lemma (37.19) and thus is a principal connection on P |U± . Now let (f± ) be a

smooth partition of unity on M which is subordinated to the open cover (U± ), and

let ω := ± (f± —¦ p)γ± . Since both requirements of lemma (37.19) are invariant

under convex linear combinations, ω is a principal connection on P .

37.22. Local descriptions of principal connections. We consider a principal

¬ber bundle (p : P ’ M, G) with some principal ¬ber bundle atlas (U± , •± :

P |U± ’ U± — G) and corresponding cocycle (•±β : U±β ’ G) of transition

functions. We consider the sections s± ∈ C ∞ (U± ← P |U± ) which are given by

•± (s± (x)) = (x, e) and satisfy s± .•±β = sβ , since we have in turn:

•± (sβ (x)) = •± •’1 (x, e) = (x, •±β (x))

β

sβ (x) = •’1 (x, e•±β (x)) = •’1 (x, e)•±β (x) = s± (x)•±β (x).

± ±

Let ¦ = ζ —¦ ω ∈ „¦1 (P ; V P ) be a principal connection with connection form ω ∈

„¦1 (P, g). We may associate the following local data to the connection:

(1) ω± := s± — ω ∈ „¦1 (U± , g), the physicists™ version of the connection.

(2) The Christo¬el forms “± ∈ „¦1 (U± ; X(G)) from (37.5), which are given by

(0x , “± (ξx , g)) = ’T (•± ).¦.T (•± )’1 (ξx , 0g ).

Lemma. These local data have the following properties and are related by the fol-

lowing formulas.

(3) The forms ω± ∈ „¦1 (U± , g) satisfy the transition formulas

ω± = Ad(•’1 )ωβ + (•β± )— κl ,

β±

37.22

390 Chapter VIII. In¬nite dimensional di¬erential geometry 37.22

where κl ∈ „¦1 (G, g) is the left Maurer-Cartan form from (36.10). Any set

of such forms with this transition behavior determines a unique principal

connection.

(4) The local expression of ω is given by

(•’1 )— ω(ξx , T µg .X) = (•’1 )— ω(ξx , 0g ) + X = Ad(g ’1 )ω± (ξx ) + X.

± ±

(5) The Christo¬el form “± and ω± are related by

“± (ξx , g) = ’Te (µg ).Ad(g ’1 )ω± (ξx ) = ’Te (µg )ω± (ξx ),

thus the Christo¬el form is right invariant: “± (ξx ) = ’Rω± (ξx ) ∈ X(G).

(6) The local expression of ¦ is given by

(((•± )’1 )— ¦)(ξx , ·g ) = ’“± (ξx , g) + ·g = Te (µg ).ω± (ξx ) + ·g

= Rω± (ξx ) (g) + ·g

for ξx ∈ Tx U± and ·g ∈ Tg G.

(7) The local expression of the curvature R is given by

((•± )’1 )— R = ’R 1

dω± + 2 [ω± ,ω± ]§

g

so that R and „¦ are indeed ˜tensorial™ 2-forms.

Proof. We start with (4).

(•’1 )— ω(ξx , 0g ) = (•’1 )— ω(ξx , Te (µg )0e ) = (ω —¦ T (•± )’1 —¦ T (IdU± —µg ))(ξx , 0e ) =

± ±

= (ω —¦ T (rg —¦ •’1 ))(ξx , 0e ) = Ad(g ’1 )ω(T (•’1 )(ξx , 0e ))

± ±

= Ad(g ’1 )(s± — ω)(ξx ) = Ad(g ’1 )ω± (ξx ).

From this we get

(•’1 )— ω(ξx , T µg .X) = (•’1 )— ω(ξx , 0g ) + (•’1 )— ω(0x , T µg .X)

± ± ±

= Ad(g ’1 )ω± (ξx ) + ω(T (•± )’1 (0x , T µg .X))

= Ad(g ’1 )ω± (ξx ) + ω(ζX (•’1 (x, g)))

±

= Ad(g ’1 )ω± (ξx ) + X.

(5) From the de¬nition of the Christo¬el forms we have

(0x , “± (ξx , g)) = ’T (•± ).¦.T (•± )’1 (ξx , 0g )

= ’T (•± ).Te (r•’1 (x,g) )ω.T (•± )’1 (ξx , 0g )

±

= ’Te (•± —¦ r•’1 (x,g) )ω.T (•± )’1 (ξx , 0g )

±

= ’(0x , Te (µg )ω.T (•± )’1 (ξx , 0g ))

= ’(0x , Te (µg )Ad(g ’1 )ω± (ξx )) = ’(0x , Te (µg )ω± (ξx )).

