _ V

􀀀

_􀀀􀀀

pr1

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; 1) and (U2; 2) are called compatible, if 1 _

􀀀1

2 is a _ber linear isomorphism, i.e. ( 1 _ 􀀀1

2 )(x; v) = (x; 1;2(x)v) for some

mapping 1;2 : U1;2 := U1 \U2 ! GL(V ). The mapping 1;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_; _)_2A for (E; p;M) is a set of pairwise compatible

vector bundle charts (U_; _) such that (U_)_2A 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 (TM; _M;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 2 M) there is a unique structure of a real vector space, induced

from any vector bundle chart (U_; _) with x 2 U_. So 0x 2 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 fx 2 M : u(x) 6= 0xg in

M. The space of all smooth sections of the bundle (E; p;M) will be denoted by

either C1(E) = C1(E; p;M) = C1(E ! M). Clearly it is a vector space with

_ber wise addition and scalar multiplication.

If (U_; _)_2A is a vector bundle atlas for (E; p;M), then any smooth mapping

f_ : U_ ! V (the standard _ber) de_nes a local section x 7! 􀀀1

_ (x; f_(x))

on U_. If (g_)_2A is a partition of unity subordinated to (U_), then a global

section can be formed by x 7!

_ g_(x) _ 􀀀1

_ (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 homomorphism

' : E ! F is a _ber respecting, _ber linear smooth mapping

E w

M w

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 category

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_)_2A of M. If (E; p;M) is a vector bundle

which admits a vector bundle atlas (U_; _) with the given open cover, then

we have _ _ 􀀀1

_ (x; v) = (x; __(x)v) for transition functions __ : U__ =

U_ \ U_ ! 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

(1)

(

__(x) _ _(x) = _(x) for each x 2 U__ = U_ \ U_ \ U;

__(x) = e for all x 2 U_:

Condition (1) is called a cocycle condition and thus we call the family ( __) the

cocycle of transition functions for the vector bundle atlas (U_; _).

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_; _) and (U_; '_) are compatible for each _, so

'_ _ 􀀀1

_ (x; v) = (x; __(x)v) for some __ : U_ ! GL(V ). But then we have

(x; __(x) __(x)v) = ('_ _ 􀀀1

_ )(x; __(x)v) =

= ('_ _ 􀀀1

_ _ _ 􀀀1

_ )(x; v) = ('_ _ 􀀀1

_ )(x; v) =

= ('_ _ '􀀀1

_ '_ _ 􀀀1

_ )(x; v) = (x; '__(x)__(x)v):

So we get

(2) __(x) __(x) = '__(x)__(x) for all x 2 U__:

We say that the two cocycles ( __) and ('__) of transition functions over

the cover (U_) are cohomologous. The cohomology classes of cocycles ( __)

over the open cover (U_) (where we identify cohomologous ones) form a set

1((U_);GL(V )), the _rst _ Cech cohomology set of the open cover (U_) with

values in the sheaf C1( ;GL(V )) =: GL(V ).

Now let (Wi)i2I be an open cover of M that re_nes (U_) with Wi _ U"(i),

where " : I ! A is some re_nement mapping. Then for any cocycle ( __)

over (U_) we de_ne the cocycle "_( __) =: ('ij) by the prescription 'ij :=

"(i);"(j)

jWij . The mapping "_ respects the cohomology relations and induces

therefore a mapping "] : _H 1((U_);GL(V )) ! _H 1((Wi);GL(V )). One can show

that the mapping "_ depends on the choice of the re_nement mapping " only up

to cohomology (use _i = "(i);_(i)

jWi if " and _ are two re_nement mappings),

so we may form the inductive limit 􀀀li!m _H 1(U;GL(V )) =: _H 1(M;GL(V )) over

all open covers of M directed by re_nement.

Theorem. There is a bijective correspondence between _H1(M;GL(V )) and the

set of all isomorphism classes of vector bundles over M with typical _ber V .

Proof. Let ( __) be a cocycle of transition functions __ : U__ ! GL(V ) over

some open cover (U_) of M. We consider the disjoint union

_2A

f_g_U_ _V

and the following relation on it: (_; x; v) _ (_; y;w) if and only if x = y and

__(x)v = w.

By the cocycle property (1) of ( __) this is an equivalence relation. The space

of all equivalence classes is denoted by E = V B( __) 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_; _) by _[(_; x; v)] = (x; v), _ : p􀀀1(U_) =: EjU_ !

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

52 Chapter II. Di_erential forms

U__V . Then the mapping __ 􀀀1

_ (x; v) = _[(_; x; v)] = _[(_; x; __(x)v)] =

(x; __(x)v) is smooth, so E becomes a smooth manifold. E is Hausdor_: let

u 6= v in E; if p(u) 6= 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 ( __) over (U_), and ('ij) over (Vi).

Then there is a common re_nement (W) 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 ( __)

and ('__) are cocycles over the same open cover (U_). If the two cocycles are

cohomologous, so ___ __ = '_____ on U__, then a _ber linear di_eomorphism _ :

V B( __) ! V B('__) is given by '__ [(_; x; v)] = (x; __(x)v). By relation (2)

this is well de_ned, so the vector bundles V B( __) and V B('__) 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

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

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_; _) be a vector bundle atlas on a vector bundle (E; p;M). Let

(ej)kj

=1 be a basis of the standard _ber V . We consider the section sj(x) :=

􀀀1

_ (x; ej) for x 2 U_. Then the sj : U_ ! E are local sections of E such that

(sj(x))kj

=1 is a basis of Ex for each x 2 U_: we say that s = (s1; : : : ; sk) is a

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; ) of E such that sj(x) = 􀀀1(x; ej ),

in the following way. We de_ne f : U _ Rk ! EjU by f(x; v1; : : : ; vk P ) := k

j=1 vjsj(x). Then f is smooth, invertible, and a _ber linear isomorphism, so

(U; = 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 2 M.

Lemma. Let ' : (E; p;M) ! (E0; q;N) be a vector bundle homomorphism

such that rank('x : Ex ! E0

'(x)) is constant in x 2 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

F(V ) = _k(V ) (the k-th exterior power), or F(V ) =

Nk V , and the like.

If (E; p;M) is a vector bundle, described by a vector bundle atlas with cocycle

of transition functions '__ : U__ ! GL(V ), where (U_) is an open cover of M,

then we may consider the smooth functions F('__) : x 7! F('__(x)), U__ !

GL(F(V )). Since F is a covariant functor, F('__) 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('__)); 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('__).

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

F(V B( __); V B('__)) := V B(F( 􀀀1

__ ; '__))

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: _kE, E _ F, E_, _E =

k_0 _kE, 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

f_E w

p_f

f_p

N w

are de_ned as follows. Let E be described by a cocycle ( __) of transition

functions over an open cover (U_) of M, E = V B( __). Then ( __ _ 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( ___f). As a manifold we have f_E = N _

(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

: F ! f_E with = g and p_f _ = '.

