d

t

X—

d

LX s := dt |0 (Flt ) s

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

of s along X.

Lemma. In this situation we have

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

t r t+r

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

X— X—

d

t

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

X— X—

t

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

t

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58 Chapter II. Di¬erential forms

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

X— X— X— X—

d d

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

dt (Flt ) s =

X— X— X—

d d

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

dt ((Flt ) s)(x) =

X X X X

d

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

=

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

d

= ’t ’r r t

((FlX )— LX s)(x),

= t

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

’t t

(3) follows from (2).

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

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

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

the pointwise tensor product.

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

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

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

Proof. Using the bilinearity of the tensor product we have

X—

d

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

X—

— (FlX )— s2 )

d

dt |0 ((Flt ) s1

= t

X— X—

d d

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

=

= LX s1 — s2 + s1 — LX s2 .

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

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

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

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

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

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

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

LX (•M s) = •M (LX s).

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

X— X

FlX )(x)

d d

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

= t

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

d

—¦

= ’t t

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 .

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60 Chapter II. Di¬erential forms

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

F F F F

(FlY —¦ FlX —¦ FlY —¦ FlX ),

‚

0= ’t ’t t t

‚t 0

YF XF YF F

1 ‚2

—¦ FlX )

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

2 ‚t

2 Y X Y X

1‚

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

=

[X,Y ]

‚

) = [X, Y ]F .

= F (Flt

‚t 0

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

result.

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

X(M ) we have

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

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

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

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

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

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

t 0

j j ’1

—¦ (•t ) —¦ •j —¦ •i .

i ’1

‚ i i j i

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

∞

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

each formal bracket expression P of lenght k we have

|0 P (•1 , . . . , •k )— s

‚

for 1 ¤ < k,

0= t t

‚t

1 ‚k k—

∈ C ∞ (F (M )).

1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

vector bundle.

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

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62 Chapter II. Di¬erential forms

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

‚

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

‚

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

‚

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

i (g.X )(g. ‚ui ) (x) =

‚

(g.X i )(x).¦(g. ‚ui )(x) = 0.

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

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

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

‚

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

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

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

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

0

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

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

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

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

for each permutation σ ∈ Sk .

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

dim M

„¦k (M )

„¦(M ) :=

k=0

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

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

(• § ψ)(X1 , . . . , Xk+ ) =

1

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

= k! !

σ∈Sk+

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

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

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

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

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

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

(1)

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

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

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

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

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

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

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

di¬eomorphisms, for each m separately.

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

7. Di¬erential forms 63

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

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

k-form • along X is de¬ned by

X—

d

LX • = dt |0 (Flt ) •.

Lemma. The Lie derivative has the following properties.

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

(2) For Yi ∈ X(M ) we have

k

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

i=1

(3) [LX , LY ]• = L[X,Y ] •.

k

T — M ’ Λk T — M , given by

Proof. The mapping Alt :

1

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

k!

σ