37.22

37.23 37. Bundles and connections 391

(3) Via (5) the transition formulas for the ω± are easily seen to be equivalent to the

transition formulas for the Christo¬el forms in lemma (37.5). A direct proof goes

as follows: We have s± (x) = sβ (x)•β± (x) = r(sβ (x), •β± (x)) and thus

ω± (ξx ) = ω(Tx (s± ).ξx )

= (ω —¦ T(sβ (x),•β± (x)) r)((Tx sβ .ξx , 0•β± (x) ) ’ (0sβ (x) , Tx •β± .ξx ))

= ω(Tsβ (x) (r•β± (x) ).Tx (sβ ).ξx ) + ω(T•β± (x) (rsβ (x) ).Tx (•β± ).ξx )

= Ad(•β± (x)’1 )ω(Tx (sβ ).ξx )

+ ω(T•β± (x) (rsβ (x) ).T (µ•β± (x) —¦ µ•β± (x)’1 )Tx (•β± ).ξx )

= Ad(•β± (x)’1 )ωβ (ξx ) + ω(Te (rsβ (x)•β± (x) ).(•β± )— κl .ξx )

= Ad(•β± (x)’1 )ωβ (ξx ) + (•β± )— κl (ξx ).

(6) This is clear from the de¬nition of the Christo¬el forms and from (5).

Second proof of (7) First note that the right trivialization or framing (κr , πG ) :

T G ’ g — G induces the isomorphism R : C ∞ (G, g) ’ X(G), given by RX (x) =

Te (µx ).X(x). For the Lie bracket we then have

[RX , RY ] = R’[X,Y ]g +dY.RX ’dX.RY ,

R’1 [RX , RY ] = ’[X, Y ]g + RX (Y ) ’ RY (X).

We write a vector ¬eld on U± — G as (ξ, RX ) where ξ : G ’ X(U± ) and X ∈

C ∞ (U± — G, g). Then the local expression of the curvature is given by

(•± ’1 )— R((ξ, RX ), (·, RY )) = (•’1 )— (R((•± )— (ξ, RX ), (•± )— (·, RY )))

±

= (•’1 )— (¦[(•± )— (ξ, RX ) ’ ¦(•± )— (ξ, RX ), . . . ])

±

= (•’1 )— (¦[(•± )— (ξ, RX ) ’ (•± )— (Rω± (ξ) + RX ), . . . ])

±

= (•’1 )— (¦(•± )— [(ξ, ’Rω± (ξ) ), (·, ’Rω± (·) )])

±

= ((•’1 )— ¦)([ξ, ·]X(U± ) ’ Rω± (ξ) (·) + Rω± (·) (ξ),

±

’ ξ(Rω± (·) ) + ·(Rω± (ξ) ) + R’[ω± (ξ),ω± (·)]+Rω± (ξ) (ω± (·))’Rω± (ξ) (ω± (·)) )

= Rω± ([ξ,·]X(U± ) ’Rω± (ξ) (·)+Rω± (·) (ξ)) ’ Rξ(ω± (·)) + R·(ω± (ξ))

+ R’[ω± (ξ),ω± (·)]+Rω± (ξ) (ω± (·))’Rω± (ξ) (ω± (·))

= ’R .

1

(dω± + 2 [ω± ,ω± ]§ )(ξ,·)

g

37.23. The covariant derivative. Let (p : P ’ M, G) be a principal ¬ber

bundle with principal connection ¦ = ζ —¦ ω. We consider the horizontal projection

χ = IdT P ’¦ : T P ’ HP , cf. (37.2), which satis¬es χ —¦ χ = χ, im χ = HP ,

ker χ = V P , and χ —¦ T (rg ) = T (rg ) —¦ χ for all g ∈ G.

If W is a convenient vector space, we consider the mapping χ— : „¦(P, W ) ’

„¦(P, W ) which is given by

(χ— •)u (X1 , . . . , Xk ) = •u (χ(X1 ), . . . , χ(Xk )).