F4 4464

f_E w

p_f

f_p

P w

N w

6.9. Theorem. Any vector bundle admits a _nite vector bundle atlas.

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

(U_; _)_2A be a vector bundle atlas. Since M is separable, by topological

dimension theory there is a re_nement of the open cover (U_)_2A of the form

(Vij)i=1;:::;m+1;j2N, such that Vij \ Vik = ; for j 6= k, see the remarks at the end

of 1.1. We de_ne the set Wi :=

j2N Vij (a disjoint union) and ijVij = _(i;j),

where _ : f1; : : : ;m + 1g _ N ! A is a re_ning map. Then (Wi; i)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 2 N.

Proof. Let (Ui; i)ni

=1 be a _nite vector bundle atlas for (E; p;M). Let (gi) be

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 : E ! M _ Rnk by

(u) =

p(u);

i=1

gi(p(u)) (`i _ pr2 _ i)(u)

;

then is smooth, _ber linear, and an embedding on each _ber, so E is a vector

sub bundle of M_Rnk via . Now we de_ne Fx = E?

x in fxg_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

mEt

: E ! E. Then (TE; _E;E), the tangent bundle of the manifold E, is itself

a vector bundle, with _ber addition denoted by +TE and scalar multiplication

denoted by mTE

t .

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

6. Vector bundles 55

If (U_; _ : EjU_ ! U_ _ V )_2A is a vector bundle atlas for E, such that

(U_; u_) is also a manifold atlas for M, then (EjU_; 0

_)_2A is an atlas for the

manifold E, where

_ := (u_ _ IdV ) _ _ : EjU_ ! U_ _ V ! u_(U_) _ V _ Rm _ V:

Hence the family (T(EjU_); T 0

_ : T(EjU_) ! T(u_(U_) _ V ) = u_(U_) _ V _

Rm _ V )_2A is the atlas describing the canonical vector bundle structure of

(TE; _E;E). The transition functions are in turn:

( _ _ 􀀀1

_ )(x; v) = (x; __(x)v) for x 2 U__

(u_ _ u􀀀1

_ )(y) = u__(y) for y 2 u_(U__)

( 0

_ ( 0

_)􀀀1)(y; v) = (u__(y); __(u􀀀1

_ (y))v)

(T 0

_ T( 0

_)􀀀1)(y; v; _;w) =

􀀀

u__(y); __(u􀀀1

_ (y))v; d(u__)(y)_;

(d( __ _ u􀀀1

_ )(y))_)v + __(u􀀀1

_ (y))w

So we see that for _xed (y; v) the transition functions are linear in (_;w) 2

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

(TE; _E;E).

For _xed (y; _) the transition functions of TE are also linear in (v;w) 2 V _V .

This gives a vector bundle structure on (TE; Tp; TM). Its _ber addition will be

denoted by T(+E) : T(E _M E) = TE _TM TE ! TE, since it is the tangent

mapping of +E. Likewise its scalar multiplication will be denoted by T(mEt

One may say that the second vector bundle structure on TE, that one over TM,

is the derivative of the original one on E.

The space f_ 2 TE : Tp:_ = 0 in TMg = (Tp)􀀀1(0) is denoted by V E and is

called the vertical bundle over E. The local form of a vertical vector _ is T 0

_:_ =

(y; v; 0;w), so the transition function looks like (T 0

_ T( 0

_)􀀀1)(y; v; 0;w) =

(u__(y); __(u􀀀1

_ (y))v; 0; __(u􀀀1

_ (y))w). They are linear in (v;w) 2 V _ V for

_xed y, so V E is a vector bundle over M. It coincides with 0_

M(TE; Tp; TM),

the pullback of the bundle TE ! TM over the zero section. We have a canonical

isomorphism vlE : E_ME ! V E, called the vertical lift, given by vlE(ux; vx) :=

j0(ux + tvx), which is _ber linear over M. The local representation of the

vertical lift is (T 0

_ vlE _ ( 0

_ 0

_)􀀀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_ME ! V F _ TF. So vl is a natural

transformation between certain functors on the category of vector bundles and

their homomorphisms.

The mapping vprE := pr2 _ vl􀀀1

E : V E ! E is called the vertical projection.

Note also the relation pr1 _ vl􀀀1

E = _EjV E.

6.12. The second tangent bundle of a manifold. All of 6.11 is valid

for the second tangent bundle T2M = TTM of a manifold, but here we have

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

_M : T2M ! T2M is de_ned locally by

(T2u _ _M _ T2u􀀀1)(x; _; _; _) = (x; _; _; _);

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

56 Chapter II. Di_erential forms

where (U; u) is a chart on M. Clearly this de_nition is invariant under changes

of charts (Tu_ equals 0

_ from 6.11).

The ip _M has the following properties:

(1) _N _ T2f = T2f _ _M for each f 2 C1(M;N).

(2) T(_M) _ _M = _TM.

(3) _TM _ _M = T(_M).

(4) _􀀀1

M = _M.

(5) _M is a linear isomorphism from the bundle (TTM; T(_M); TM) to

(TTM; _TM; TM), so it interchanges the two vector bundle structures

on TTM.

(6) It is the unique smooth mapping TTM ! TTM which satis_es

@s c(t; s) = _M

@t c(t; s) for each c : R2 ! 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 2 X(M) we have

[X; Y ] = vprTM _ (TY _ X 􀀀 _M _ TX _ 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; d _ Y (x):_X (x) 􀀀

d_X (x): _ Y (x)), and

vprTM _ (TY _ X 􀀀 _M _ TX _ Y )(x) =

= vprTM _ (TY:(x; _X (x)) 􀀀 _M _ TX:(x; _ Y (x))) =

= vprTM

􀀀

(x; _ Y (x); _X (x); d _ Y (x):_X (x))􀀀

􀀀 _M((x; _X (x); _ Y (x); d_X (x): _ Y (x))

= vprTM(x; _ Y (x); 0; d _ Y (x):_X (x) 􀀀 d_X (x): _ Y (x)) =

= (x; d _ Y (x):_X (x) 􀀀 d_X (x): _ Y (x)): _

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

smooth manifolds and local di_eomorphisms (i.e. immersions) between

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

F(M) w

F(f)

F(N)

M w

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

6. Vector bundles 57

to each f : M ! N in Mfm, which covers f and is _berwise a linear isomorphism.

We also require that for smooth f : R _ M ! N the mapping

(t; x) 7! 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. TM, 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 := ((Txf)􀀀1)_ : T_

xM ! T_

f(x)N. This functor is de_ned on Mfm

only.

3. _kT_M, _T_M =

k_0 _kT_M.

