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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


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.
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




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
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 .

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
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.

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 ’ 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

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


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