37.23

392 Chapter VIII. In¬nite dimensional di¬erential geometry 37.23

The mapping χ— is a projection onto the subspace of horizontal di¬erential forms,

i.e. the space „¦hor (P, W ) := {ψ ∈ „¦(P, W ) : iX ψ = 0 for X ∈ V P }. The notion of

horizontal form is independent of the choice of a connection.

The projection χ— has the following properties: χ— (• § ψ) = χ— • § χ— ψ if one of

the two forms has real values, χ— —¦ χ— = χ— , χ— —¦ (rg )— = (rg )— —¦ χ— for all g ∈ G,

χ— ω = 0, and χ— —¦ L(ζX ) = L(ζX ) —¦ χ— . All but the last easily follow from the

corresponding properties of χ. The last property uses that for a smooth curve g(t)

‚ ‚

in G with g(0) = e and ‚t 0 g(t) = X by (33.19) we have LζX = ‚t 0 rg(t) .

We de¬ne the covariant exterior derivative dω : „¦k (P, W ) ’ „¦k+1 (P, W ) by the

prescription dω := χ— —¦ d.

Theorem. The covariant exterior derivative dω has the following properties.

dω (• § ψ) = dω (•) § χ— ψ + (’1)deg • χ— • § dω (ψ) if • or ψ is real valued.

(1)

L(ζX ) —¦ dω = dω —¦ L(ζX ) for each X ∈ g.

(2)

(rg )— —¦ dω = dω —¦ (rg )— for each g ∈ G.

(3)

dω —¦ p— = d —¦ p— = p— —¦ d : „¦(M, W ) ’ „¦hor (P, W ).

(4)

(5)

dω ω = „¦, the curvature form.

(6)

dω „¦ = 0, the Bianchi identity.

dω —¦ χ— ’ dω = χ— —¦ i(R), where R is the curvature.

(7)

dω —¦ dω = χ— —¦ i(R) —¦ d.

(8)

Let „¦hor (P, g)G be the algebra of all horizontal G-equivariant g-valued forms,

(9)

i.e., (rg )— ψ = Ad(g ’1 )ψ. Then for any ψ ∈ „¦hor (P, g)G we have dω ψ =

dψ + [ω, ψ]§ .

(10) The mapping ψ ’ ζψ , where ζψ (X1 , . . . , Xk )(u) = ζψ(X1 ,...,Xk )(u) (u), is

an isomorphism between „¦hor (P, g)G and the algebra „¦hor (P, V P )G of all

horizontal G-equivariant forms with values in the vertical bundle V P . Then

we have ζdω ψ = ’[¦, ζψ ].

Proof. (1) through (4) follow from the properties of χ— .

(5) We have

(dω ω)(ξ, ·) = (χ— dω)(ξ, ·) = dω(χξ, χ·)

= (χξ)ω(χ·) ’ (χ·)ω(χξ) ’ ω([χξ, χ·])

= ’ω([χξ, χ·]) and

’ζ(„¦(ξ, ·)) = R(ξ, ·) = ¦[χξ, χ·] = ζω([χξ,χ·]) .

(6) Using (37.20) we have

dω „¦ = dω (dω + 1 [ω, ω]§ )

2

= χ— ddω + 1 χ— d[ω, ω]§

2

= 1 χ— ([dω, ω]§ ’ [ω, dω]§ ) = χ— [dω, ω]§

2

= [χ— dω, χ— ω]§ = 0, since χ— ω = 0.

37.23

37.23 37. Bundles and connections 393

(7) For • ∈ „¦(P, W ) we have

(dω χ— •)(X0 , . . . , Xk ) = (dχ— •)(χ(X0 ), . . . , χ(Xk ))

(’1)i χ(Xi )((χ— •)(χ(X0 ), . . . , χ(Xi ), . . . , χ(Xk )))

=

0¤i¤k

(’1)i+j (χ— •)([χ(Xi ), χ(Xj )],

+

i<j

χ(X0 ), . . . , χ(Xi ), . . . , χ(Xj ), . . . , χ(Xk ))

(’1)i χ(Xi )(•(χ(X0 ), . . . , χ(Xi ), . . . , χ(Xk )))

=

0¤i¤k

(’1)i+j •([χ(Xi ), χ(Xj )] ’ ¦[χ(Xi ), χ(Xj )],

+

i<j

χ(X0 ), . . . , χ(Xi ), . . . , χ(Xj ), . . . , χ(Xk ))

= (d•)(χ(X0 ), . . . , χ(Xk )) + (iR •)(χ(X0 ), . . . , χ(Xk ))

= (dω + χ— iR )(•)(X0 , . . . , Xk ).