Nk T_M

N` TM = T_M _ _ _ T_M TM _ _ _ TM, where the

action on morphisms involves Tf􀀀1 in the T_M-parts and Tf in the TM-parts.

5. F(TM), where F is any smooth functor on the category of _nite dimensional

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 2 X(M) be a vector _eld on M. Then

the ow FlXt

, for _xed t, is a di_eomorphism de_ned on an open subset of M,

which we do not specify. The mapping

F(M) w

F(FlXt

)

F(M)

M w

FlXt

is then a vector bundle isomorphism, de_ned over an open subset of M.

We consider a section s 2 C1(F(M)) of the vector bundle (F(M); pM;M)

and we de_ne for t 2 R

(FlXt

)_s := F(FlX􀀀

t) _ s _ FlXt

This is a local section of the vector bundle F(M). For each x 2 M the value

((FlXt

)_s)(x) 2 F(M)x is de_ned, if t is small enough. So in the vector space

F(M)x the expression d

j0((FlXt

)_s)(x) makes sense and therefore the section

LXs := d

j0(FlXt

)_s

is globally de_ned and is an element of C1(F(M)). It is called the Lie derivative

of s along X.

Lemma. In this situation we have

(1) (FlXt

)_(FlXr

)_s = (FlX t+r)_s, whenever de_ned.

(2) d

dt (FlXt

)_s = (FlXt

)_LXs = LX(FlXt)_s, so

[LX; (FlXt

)_] := LX _ (FlXt

)_ 􀀀 (FlXt

)_ _ LX = 0, whenever de_ned.

(3) (FlXt

)_s = s for all relevant t if and only if LXs = 0.

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

58 Chapter II. Di_erential forms

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

dt (FlXt

)_s = d

j0(FlXr

)_(FlXt

)_s = LX(FlXt

)_s:

dt ((FlXt

)_s)(x) = d

j0((FlXt

)_(FlXr

)_s)(x)

= d

j0F(FlX􀀀

t)(F(FlX􀀀

r) _ s _ FlXr

)(FlXt

(x))

= F(FlX􀀀

t) d

j0(F(FlX􀀀

r) _ s _ FlXr

)(FlXt

(x))

= ((FlXt

)_LXs)(x);

since F(FlX􀀀

t) : F(M)FlXt

(x)

! F(M)x is linear.

(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 2 C1(Fi(M)) there is a section s1 s2 2 C1((F1 F2)(M)), given by

the pointwise tensor product.

Lemma. In this situation, for X 2 X(M) we have

LX(s1 s2) = LXs1 s2 + s1 LXs2:

In particular, for f 2 C1(M;R) we have LX(fs) = df(X) s + f LXs.

Proof. Using the bilinearity of the tensor product we have

LX(s1 s2) = d

j0(FlXt

)_(s1 s2)

= d

j0((FlXt

)_s1 (FlXt

)_s2)

= d

j0(FlXt

)_s1 s2 + s1 d

j0(FlXt

)_s2

= LXs1 s2 + s1 LXs2: _

6.17. Let ' : F1 ! F2 be a linear natural transformation between vector bundle

functors on Mfm, i.e. for each M 2 Mfm we have a vector bundle homomorphism

'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 2 C1(F1(M)) and X 2 X(M), we have

LX('M s) = 'M(LXs).

Proof. Since 'M is _ber linear and natural we can compute as follows.

LX('M s)(x) = d

j0((FlXt

)_('M s))(x) = d

j0(F2(FlX􀀀

t) _ 'M _ s _ FlXt

)(x)

= 'M _ d

j0(F1(FlX􀀀

t) _ s _ FlXt

)(x) = ('M LXs)(x): _

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

6. Vector bundles 59

6.18. A tensor _eld of type

􀀀p

is a smooth section of the natural bundle Nq T_M

Np TM. 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 2 X(M) be a

vector _eld. We consider the local vector bundle homomorphism F(FlXt

) on

F(M). Since F(FlXt

) _ F(FlXs

) = F(FlX t+s) and F(FlX0

) = IdF(M) we have

dtF(FlXt

) = d

j0F(FlXs

) _ F(FlXt

) = XF _ F(FlXt

), so we get F(FlXt

) = FlXF

t ,

where XF = d

j0F(FlXs

) 2 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 XF = FX.

Lemma.

(1) XT = _M _ TX.

(2) [X; Y ]F = [XF ; Y F ].

(3) XF : (F(M); pM;M) ! (TF(M); T(pM); TM) is a vector bundle homomorphism

for the T(+)-structure.

(4) For s 2 C1(F(M)) and X 2 X(M) we have

LXs = vprF(M)(Ts _ X 􀀀 XF _ s).

(5) LXs is linear in X and s.

Proof. (1) is an easy computation. F(FlXt

) is _ber linear and this implies (3).

(4) is seen as follows:

(LXs)(x) = d

j0(F(FlX􀀀

t) _ s _ FlXt

)(x) in F(M)x

= vprF(M)( d

j0(F(FlX􀀀

t) _ s _ FlXt

)(x) in V F(M))

= vprF(M)(􀀀XF _ s _ FlX0

(x) + T(F(FlX0

)) _ Ts _ X(x))

= vprF(M)(Ts _ X 􀀀 XF _ s)(x):

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

mapping X(M) ! C1(F(M)), so it is linear. See [Frolicher, Kriegl, 88] for the

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

0 = @

0 (FlY

􀀀t

_ FlX􀀀

_ FlYt

_ FlXt

);

[X; Y ] = 1

@t2 j0(FlY

􀀀t

_ FlX􀀀

_ FlYt

_ FlXt

)

= @

0 Fl[X;Y ]

t :

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

60 Chapter II. Di_erential forms

Applying F to these curves (of local di_eomorphisms) we get

0 = @

0 (FlY F

􀀀t

_ FlXF

􀀀t

_ FlY F

_ FlXF

t );

[XF ; Y F ] = 1

@t2 j0(FlY F

􀀀t

_ FlXF

􀀀t

_ FlY F

_ FlXF

t )

= 1

@t2 j0F(FlY

􀀀t

_ FlX􀀀

_ FlYt

_ FlXt

)

= @

0 F(Fl[X;Y ]

t ) = [X; Y ]F :

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 2

X(M) we have

[LX;LY ] := LX _ LY 􀀀 LY _ LX = L

[X;Y ] : C1(F(M)) ! C1(F(M)):

So L : X(M) ! EndC1(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 f0g _M

in R _M, such that each 'i

t is a di_eomorphism on its domain, 'i

0 = IdM, and

0 'i

t = Xi 2 X(M). We put ['i; 'j ]t = ['i

t; 'j

t ] := ('j

t )􀀀1 _ ('i

t)􀀀1 _ 'j

_ 'i

Let F be a vector bundle functor, let s 2 C1(F(M)) be a section. Then for

each formal bracket expression P of lenght k we have

0 = @`

@t`

j0P('1t

; : : : ; 'kt

)_s for 1 _ ` < k;

P(X1;:::;Xk)s = 1

@tk

j0P('1t

; : : : ; 'kt

)_s 2 C1(F(M)):

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) 7! p + v for p 2 A and v 2 ~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 : EjU = p􀀀1(U) ! U _ V on E

there exists a _ber respecting di_eomorphism ' : ZjU = q􀀀1(U) ! U _ A such

that 'x : Zx ! A is an a_ne morphism satisfying ~'x = x : Ex ! V for each

x 2 U. We also write E = ~Z to have a functorial assignment of the modelling

vector bundle.

