±

= ’Te ( s )ω± (ξx ) by (2)

= ’ζω± (ξx ) (s).

37.25

396 Chapter VIII. In¬nite dimensional di¬erential geometry 37.26

So the condition is necessary.

For the converse let us suppose that a connection Ψ on P [S] is given such that the

Christo¬el forms “± with respect to a ¬ber bundle atlas of the G-structure have

Ψ

values in Xfund (S). Then unique g-valued forms ω± ∈ „¦1 (U± , g) are given by the

equation

“± (ξx ) = ζ(ω± (ξx )),

Ψ

since the action is in¬nitesimally e¬ective. From the transition formulas (37.5)

for the “± follow the transition formulas (37.22.3) for the ω ± , so that they they

Ψ

combine to a unique principal connection on P , which by the ¬rst part of the proof

induces the given connection Ψ on P [S].

37.26. Inducing principal connections on associated vector bundles. Let

(p : P ’ M, G) be a principal ¬ber bundle, and let ρ : G ’ GL(W ) be a represen-

tation of the structure group G on a convenient vector space W . See the beginning

of section (49) for a discussion of such representations. We consider the associated

vector bundle (p : E := P [W, ρ] ’ M, W ), see (37.12).

Recall from (29.9) that the tangent bundle T E = T P —T G T W has two vector

bundle structures, with the projections

πE : T E = T P —T G T W ’ P —G W = E,

T p —¦ pr1 : T E = T P —T G T W ’ T M,

respectively. Recall the vertical bundle V E = ker(T p) which is a vector subbundle

of πE : T E ’ E, and recall the vertical lift mapping vlE : E —M E ’ V E, which

is an isomorphism, (pr1 “πE )“¬berwise linear and also (p“T p)“¬berwise linear.

Now let ¦ = ζ —¦ ω ∈ „¦1 (P ; T P ) be a principal connection on P . We consider

¯

the induced connection ¦ ∈ „¦1 (E; T E) on the associated bundle E from (37.24).

A glance at the following diagram shows that the induced connection is linear in

both vector bundle structures. This property is expressed by calling it a linear

connection, see (37.27), on the associated vector bundle.

w TP — TW

&& ¦ — Id

xx

TP — TW TP — W — W

&&&

xxπ

π(

x

P —W

u

q

Tq Tq

x

x

P —G W = E

π

u xu

πE

E

xw T P —

TP — TW &

&& TW TE

x

TG TG

¯

¦

&( xxT p —¦ pr

&

x

T p —¦ pr 1 1

TM

37.26

37.28 37. Bundles and connections 397

¯

Now we de¬ne the connector K of the linear connection ¦ by

¯

K := pr2 —¦ (vlE )’1 —¦ ¦ : T E ’ V E ’ E —M E ’ E.

Lemma. The connector K : T E ’ E is a vector bundle homomorphism for both

vector bundle structures on T E and satis¬es K —¦ vlE = pr2 : E —M E ’ T E ’ E.

So K is πE “p“¬berwise linear and T p“p“¬berwise linear.

Proof. This follows from the ¬berwise linearity of the parts of K and from its

de¬nition.

37.27. Linear connections. If p : E ’ M is a vector bundle, a connection

Ψ ∈ „¦1 (E; T E) such that Ψ : T E ’ V E ’ T E is additionally T p“T p“¬berwise

linear is called a linear connection.

Equivalently, a linear connection may be speci¬ed by a connector K : T E ’ E

with the three properties of lemma (37.26). For then HE := {ξu : K(ξu ) = 0p(u) }

is a complement to V E in T E which is T p“¬berwise linearly chosen.

37.28. Covariant derivative on vector bundles. Let p : E ’ M be a vector

bundle with a linear connection, given by a connector K : T E ’ E with the

properties in lemma (37.26).

For any manifold N , smooth mapping s : N ’ E, and kinematic vector ¬eld

X ∈ X(N ) we de¬ne the covariant derivative of s along X by

:= K —¦ T s —¦ X : N ’ T N ’ T E ’ E.

