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

(M ; T M ) we have

[LK , iL ] = i([K, L]) ’ (’1)k L(iL K), or

[iL , LK ] = L(iL K) + (’1)k i([L, K]).

This generalizes 7.7.2.

Proof. For f ∈ C ∞ (M, R) we have [iL , LK ]f = iL iK df ’ 0 = iL (df —¦ K) =

df —¦ (iL K) = L(iL K)f . So [iL , LK ] ’ L(iL K) is an algebraic derivation.