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

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 7! ~ 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 manifoldM is the vector bundle T_M := (TM)_,

the (real) dual of the tangent bundle.

If (U; u) is a chart on M, then ( @

@u1 ; : : : ; @

@um ) is the associated frame _eld

over U of TM. Since @

@ui

jx(uj) = duj( @

@ui

jx) = _j

i 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.

7.2. According to 6.18 a tensor _eld of type

􀀀p

on a manifold M is a smooth

section of the vector bundle

T_M = TM

p times z }| {

_ _ _TM T_M

q times z }| {

_ _ _T_M:

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

􀀀 @

@ui1

@ui2

_ _ _ @

@uip

duj1 _ _ _ dujq

i2f1;::: ;mgp;j2f1;::: ;mgq

over U of this tensor bundle, and for any

􀀀p

-tensor _eld A we have

A j U =

i;j

Ai1:::ip

j1:::jq

@ui1

_ _ _ @

@uip

duj1 _ _ _ dujq :

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 ! C1(

N` TM) be a

mapping which is k-linear over C1(M;R) then _ is given by a

􀀀`

-tensor _eld.

Proof. For simplicity's sake we put k = 1, ` = 0, so _ : X(M) ! C1(M;R) is a

C1(M;R)-linear mapping: _(f:X) = f:_(X).

Claim 1. If X j U = 0 for some open subset U _ M, then we have _(X) j

U = 0.

Let x 2 U. We choose f 2 C1(M;R) with f(x) = 0 and f j M n 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 2 V _ _ V _ U. Then

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

62 Chapter II. Di_erential forms

X j U =

Xi @

@ui and Xi(x) = 0. We choose g 2 C1(M;R) with g j V _ 1 and

supp g _ U. Then (g2:X) j V = X j V and by claim 1 _(X) j V depends only on

X j V and g2:X =

i(g:Xi)(g: @

@ui ) is a decomposition which is globally de_ned

on M. Therefore we have _(X)(x) = _(g2:X)(x) = _

􀀀P

i(g:Xi)(g: @

@ui )

P (x) =

(g:Xi)(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 2 M. So there is a linear map 'x : TxM ! R for

each x 2 M with _(X)(x) = 'x(X(x)). Then ' : M ! T_M is smooth since

' j 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 _kT_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

-tensor _elds, i.e. (by 7.3) the space of all mappings

_ : X(M) _ _ _ _ _ X(M) = X(M)k ! C1(M;R);

which are k-linear over C1(M;R) and are skew symmetric:

_(X_1; : : : ;X_k) = sign _ _ _(X1; : : : ;Xk)

for each permutation _ 2 Sk.

We put 0(M) := C1(M;R). Then the space

(M) :=

dMimM

k=0

k(M)

is an algebra with the following product. For ' 2 k(M) and 2 `(M) and

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

(' ^ )(X1; : : : ;Xk+`) =

= 1

k! `!

_2Sk+`

sign _ _ '(X_1; : : : ;X_k): (X_(k+1); : : : ;X_(k+`)):

This product is de_ned _ber wise, i.e. (' ^ )x = 'x ^ x for each x 2 M. It

is also associative, i.e. (' ^ ) ^ _ = ' ^ ( ^ _ ), and graded commutative, i.e.

' ^ = (􀀀1)k` ^ '. These properties are proved in multilinear algebra.

7.5. If f : N ! M is a smooth mapping and ' 2 k(M), then the pullback

f_' 2 k(N) is de_ned for Xi 2 TxN by

(1) (f_')x(X1; : : : ;Xk) := 'f(x)(Txf:X1; : : : ; Txf:Xk):

Then we have f_('^ ) = f_'^f_ , 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 7! (M) = C1(_T_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 7! _T_M is a covariant vector bundle functor

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 7! _kT_M is a

vector bundle functor on Mfm, by 6.15 for X 2 X(M) the Lie derivative of a

k-form ' along X is de_ned by

LX' = d

j0(FlXt

)_':

Lemma. The Lie derivative has the following properties.

(1) LX(' ^ ) = LX' ^ + ' ^ LX , so LX is a derivation.

(2) For Yi 2 X(M) we have

(LX')(Y1; : : : ; Yk) = X('(Y1; : : : ; Yk)) 􀀀

i=1

'(Y1; : : : ; [X; Yi]; : : : ; Yk):

(3) [LX;LY ]' = L

[X;Y ]'.

Proof. The mapping Alt :

Nk T_M ! _kT_M, given by

(AltA)(Y1; : : : ; Yk) := 1

sign(_)A(Y_1; : : : ; Y_k);

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

homomorphism from the tensor algebra

k_0 C1(

Nk 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) + ' (

i Y1 _ _ _ LXYi _ _ _ Yk)

Now we use LXYi = [X; Yi].

(3) is a special case of 6.20. _

7.7. The insertion operator. For a vector _eld X 2 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(' ^ ) = iX' ^ + (􀀀1)deg '' ^ iX .

(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 ' 2 k(M) and 2 `(M) we have

(iX1 (' ^ ))(X2; : : : ;Xk+`) = (' ^ )(X1; : : : ;Xk+`) =

= 1

k! `!

sign(_) '(X_1; : : : ;X_k) (X_(k+1); : : : ;X_(k+`)):

(iX1' ^ + (􀀀1)k' ^ iX1 )(X2; : : : ;Xk+`) =

= 1

(k􀀀1)! `!

sign(_) '(X1;X_2; : : : ;X_k) (X_(k+1); : : : ;X_(k+`))+

(􀀀1)k

k! (` 􀀀 1)!

sign(_) '(X_2; : : : ;X_(k+1)) (X1;X_(k+2); : : : ):

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

adding an appropriate sign. These are k+` summands. Since 1

(k􀀀1)! `!+ 1

k! (`􀀀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:

LXiY ' = LX Trace1(Y ') = Trace1 LX(Y ')

= Trace1(LXY ' + 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 ' 2 k(Rn). Then we may view ' as an element

of C1(U;Lk

alt(Rn;R)). We consider D' 2 C1(U;L(Rn;Lk

alt(Rn;R))), and we

take its canonical image Alt(D') in C1(U;Lk+1

alt (Rn;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

(d')x(X0; : : : ;Xk) = 1

sign(_)D'((1) x)(X_0)(X_1; : : : ;X_k)

i=0

(􀀀1)iD'(x)(Xi)(X0; : : : ;cXi; : : : ;Xk);

where the hat over a symbol means that this is to be omitted, and where Xi 2 Rn.