(1) Xs

∞

If f : N ’ M is a ¬xed smooth mapping, let us denote by Cf (N, E) the vector

space of all smooth mappings s : N ’ E with p —¦ s = f ” they are called sections

of E along f . It follows from the universal property of the pullback that the vector

space Cf (N, E) is canonically linearly isomorphic to the space C ∞ (N ← f — E) of

∞

sections of the pullback bundle. Then the covariant derivative may be viewed as a

bilinear mapping

∞ ∞

: X(N ) — Cf (N, E) ’ Cf (N, E).

(2)

In particular, for f = IdM we have

: X(M ) — C ∞ (M ← E) ’ C ∞ (M ← E).

Lemma. This covariant derivative has the following properties:

is C ∞ (N, R)-linear in X ∈ X(N ). Moreover, for a tangent vector

(3) Xs

∞

Xx ∈ Tx N the mapping Xx : Cf (N, E) ’ Ef (x) makes sense, and we

have ( X s)(x) = X(x) s. Thus, s ∈ „¦1 (N ; f — E).

∞

(4) X s is R-linear in s ∈ Cf (N, E).

(5) X (h.s) = dh(X).s + h. X s for h ∈ C ∞ (N, R), the derivation property of

X.

(6) For any manifold Q, smooth mapping g : Q ’ N , and Yy ∈ Ty Q we have

Yy (s —¦ g). If Y ∈ X(Q) and X ∈ X(N ) are g-related, then we

T g.Yy s =

have Y (s —¦ g) = ( X s) —¦ g.

37.28

398 Chapter VIII. In¬nite dimensional di¬erential geometry 37.29

Proof. All these properties follow easily from de¬nition (1).

For vector ¬elds X, Y ∈ X(M ) and a section s ∈ C ∞ (M ← E) an easy computation

shows that

RE (X, Y )s := s’ ’

Xs [X,Y ] s

X Y Y

]’

= ([ X, [X,Y ] )s

Y

is C ∞ (M, R)-linear in X, Y , and s. By the method of (14.3), it follows that RE is a

2-form on M with values in the vector bundle L(E, E), i.e. RE ∈ „¦2 (M ; L(E, E)).

It is called the curvature of the covariant derivative.

∞

For f : N ’ M , vector ¬elds X, Y ∈ X(N ), and a section s ∈ Cf (N, E) along f

one can prove that

= (f — RE )(X, Y )s := RE (T f.X, T f.Y )s.

s’ ’

Xs [X,Y ] s

X Y Y

37.29. Covariant exterior derivative. Let p : E ’ M be a vector bundle with

a linear connection, given by a connector K : T E ’ E.

For a smooth mapping f : N ’ M let „¦(N ; f — E) be the vector space of all forms

on N with values in the vector bundle f — E. We can also view them as forms on N

with values along f in E, but we do not introduce an extra notation for this.

As in (32.1), (33.2), and (35.1) we have to assume the

Convention. We consider each derivation and homomorphism to be a sheaf mor-

phism (compare (32.1) and the de¬nition of modular 1-forms in (33.2)), or we

assume that all manifolds in question are smoothly regular.

The graded space „¦(N ; f — E) is a graded „¦(N )-module via

(• § ¦)(X1 , . . . , Xp+q ) =

1

= sign(σ) •(Xσ1 , . . . , Xσp )¦(Xσ(p+1) , . . . , Xσ(p+q) ).

p! q!

σ

Any A ∈ „¦p (N ; f — L(E, E)) de¬nes a graded module homomorphism

µ(A) : „¦(N ; f — E) ’ „¦(N ; f — E),

(1)

(µ(A)¦)(X1 , . . . , Xp+q ) =

1

= sign(σ) A(Xσ1 , . . . , Xσp )(¦(Xσ(p+1) , . . . , Xσ(p+q) )),

p! q!

σ

deg A. deg •

µ(A)(• § ¦) = (’1) • § µ(A)(¦).

But in general not all graded module homomorphisms are of this form, recall the

distinction between modular di¬erential forms and di¬erential forms in (33.2). This

is only true if the modeling spaces of N have the bornological approximation prop-

erty; the proof is as in (33.5).

37.29

37.29 37. Bundles and connections 399

The covariant exterior derivative is given by

d : „¦p (N ; f — E) ’ „¦p+1 (N ; f — E)

(2)

p

(’1)i

(d ¦)(X0 , . . . , Xp ) = Xi ¦(X0 , . . . , Xi , . . . , Xp )+

i=0

(’1)i+j ¦([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xp ),

+

0¤i<j¤p

where the Xi are vector ¬elds on N . It will be shown below that it is indeed well

de¬ned, i.e. that d ¦ ∈ „¦p+1 (N ; f — E). Now we only see that it is a modular

di¬erential form.