(8) dω dω = dω χ— d = (χ— iR + χ— d)d = χ— iR d holds by (7).

(9) If we insert one vertical vector ¬eld, say ζX for X ∈ g, into dω ψ, we get 0

by de¬nition. For the right hand side we use iζX ψ = 0, and that by (33.19) for

‚

a smooth curve g(t) in G with g(0) = e and ‚t 0 g(t) = X we have LζX ψ =

g(t) —

) ψ = ‚t 0 Ad(g(t)’1 )ψ = ’ad(X)ψ in the computation

‚ ‚

‚t 0 (r

iζX (dψ + [ω, ψ]§ ) = iζX dψ + diζX ψ + [iζX ω, ψ] ’ [ω, iζX ψ]

= LζX ψ + [X, ψ] = ’ad(X)ψ + [X, ψ] = 0.

Let now all vector ¬elds ξi be horizontal. Then we get

(dω ψ)(ξ0 , . . . , ξk ) = (χ— dψ)(ξ0 , . . . , ξk ) = dψ(ξ0 , . . . , ξk ),

(dψ + [ω, ψ]§ )(ξ0 , . . . , ξk ) = dψ(ξ0 , . . . , ξk ).

(10) We proceed in a similar manner. Let Ψ be in the space „¦hor (P, V P )G of all

horizontal G-equivariant forms with vertical values. Then for each X ∈ g we have

iζX Ψ = 0. Furthermore, the G-equivariance (rg )— Ψ = Ψ implies that LζX Ψ =

[ζX , Ψ] = 0 by (35.14.5). Using formula (35.9.2) we have

iζX [¦, Ψ] = [iζX ¦, Ψ] ’ [¦, iζX Ψ] + i([¦, ζX ])Ψ + i([Ψ, ζX ])¦

= [ζX , Ψ] ’ 0 + 0 + 0 = 0.

Let now all vector ¬elds ξi again be horizontal, then from the huge formula (35.5.1)

for the Fr¨licher-Nijenhuis bracket only the following terms in the fourth and ¬fth

o

line survive:

[¦, Ψ](ξ1 , . . . , ξ +1 ) =

(’1)

= sign σ ¦([Ψ(ξσ1 , . . . , ξσ ), ξσ( +1) ])

!

σ

1

+ sign σ ¦(Ψ([ξσ1 , ξσ2 ], ξσ3 , . . . , ξσ( +1) )).

( ’1)! 2!

σ

37.23

394 Chapter VIII. In¬nite dimensional di¬erential geometry 37.24

For f : P ’ g and horizontal ξ we have ¦[ξ, ζf ] = ζξ(f ) = ζdf (ξ) : It is C ∞ (P, R)-

linear in ξ; or imagine it in local coordinates. So the last expression becomes

’ζdψ(ξ0 ,...,ξk ) = ’ζ(dψ+[ω,ψ]§ )(ξ0 ,...,ξk ) ,

as required.

37.24. Inducing principal connections on associated bundles.

Let (p : P ’ M, G) be a principal bundle with principal right action r : P —G ’ P ,

and let : G — S ’ S be a left action of the structure group G on some manifold

S. Then we consider the associated bundle P [S] = P [S, ] = P —G S, constructed

in (37.12). Recall from (37.18) that its tangent and vertical bundles are given by

T (P [S, ]) = T P [T S, T ] = T P —T G T S and V (P [S, ]) = P [T S, T2 ] = P —G T S.

Let ¦ = ζ —¦ ω ∈ „¦1 (P ; T P ) be a principal connection on the principal bundle P .

¯

We construct the induced connection ¦ ∈ „¦1 (P [S]; T (P [S])) by factorizing as in

the following diagram:

w TP — TS w T (P — S)

¦ — Id =

TP — TS

u u u

Tq = q q Tq

w TP — w T (P —

¯

¦ =

T P —T G T S TS S).

TG G

Let us ¬rst check that the top mapping ¦ — Id is T G-equivariant. For g ∈ G and

X ∈ g the inverse of Te (µg )X in the Lie group T G from (38.10) is denoted by

(Te (µg )X)’1 . Furthermore, by (36.12) we have

T r(ξu , Te (µg )X) = Tu (rg )ξu + Tg (ru )(Te (µg )X) = Tu (rg )ξu + ζX (ug).