Now we pass to an arbitrary manifold M. For a k-form ' 2 k(M) and

vector _elds Xi 2 X(M) we try to replace D'(x)(Xi)(X0; : : : ) in formula (1)

Pby Lie derivatives. We di_erentiate Xi('(x)(X0; : : : )) = D'(x)(Xi)(X0; : : : ) +

0_j_k;j6=i '(x)(X0; : : : ;DXj(x)Xi; : : : ), so inserting this expression into formula

(1) we get (cf. 3.4) our working de_nition

d'(X0; : : : ;Xk) :=

i=0

(2) (􀀀1)iXi('(X0; : : : ;Xci; : : : ;Xk))

i<j

(􀀀1)i+j'([Xi;Xj ];X0; : : : ;cXi; : : : ; cXj ; : : : ;Xk):

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 C1(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

'jU =

i1<___<ik

'i1;::: ;ikdui1 ^ _ _ _ ^ duik ;

where 'i1;::: ;ik = '( @

@ui1 ; : : : ; @

@uik ). An easy computation shows that (2) leads

(3) d'jU =

i1<___<ik

d'i1;::: ;ik

^ dui1 ^ _ _ _ ^ duik ;

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 following

properties:

(1) d('^ ) = d'^ +(􀀀1)deg ''^d , so d is a graded derivation of degree

(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

2 [d; d] = 0.

(4) [f_; d] = 0.

(5) [LX; d] = 0.

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

Proof. (2) For ' 2 k(M) and Xi 2 X(M) we have

(LX0')(X1; : : : ;Xk) = X0('(X1; : : : ;Xk))+

j=1

(􀀀1)0+j'([X0;Xj ];X1; : : : ; cXj ; : : : ;Xk) by 7.6.2;

(iX0d')(X1; : : : ;Xk) = d'(X0; : : : ;Xk)

i=0

(􀀀1)iXi('(X0; : : : ;cXi; : : : ;Xk))+

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

66 Chapter II. Di_erential forms

0_i<j

(􀀀1)i+j'([Xi;Xj ];X0; : : : ;cXi; : : : ; cXj ; : : : ;Xk):

(diX0')(X1; : : : ;Xk) =

i=1

(􀀀1)i􀀀1Xi((iX0')(X1; : : : ;cXi; : : : ;Xk))+

1_i<j

(􀀀1)i+j􀀀1(iX0')([Xi;Xj ];X1; : : : ;cXi; : : : ; cXj ; : : : ;Xk)

= 􀀀

i=1

(􀀀1)iXi('(X0;X1; : : : ;cXi; : : : ;Xk))􀀀

􀀀

1_i<j

(􀀀1)i+j'([Xi;Xj ];X0;X1; : : : ;cXi; : : : ; cXj ; : : : ;Xk):

By summing up the result follows.

(1) Let ' 2 p(M) and 2 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 2 X(M) we have by part (2)

and 7.6, 7.7 and by induction

iX d(' ^ ) = LX(' ^ ) 􀀀 d iX(' ^ )

= LX' ^ + ' ^ LX 􀀀 d(iX' ^ + (􀀀1)p' ^ iX )

= iXd' ^ + diX' ^ + ' ^ iXd + ' ^ diX 􀀀 diX' ^

􀀀 (􀀀1)p􀀀1iX' ^ d 􀀀 (􀀀1)pd' ^ iX 􀀀 ' ^ diX

= iX(d' ^ + (􀀀1)p' ^ d ):

Since X is arbitrary, (1) follows.

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

2 [d; d] is a graded

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

generated as an algebra by C1(M;R) and fdf : f 2 C1(M;R)g, it su_ces to

show that d2f = 0 for each f 2 C1(M;R) (d3f = 0 is a consequence). But this is

easy: d2f(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 2 C1(M;R). We have (f_dg)y(Y ) = (dg)f(y)(Tyf:Y ) = (Tyf: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 = diXd + iXdd = LXd. _

7.10. A di_erential form ! 2 k(M) is called closed if d! = 0, and it is called

exact if ! = d' for some ' 2 k􀀀1(M). Since d2 = 0, any exact form is closed.

The quotient space

Hk(M) :=

ker(d : k(M) ! k+1(M))

im(d : k􀀀1(M) ! k(M))

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

in this book, and we _nish with the

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

8. Derivations on the algebra of di_erential forms and the Frolicher-Nijenhuis bracket 67

Lemma of Poincar_e. A closed di_erential form is locally exact. More precisely:

let ! 2 k(M) with d! = 0. Then for any x 2 M there is an open

neighborhood U of x in M and a ' 2 k􀀀1(U) with d' = !jU.

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 2 X(Rm)

be the vector _eld I(x) = x, then _(et; x) = FlIt

(x). So for t > 0 we have

dt__

t ! = d

dt (FlI

log t)_! = 1

t (FlI

log t)_LI!

= 1

t __

t (iId! + diI!) = 1

t d__

t iI!:

Note that Tx(_t) = t: Id. Therefore

( 1

t __

t iI!)x(X2; : : : ;Xk) = 1

t (iI!)tx(tX2; : : : ; tXk)

= 1

t !tx(tx; tX2; : : : ; tXk) = !tx(x; tX2; : : : ; tXk):

So if k _ 1, the (k􀀀1)-form 1

t __

t iI! is de_ned and smooth in (t; x) for all t 2 R.

Clearly __

1! = ! and __

0! = 0, thus

! = __

1! 􀀀 __

0! =

Z 1

dt__

t !dt

Z 1

d( 1

t __

t iI!)dt = d

_Z 1

t __

t iI!dt

= d': _

7.11. Vector bundle valued di_erential forms. Let (E; p;M) be a vector

bundle. The space of smooth sections of the bundle _kT_ME 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 Frolicher-Nijenhuis bracket

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

commutative algebra (M) =

LdimM

k=0 k(M) =

k=􀀀1 k(M) of di_erential

forms on M, where we put k(M) = 0 for k < 0 and k > dimM.

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(' ^ ) = D(') ^ + (􀀀1)k`' ^ D( ) for ' 2 `(M).