The covariant Lie derivative along a vector ¬eld X ∈ X(N ) is given by

LX : „¦p (N ; f — E) ’ „¦p (N ; f — E)

(3)

(LX ¦)(X1 , . . . , Xp ) = X (¦(X1 , . . . , Xp ))’

’ ¦(X1 , . . . , [X, Xi ], . . . , Xp ).

i

Again we will show below that it is well de¬ned. Finally we recall the insertion

operator

iX : „¦p (N ; f — E) ’ „¦p’1 (N ; f — E)

(4)

(LX ¦)(X1 , . . . , Xp’1 ) = ¦(X, X1 , . . . , Xp’1 )

Theorem. The covariant exterior derivative d , and the covariant Lie derivative

are well de¬ned and have the following properties.

For s ∈ C ∞ (N ← f — E) = „¦0 (N ; f — E) we have (d s)(X) = X s.

(5)

d (• § ¦) = d• § ¦ + (’1)deg • • § d ¦.

(6)

For smooth g : Q ’ N and ¦ ∈ „¦(N ; f — E) we have d (g — ¦) = g — (d ¦).

(7)

d d ¦ = µ(f — RE )¦.

(8)

iX (• § ¦) = iX • § ¦ + (’1)deg • • § iX ¦.

(9)

LX (• § ¦) = LX • § ¦ + • § LX ¦.

(10)

[LX , iY ] = LX —¦ iY ’ iY —¦ LX = i[X,Y ] .

(11)

[iX , d ] = iX —¦ d + d —¦ iX = LX .

(12)

Proof. By the formula above d ¦ is a priori de¬ned as a modular di¬erential form

and we have to show that it really lies in „¦(M ; f — E). For that let s— ∈ C ∞ (N ←

f — E ) be a local smooth section on U ⊆ N along f |U : U ’ M of the dual vector

bundle E ’ M . Then ¦, s— ∈ „¦k (N ), and for the canonical covariant derivative

on the dual bundle (write down its connector!) we have

d ¦, s— = d ¦, s— + (’1)k ¦, s—

E

§,

which shows that d ¦ ∈ „¦k (N, f — E) since d respects „¦— (N ) by (33.12).

37.29

400 Chapter VIII. In¬nite dimensional di¬erential geometry 37.29

(5) is just (33.11). (7) follows from (37.28.6).

(11) Take the di¬erence of the following two expressions:

(LX iY ¦)(Z1 , . . . , Zk ) = X ((iY ¦)(Z1 , . . . , Zk ))’

’ (iY ¦)(Z1 , . . . , [X, Zi ], . . . , Zk )

i

’

= X (¦(Y, Z1 , . . . , Zk )) ¦(Y, Z1 , . . . , [X, Zi ], . . . , Zk )

i

(iY LX ¦)(Z1 , . . . , Zk ) = LX ¦(Y, Z1 , . . . , Zk )

’ ¦([X, Y ], Z1 , . . . , Zk )’

= X (¦(Y, Z1 , . . . , Zk ))

’ ¦(Y, Z1 , . . . , [X, Zi ], . . . , Zk ).

i

(10) Let • be of degree p and ¦ of degree q. We prove the result by induction on

p + q. Suppose that (5) is true for p + q < k. Then for X we have by (9), by (11),

and by induction

(iY LX )(• § ¦) = (LX iY )(• § ¦) ’ i[X,Y ] (• § ¦)

= LX (iY • § ¦ + (’1)p • § iY ¦) ’ i[X,Y ] • § ¦ ’ (’1)p • § i[X,Y ] ¦

= LX iY • § ¦ + iY • § LX ¦ + (’1)p LX • § iY ¦+

+ (’1)p • § LX iY ¦ ’ i[X,Y ] • § ¦ ’ (’1)p • § i[X,Y ] ¦

iY (LX • § ¦ + • § LX ¦) = iY LX • § ¦ + (’1)p LX • § iY ¦+

+ iY • § LX ¦ + (’1)p • § iY LX ¦.