We compute

(¦ — Id)(T r(ξu , Te (µg )X), T ((Te (µg )X)’1 , ·s ))

= (¦(Tu (rg )ξu + ζX (ug)), T ((Te (µg )X)’1 , ·s ))

= (¦(Tu (rg )ξu ) + ¦(ζX (ug)), T ((Te (µg )X)’1 , ·s ))

= (Tu (rg )¦ξu + ζX (ug), T ((Te (µg )X)’1 , ·s ))

= (T r(¦(ξu ), Te (µg )X), T ((Te (µg )X)’1 , ·s )).

¯

So the mapping ¦ — Id factors to ¦ as indicated in the diagram, and we have

¯¯ ¯ ¯

¦ —¦ ¦ = ¦ from (¦ — Id) —¦ (¦ — Id) = ¦ — Id. The mapping ¦ is ¬berwise linear,

¯

since ¦ — Id and q = T q are. The image of ¦ is

q (V P — T S) = q (ker(T p : T P — T S ’ T M ))

= ker(T p : T P —T G T S ’ T M ) = V (P [S, ]).

¯

Thus, ¦ is a connection on the associated bundle P [S]. We call it the induced

connection.

37.24

37.25 37. Bundles and connections 395

From the diagram also follows that the vector valued forms ¦—Id ∈ „¦1 (P —S; T P —

¯

T S) and ¦ ∈ „¦1 (P [S]; T (P [S])) are (q : P — S ’ P [S])-related. So by (35.13) we

have for the curvatures

R¦—Id = 2 [¦ — Id, ¦ — Id] = 1 [¦, ¦] — 0 = R¦ — 0,

1

2

¯¯

R¦ = 1 [¦, ¦]

¯ 2

that they are also q-related, i.e., T q —¦ (R¦ — 0) = R¦ —¦ (T q —M T q).

¯

37.25. Recognizing induced connections. Let (p : P ’ M, G) be a principal

¬ber bundle, and let : G — S ’ S be a left action. We consider a connection

Ψ ∈ „¦1 (P [S]; T (P [S])) on the associated bundle P [S] = P [S, ]. Then the following

question arises: When is the connection Ψ induced by a principal connection on P ?

If this is the case, we say that Ψ is compatible with the G-structure on P [S]. The

answer is given in the following

Theorem. Let Ψ be a (general) connection on the associated bundle P [S]. Let us

suppose that the action is in¬nitesimally e¬ective, i.e. the fundamental vector

¬eld mapping ζ : g ’ X(S) is injective.

Then the connection Ψ is induced from a principal connection ω on P if and only

if the following condition is satis¬ed:

In some (equivalently any) ¬ber bundle atlas (U± , ψ± ) of P [S] belonging

to the G-structure of the associated bundle the Christo¬el forms “± ∈

„¦1 (U± ; X(S)) have values in the sub Lie algebra Xf und (S) of fundamental

vector ¬elds for the action .

Proof. Let (U± , •± : P |U± ’ U± —G) be a principal ¬ber bundle atlas for P . Then

by the proof of theorem (37.12) the induced ¬ber bundle atlas (U± , ψ± : P [S]|U± ’

U± — S) is given by

ψ± (x, s) = q(•’1 (x, e), s),

’1

(1) ±

(ψ± —¦ q)(•’1 (x, g), s) = (x, g.s).

(2) ±

¯

Let ¦ = ζ —¦ ω be a principal connection on P , and let ¦ be the induced connection

on the associated bundle P [S]. By (37.5), its Christo¬el symbols are given by

¯ ’1

(0x , “± (ξx , s)) = ’(T (ψ± ) —¦ ¦ —¦ T (ψ± ))(ξx , 0s )

¯

¦

¯

= ’(T (ψ± ) —¦ ¦ —¦ T q —¦ (T (•’1 ) — Id))(ξx , 0e , 0s ) by (1)

±

= ’(T (ψ± ) —¦ T q —¦ (¦ — Id))(T (•’1 )(ξx , 0e ), 0s ) by (37.24)

±

= ’(T (ψ± ) —¦ T q)(¦(T (•’1 )(ξx , 0e )), 0s )

±

= (T (ψ± ) —¦ T q)(T (•’1 )(0x , “± (ξx , e)), 0s ) by (37.22.2)

± ¦