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68 Chapter II. Di_erential forms

Lemma. Then the space Der (M) =

k Derk (M) is a graded Lie algebra

with the graded commutator [D1;D2] := D1 _ D2 􀀀 (􀀀1)k1k2D2 _ D1 as

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

􀀀(􀀀1)k1k2 [D2;D1], and satis_es the graded Jacobi identity [D1; [D2;D3]] =

[[D1;D2];D3] + (􀀀1)k1k2 [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 2 Derk (M) is called algebraic if D j 0(M) = 0. Then

D(f:!) = f:D(!) for f 2 C1(M;R), so D is of tensorial character by 7.3. So D

induces a derivation Dx 2 Derk _T_

xM for each x 2 M. It is uniquely determined

by its restriction to 1-forms DxjT_

xM : T_

xM ! _k+1T_M which we may view as

an element Kx 2 _k+1T_

xM TxM depending smoothly on x 2 M. To express

this dependence we write D = iK = i(K), where K 2 C1(_k+1T_M TM) =:

k+1(M; TM). Note the de_ning equation: iK(!) = ! _ K for ! 2 1(M). We

call (M; TM) =

LdimM

k=0 k(M; TM) the space of all vector valued di_erential

forms.

Theorem. (1) For K 2 k+1(M; TM) the formula

(iK!)(X1; : : : ;Xk+`) =

= 1

(k+1)! (`􀀀1)!

_2Sk+`

sign _ :!(K(X_1; : : : ;X_(k+1));X_(k+2); : : : )

for ! 2 `(M), Xi 2 X(M) (or TxM) de_nes an algebraic graded derivation

iK 2 Derk (M) and any algebraic derivation is of this form.

(2) By i([K;L]^) := [iK; iL] we get a bracket [ ; ]^ on _+1(M; TM)

which de_nes a graded Lie algebra structure with the grading as indicated, and

for K 2 k+1(M; TM), L 2 `+1(M; TM) we have

[K;L]^ = iKL 􀀀 (􀀀1)k`iLK;

where iK(! X) := iK(!) X.

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

see [Nijenhuis-Richardson, 67].

Proof. Since _T_

xM is the free graded commutative algebra generated by the

vector space T_

xM any K 2 k+1(M; TM) 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; TM) ! Der_((M)) induces a graded Lie bracket on

_+1(M; TM) whose form can be seen by applying it to a 1-form. _

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8. Derivations on the algebra of di_erential forms and the Frolicher-Nijenhuis bracket 69

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 2 k(M; TM)

the Lie derivation LK = L(K) 2 Derk (M) by LK := [iK; d].

Then the mapping L : (M; TM) ! Der (M) is injective, since LKf =

iKdf = df _ K for f 2 C1(M;R).

Theorem. For any graded derivation D 2 Derk (M) there are unique K 2

k(M; TM) and L 2 k+1(M; TM) 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 2 X(M) be vector _elds. Then f 7! (Df)(X1; : : : ;Xk) is a

derivation C1(M;R) ! C1(M;R), so by 3.3 there is a unique vector _eld

K(X1; : : : ;Xk) 2 X(M) such that

(Df)(X1; : : : ;Xk) = K(X1; : : : ;Xk)f = df(K(X1; : : : ;Xk)):

Clearly K(X1; : : : ;Xk) is C1(M;R)-linear in each Xi and alternating, so K is

tensorial by 7.3, K 2 k(M; TM).

The de_ning equation for K is Df = df _K = iKdf = LKf for f 2 C1(M;R).

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

L 2 k+1(M; TM).

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 7!

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

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

i(IdTM)! = k! for ! 2 k(M). Thus we have L(IdTM)! = i(IdTM)d! 􀀀

d i(IdTM)! = (k + 1)d! 􀀀 kd! = d!. Thus L(IdTM) = d.

8.5. Let K 2 k(M; TM) and L 2 `(M; TM). Then obviously [[LK;LL]; d] =

0, so we have

[L(K);L(L)] = L([K;L])

for a uniquely de_ned [K;L] 2 k+`(M; TM). This vector valued form [K;L] is

called the Frolicher-Nijenhuis bracket of K and L.

Theorem. The space (M; TM) =

LdimM

k=0 k(M; TM) with its usual grading

is a graded Lie algebra for the Frolicher-Nijenhuis bracket. So we have

[K;L] = 􀀀(􀀀1)k`[L;K]

[K1; [K2;K3]] = [[K1;K2];K3] + (􀀀1)k1k2 [K2; [K1;K3]]

IdTM 2 1(M; TM) is in the center, i.e. [K; IdTM] = 0 for all K.

L : ((M; TM); [ ; ]) ! Der (M) is an injective homomorphism of graded

Lie algebras. For vector _elds the Frolicher-Nijenhuis bracket coincides with

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 2 k(M; TM) and L 2 `+1(M; TM) we have

[LK; iL] = i([K;L]) 􀀀 (􀀀1)k`L(iLK), or

[iL;LK] = L(iLK) + (􀀀1)k i([L;K]):

This generalizes 7.7.2.

Proof. For f 2 C1(M;R) we have [iL;LK]f = iL iK df 􀀀 0 = iL(df _ K) =

df _ (iLK) = L(iLK)f. So [iL;LK] 􀀀 L(iLK) is an algebraic derivation.

[[iL;LK]; d] = [iL; [LK; d]] 􀀀 (􀀀1)k`[LK; [iL; d]] =

= 0 􀀀 (􀀀1)k`L([K;L]) = (􀀀1)k[i([L;K]); d]:

Since [ ; d] kills the L's and is injective on the i's, the algebraic part of [iL;LK]

is (􀀀1)k i([L;K]). _

8.7. The space Der (M) is a graded module over the graded algebra (M)

with the action (! ^ D)' = ! ^ D('), because (M) is graded commutative.

Theorem. Let the degree of ! be q, of ' be k, and of be `. Let the other

degrees be as indicated. Then we have:

[! ^ D1;D2] = ! ^ [D1;D2] 􀀀 (1) (􀀀1)(q+k1)k2D2(!) ^ D1:

(2) i(! ^ L) = ! ^ i(L)

! ^ L(3) K = L(! ^ K) + (􀀀1)q+k􀀀1i(d! ^ K):

[! ^ L1;L2]^ = ! ^ [L1;L2]^(4) 􀀀

􀀀 (􀀀1)(q+`1􀀀1)(`2􀀀1)i(L2)! ^ L1:

(5) [! ^ K1;K2] = ! ^ [K1;K2] 􀀀 (􀀀1)(q+k1)k2L(K2)! ^ K1

+ (􀀀1)q+k1d! ^ i(K1)K2:

(6) [' X; Y ] = ' ^ [X; Y ]

􀀀

iY d' ^ X 􀀀 (􀀀1)k`iXd ^ ' Y

􀀀

d(iY ' ^ ) X 􀀀 (􀀀1)k`d(iX ^ ') Y

= ' ^ [X; Y ] + ' ^ LX Y 􀀀 LY ' ^ X

+ (􀀀1)k (d' ^ iX Y + iY ' ^ d X) :

Proof. For (1), (2), (3) write out the de_nitions. For (4) compute i([!^L1;L2]^).