Using again (11), we get the result since the iY for all local vector ¬elds Y to-

gether act point separating on each space of di¬erential forms, in both cases of the

convention.

(12) We write out all relevant expressions.

(LX0 ¦)(X1 , . . . , Xk ) = X0 (¦(X1 , . . . , Xk ))+

k

(’1)0+j ¦([X0 , Xj ], X1 , . . . , Xj , . . . , Xk )

+

j=1

(iX0 d ¦)(X1 , . . . , Xk ) = d ¦(X0 , . . . , Xk )

k

(’1)i

= Xi (¦(X0 , . . . , Xi , . . . , Xk )) +

i=0

(’1)i+j ¦([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xk ).

+

0¤i<j

k

(’1)i’1

(d iX0 ¦)(X1 , . . . , Xk ) = Xi ((iX0 ¦)(X1 , . . . , Xi , . . . , Xk )) +

i=1

(’1)i+j’2 (iX0 ¦)([Xi , Xj ], X1 , . . . , Xi , . . . , Xj , . . . , Xk )

+

1¤i<j

37.29

37.30 37. Bundles and connections 401

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.

(6) We prove the result again by induction on p + q. Suppose that (6) is true for

p + q < k. Then for each local vector ¬eld X we have by (10), (9), (12), and by

induction

iX d (• § ¦) = LX (• § ¦) ’ d iX (• § ¦)

= LX • § ¦ + • § LX ¦ ’ d (iX • § ¦ + (’1)p • § iX ¦)

= iX d• § ¦ + d iX • § ¦ + • § iX d¦ + • § d iX ¦ ’ d iX • § ¦

’ (’1)p’1 iX • § d ¦ ’ (’1)p d • § iX ¦ ’ • § d iX ¦

= iX (d • § ¦ + (’1)p • § d ¦).

Since X is arbitrary, the result follows.

(8) follows from a direct computation. The usual fast proofs are not conclusive

in in¬nite dimensions. The computation is similar to the one for the proof of

(33.18.4), and only the de¬nitions (2) of d and (37.28) of RE , and the Jacobi

identity enter.

37.30. Let (p : P ’ M, G) be a principal ¬ber bundle, and let ρ : G ’ GL(W )

be a representation of the structure group G on a convenient vector space W , as in

(49.1).

Theorem. There is a canonical isomorphism from the space of P [W, ρ]-valued dif-

ferential forms on M onto the space of horizontal G-equivariant W -valued di¬er-

ential forms on P :

q : „¦(M ; P [W, ρ]) ’ „¦hor (P, W )G := {• ∈ „¦(P, W ) : iX • = 0

for all X ∈ V P, (rg )— • = ρ(g ’1 ) —¦ • for all g ∈ G},

In particular, for W = R with trivial representation we see that

p— : „¦(M ) ’ „¦hor (P )G = {• ∈ „¦hor (P ) : (rg )— • = •}

also is an isomorphism. We have q (• § ¦) = p— • § q ¦ for • ∈ „¦(M ) and

¦ ∈ „¦(M ; P [W ]).

The isomorphism

q : „¦0 (M ; P [W ]) = C ∞ (M ← P [W ]) ’ „¦0 (P, W )G = C ∞ (P, W )G

hor

is a special case of the one from (37.16).

37.30

402 Chapter VIII. In¬nite dimensional di¬erential geometry 37.31

Proof. Let • ∈ „¦k (P, W )G , X1 , . . . , Xk ∈ Tu P , and X1 , . . . , Xk ∈ Tu P such

hor

that Tu p.Xi = Tu p.Xi for each i. Then there is a g ∈ G such that ug = u :

q(u, •u (X1 , . . . , Xk )) = q(ug, ρ(g ’1 )•u (X1 , . . . , Xk ))

= q(u , ((rg )— •)u (X1 , . . . , Xk ))

= q(u , •ug (Tu (rg ).X1 , . . . , Tu (rg ).Xk ))

= q(u , •u (X1 , . . . , Xk )), since Tu (rg )Xi ’ Xi ∈ Vu P.

Thus, a vector bundle valued form ¦ ∈ „¦k (M ; P [W ]) is uniquely determined by

¦p(u) (Tu p.X1 , . . . , Tu p.Xk ) := q(u, •u (X1 , . . . , Xk )).