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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

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 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.

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