For (5) compute L([! ^ K1;K2]). For (6) use (5) . _

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

8. Derivations on the algebra of di_erential forms and the Frolicher-Nijenhuis bracket 71

8.8. Theorem. For K 2 k(M; TM) and ! 2 `(M) the Lie derivative of !

along K is given by the following formula, where the Xi are vector _elds on M.

(LK!)(X1; : : : ;Xk+`) =

= 1

k! `!

sign _ L(K(X_1; : : : ;X_k))(!(X_(k+1); : : : ;X_(k+`)))

+ 􀀀1

k! (`􀀀1)!

sign _ !([K(X_1; : : : ;X_k);X_(k+1)];X_(k+2); : : : )

+ (􀀀1)k􀀀1

(k􀀀1)! (`􀀀1)! 2!

sign _ !(K([X_1;X_2];X_3; : : : );X_(k+2); : : : ):

Proof. It su_ces to consider K = ' X. Then by 8.7.3 we have L(' X) =

' ^ LX 􀀀 (􀀀1)k􀀀1d' ^ iX. Now use the global formulas of section 7 to expand

this. _

8.9. Theorem. For K 2 k(M; TM) and L 2 `(M; TM) we have for the

Frolicher-Nijenhuis bracket [K;L] the following formula, where the Xi are vector

_elds on M.

[K;L](X1; : : : ;Xk+`) =

= 1

k! `!

sign _ [K(X_1; : : : ;X_k);L(X_(k+1); : : : ;X_(k+`))]

+ 􀀀1

k! (`􀀀1)!

sign _ L([K(X_1; : : : ;X_k);X_(k+1)];X_(k+2); : : : )

+ (􀀀1)k`

(k􀀀1)! `!

sign _ K([L(X_1; : : : ;X_`);X_(`+1)];X_(`+2); : : : )

+ (􀀀1)k􀀀1

(k􀀀1)! (`􀀀1)! 2!

sign _ L(K([X_1;X_2];X_3; : : : );X_(k+2); : : : )

+ (􀀀1)(k􀀀1)`

(k��1)! (`􀀀1)! 2!

sign _ K(L([X_1;X_2];X_3; : : : );X_(`+2); : : : ):

Proof. It su_ces to consider K = 'X and L = Y , then for ['X; Y ]

we may use 8.7.6 and evaluate that at (X1; : : : ;Xk+`). After some combinatorial

computation we get the right hand side of the above formula for K = 'X and

L = Y . _

There are more illuminating ways to prove this formula, see [Michor, 87].

8.10. Local formulas. In a local chart (U; u) on the manifold M we put

K j U =

Ki_ d_ @i, L j U =

_d_ @j , and ! j U =

!d, where

_ = (1 _ _1 < _2 < _ _ _ < _k _ dimM) is a form index, d_ = du_1 ^ : : : ^ du_k ,

@i = @

@ui and so on.

Plugging Xj = @ij into the global formulas 8.2, 8.8, and 8.9, we get the

following local formulas:

iK! j U =

Ki_1:::_k!i_k+1:::_k+`􀀀1 d_

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

72 Chapter II. Di_erential forms

[K;L]^ j U =

Ki_1:::_k Lj

i_k+1:::_k+`

􀀀 (􀀀1)(k􀀀1)(`􀀀1)Li

_1:::_` Kj

i_`+1:::_k+`

d_ @j

LK! j U =

Ki_1:::_k @i!_k+1:::_k+`

+ (􀀀1)k(@_1Ki_2:::_k+1) !i_k+2:::_k+`

[K;L] j U =

Ki_1:::_k @iLj

_k+1:::_k+`

􀀀 (􀀀1)k`Li

_1:::_` @iKj_`+1:::_k+`

􀀀 kKj

_1:::_k􀀀1i @_kLi

_k+1:::_k+`

+ (􀀀1)k``Lj

_1:::_`􀀀1i @_`Ki_`+1:::_k+`

d_ @j

8.11. Theorem. For Ki 2 ki (M; TM) and Li 2 ki+1(M; TM) we have

(1) [LK1 + iL1 ;LK2 + iL2 ] =

= L

􀀀

[K1;K2] + iL1K2 􀀀 (􀀀1)k1k2 iL2K1

+ i

􀀀

[L1;L2]^ + [K1;L2] 􀀀 (􀀀1)k1k2 [K2;L1]

Each summand of this formula looks like a semidirect product of graded Lie

algebras, but the mappings

i : (M; TM) ! End((M; TM); [ ; ])

ad : (M; TM) ! End((M; TM); [ ; ]^)

do not take values in the subspaces of graded derivations. We have instead for

K 2 k(M; TM) and L 2 `+1(M; TM) the following relations:

(2) iL[K1;K2] = [iLK1;K2] + (􀀀1)k1`[K1; iLK2]

􀀀

(􀀀1)k1`i([K1;L])K2 􀀀 (􀀀1)(k1+`)k2 i([K2;L])K1

[K; [L1;L2]^] = [[K;L1];L2]^ + (􀀀1)kk1 [L1; [K;L2]]^(3) 􀀀

􀀀

(􀀀1)kk1 [i(L1)K;L2] 􀀀 (􀀀1)(k+k1)k2 [i(L2)K;L1]

The algebraic meaning of the relations of this theorem and its consequences in

group theory have been investigated in [Michor, 89]. The corresponding product

of groups is well known to algebraists under the name `Zappa-Szep'-product.

Proof. Equation (1) is an immediate consequence of 8.6. Equations (2) and (3)

follow from (1) by writing out the graded Jacobi identity, or as follows: Consider

L(iL[K1;K2]) and use 8.6 repeatedly to obtain L of the right hand side of (2).

Then consider i([K; [L1;L2]^]) and use again 8.6 several times to obtain i of the

right hand side of (3). _

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

8. Derivations on the algebra of di_erential forms and the Frolicher-Nijenhuis bracket 73

8.12. Corollary (of 8.9). For K, L 2 1(M; TM) we have

[K;L](X; Y ) = [KX; LY ] 􀀀 [KY;LX]

􀀀 L([KX; Y ] 􀀀 [KY;X])

􀀀 K([LX; Y ] 􀀀 [LY;X])

+ (LK + KL)[X; Y ]:

8.13. Curvature. Let P 2 1(M; TM) satisfy P _ P = P, i.e. P is a projection

in each _ber of TM. This is the most general case of a (_rst order)

connection. We may call ker P the horizontal space and im P the vertical space

of the connection. If P is of constant rank, then both are sub vector bundles of

TM. If im P is some primarily _xed sub vector bundle or (tangent bundle of) a

foliation, P can be called a connection for it. Special cases of this will be treated

extensively later on. The following result is immediate from 8.12.

Lemma. We have

[P; P] = 2R + 2 _ R;

where R, _R 2 2(M; TM) are given by R(X; Y ) = P[(Id􀀀P)X; (Id􀀀P)Y ] and

(X; Y ) = (Id􀀀P)[PX; PY ].

If P has constant rank, then R is the obstruction against integrability of the

horizontal bundle ker P, and _R is the obstruction against integrability of the

vertical bundle im P. Thus we call R the curvature and _R the cocurvature of the

connection P. We will see later, that for a principal _ber bundle R is just the

negative of the usual curvature.

8.14. Lemma (Bianchi identity). If P 2 1(M; TM) is a connection (_ber

projection) with curvature R and cocurvature _R, then we have

[P;R + _R] = 0

[R; P] = iR_R + i_R R:

Proof. We have [P; P] = 2R + 2_R by 8.13 and [P; [P; P]] = 0 by the graded

Jacobi identity. So the _rst formula follows. We have 2R = P _ [P; P] = i[P;P ]P.

By 8.11.2 we get i[P;P ][P; P] = 2[i[P;P ]P; P] 􀀀 0 = 4[R; P]. Therefore [R; P] =

4 i[P;P ][P; P] = i(R + _R)(R + _R) = iR_R + i_R R since R has vertical values and

kills vertical vectors, so iRR = 0; likewise for _R . _

8.15. f-relatedness of the Frolicher-Nijenhuis bracket. Let f : M !

N be a smooth mapping between manifolds. Two vector valued forms K 2

k(M; TM) and K0 2 k(N; TN) are called f-related or f-dependent, if for all

Xi 2 TxM we have

(1) K0

f(x)(Txf _ X1; : : : ; Txf _ Xk) = Txf _ Kx(X1; : : : ;Xk):

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

74 Chapter II. Di_erential forms

Theorem.

(2) If K and K0 as above are f-related then iK _ f_ = f_ _ iK0 : (N) !

(M).

(3) If iK _ f_ j B1(N) = f_ _ iK0 j B1(N), then K and K0 are f-related,

where B1 denotes the space of exact 1-forms.

(4) If Kj and K0

j are f-related for j = 1; 2, then iK1K2 and iK01

2 are

f-related, and also [K1;K2]^ and [K0

1;K0

2]^ are f-related.

(5) If K and K0 are f-related then LK _ f_ = f_ _ LK0 : (N) ! (M).

(6) If LK _ f_ j 0(N) = f_ _ LK0 j 0(N), then K and K0 are f-related.

(7) If Kj and K0

j are f-related for j = 1; 2, then their Frolicher-Nijenhuis

brackets [K1;K2] and [K0

1;K0

2] are also f-related.

Proof. (2) By 8.2 we have for ! 2 q(N) and Xi 2 TxM:

(iKf_!)x(X1; : : : ;Xq+k􀀀1) =

= 1

k! (q􀀀1)!

sign _ (f_!)x(Kx(X_1; : : : ;X_k);X_(k+1); : : : )

= 1

k! (q􀀀1)!

sign _ !f(x)(Txf _ Kx(X_1; : : : ); Txf _ X_(k+1); : : : )

= 1

k! (q􀀀1)!

sign _ !f(x)(K0

f(x)(Txf _ X_1; : : : ); Txf _ X_(k+1); : : : )

= (f_iK0!)x(X1; : : : ;Xq+k􀀀1)

(3) follows from this computation, since the df, f 2 C1(M;R) separate

points.

(4) follows from the same computation for K2 instead of !, the result for the

bracket then follows 8.2.2.

(5) The algebra homomorphism f_ intertwines the operators iK and iK0 by

(2), and f_ commutes with the exterior derivative d. Thus f_ intertwines the

commutators [iK; d] = LK and [iK0 ; d] = LK0 .

(6) For g 2 0(N) we have LK f_ g = iK d f_ g = iK f_ dg and f_ LK0 g =

f_ iK0 dg. By (3) the result follows.

(7) The algebra homomorphism f_ intertwines LKj and LK0

, thus also their

graded commutators, which are equal to L([K1;K2]) and L([K0

1;K0

2]), respectively.

Then use (6). _

8.16. Let f : M ! N be a local di_eomorphism. Then we can consider the

pullback operator f_ : (N; TN) ! (M; TM), given by

(1) (f_K)x(X1; : : : ;Xk) = (Txf)􀀀1Kf(x)(Txf _ X1; : : : ; Txf _ Xk):

Note that this is a special case of the pullback operator for sections of natural

vector bundles in 6.15. Clearly K and f_K are then f-related.

Theorem. In this situation we have:

(2) f_ [K;L] = [f_K; f_L].

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

Remarks 75

(3) f_ iKL = if_Kf_L.

(4) f_ [K;L]^ = [f_K; f_L]^.

(5) For a vector _eld X 2 X(M) and K 2 (M; TM) by 6.15 the Lie

derivative LXK = @

0 (FlXt

)_K is de_ned. Then we have LXK =

[X;K], the Frolicher-Nijenhuis-bracket.

This is sometimes expressed by saying that the Frolicher-Nijenhuis bracket,

[ ; ]^, etc. are natural bilinear concomitants.

Proof. (2) { (4) are obvious from 8.15. They also follow directly from the geometrical

constructions of the operators in question. (5) Obviously LX is R-linear,

so it su_ces to check this formula for K = Y , 2 (M) and Y 2 X(M).

But then

LX( Y ) = LX Y + LXY by 6.16

= LX Y + [X; Y ]

= [X; Y ] by 8.7.6: _

8.17. Remark. At last we mention the best known application of the Frolicher-

Nijenhuis bracket, which also led to its discovery. A vector valued 1-form J 2

1(M; TM) with J _ J = 􀀀Id is called a almost complex structure; if it exists,

dimM is even and J can be viewed as a _ber multiplication with

􀀀1 on TM.

By 8.12 we have

[J; J](X; Y ) = 2([JX; JY ] 􀀀 [X; Y ] 􀀀 J[X; JY ] 􀀀 J[JX; Y ]):

The vector valued form 1

2 [J; J] is also called the Nijenhuis tensor of J, because

we have the following result:

A manifold M with an almost complex structure J is a complex

manifold (i.e., there exists an atlas for M with holomorphic chartchange

mappings) if and only if [J; J] = 0. See [Newlander-Nirenberg,

57].

Remarks

The material on the Lie derivative on natural vector bundles 6.14{6.20 appears

here for the _rst time. Most of section 8 is due to [Frolicher-Nijenhuis, 56], the

formula in 8.9 was proved by [Mangiarotti-Modugno, 84] and [Michor, 87]. The

Bianchi identity 8.14 is from [Michor, 89a].

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

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