of horizontal form is independent of the choice of a connection.

The projection χ— has the following properties where in the ¬rst assertion one

of the two forms has values in R:

χ— (• § ψ) = χ— • § χ— ψ,

χ— —¦ χ— = χ— ,

χ— —¦ (rg )— = (rg )— —¦ χ— for all g ∈ G,

—

χ ω=0

χ— —¦ L(ζX ) = L(ζX ) —¦ χ— .

They follow easily from the corresponding properties of χ, the last property uses

ζ(X)

= rexp tX .

that Flt

Now 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

(9)

forms, 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 = ’[¦, ζω ].

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

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104 Chapter III. Bundles and connections

(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 11.2 we have

dω „¦ = dω (dω + 1 [ω, ω]§ )

2

= χ— ddω + 1 χ— d[ω, ω]§

2

= 2 χ— ([dω, ω]§ ’ [ω, dω]§ ) = χ— [dω, ω]§

1

= [χ— dω, χ— ω]§ = 0, since χ— ω = 0.

(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 )], χ(X0 ), . . .

+

i<j

. . . , χ(Xi ), . . . , χ(Xj ), . . . )

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

=

0¤i¤k

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

+

i<j

. . . , χ(Xi ), . . . , χ(Xj ), . . . )

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

= (dω + χ— iR )(•)(X0 , . . . , Xk ).

(8) dω dω = χ— dχ— 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 LζX ψ =

ζX — exp tX —

‚ ‚ ‚

) ψ = ‚t 0 Ad(exp(’tX))ψ = ’ ad(X)ψ to get

‚t 0 (Flt ) ψ = ‚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 ).

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11. Principal and induced connections 105

So the ¬rst formula holds.

(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 8.16.(5). Using formula 8.11.(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 8.9

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

o

line survive:

(’1)

[¦, Ψ](ξ1 , . . . , ξ +1 ) = sign σ ¦([Ψ(ξσ1 , . . . , ξσ ), ξσ( +1) ])

!

σ

1

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

( ’1)! 2!

σ

For f : P ’ g and horizontal ξ we have ¦[ξ, ζf ] = ζξ(f ) = ζdf (ξ) since it is

C ∞ (P, R)-linear in ξ. So the last expression becomes

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

as required.

11.6. Theorem. Let (P, p, M, G) be a principal ¬ber bundle with principal

connection ω. Then the parallel transport for the principal connection is globally

de¬ned and G-equivariant.

In detail: For each smooth curve c : R ’ M there is a smooth mapping

Ptc : R — Pc(0) ’ P such that the following holds:

d

(1) Pt(c, t, u) ∈ Pc(t) , Pt(c, 0) = IdPc(0) , and ω( dt Pt(c, t, u)) = 0.

(2) Pt(c, t) : Pc(0) ’ Pc(t) is G-equivariant, i.e. Pt(c, t, u.g) = Pt(c, t, u).g

holds for all g ∈ G and u ∈ P . Moreover we have Pt(c, t)— (ζX |Pc(t) ) =

ζX |Pc(0) for all X ∈ g.

(3) For any smooth function f : R ’ R we have

Pt(c, f (t), u) = Pt(c —¦ f, t, Pt(c, f (0), u)).

Proof. By 11.4 the Christo¬el forms “± ∈ „¦1 (U± , X(G)) of the connection ω with

respect to a principal ¬ber bundle atlas (U± , •± ) are given by “± (ξx ) = Rω± (ξx ) ,

so they take values in the Lie subalgebra XR (G) of all right invariant vector

¬elds on G, which are bounded with respect to any right invariant Riemannian

metric on G. Each right invariant metric on a Lie group is complete. So the

connection is complete by the remark in 9.9.

Properties (1) and (3) follow from theorem 9.8, and (2) is seen as follows:

ω( dt Pt(c, t, u).g) = Ad(g ’1 )ω( dt Pt(c, t, u)) = 0

d d

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106 Chapter III. Bundles and connections

implies that Pt(c, t, u).g = Pt(c, t, u.g). For the second assertion we compute for

u ∈ Pc(0) :

Pt(c, t)— (ζX |Pc(t) )(u) = T Pt(c, t)’1 ζX (Pt(c, t, u)) =

= T Pt(c, t)’1 ds |0 Pt(c, t, u). exp(sX) =

d

= T Pt(c, t)’1 ds |0 Pt(c, t, u. exp(sX)) =

d

’1

d

ds |0 Pt(c, t)

= Pt(c, t, u. exp(sX))

d

ds |0 u. exp(sX) = ζX (u).

=

11.7. Holonomy groups. Let (P, p, M, G) be a principal ¬ber bundle with

principal connection ¦ = ζ —¦ ω. We assume that M is connected and we ¬x

x0 ∈ M .

In 9.10 we de¬ned the holonomy group Hol(¦, x0 ) ‚ Di¬(Px0 ) as the group

of all Pt(c, 1) : Px0 ’ Px0 for c any piecewise smooth closed loop through

x0 . (Reparametrizing c by a function which is ¬‚at at each corner of c we may

assume that any c is smooth.) If we consider only those curves c which are

nullhomotopic, we obtain the restricted holonomy group Hol0 (¦, x0 ).

Now let us ¬x u0 ∈ Px0 . The elements „ (u0 , Pt(c, t, u0 )) ∈ G form a subgroup

of the structure group G which is isomorphic to Hol(¦, x0 ); we denote it by

Hol(ω, u0 ) and we call it also the holonomy group of the connection. Considering

only nullhomotopic curves we get the restricted holonomy group Hol0 (ω, u0 ) a

normal subgroup of Hol(ω, u0 ).

Theorem. The main results for the holonomy are as follows:

(1) We have Hol(ω, u0 .g) = conj(g ’1 ) Hol(ω, u0 ) and

Hol0 (ω, u0 .g) = conj(g ’1 ) Hol0 (ω, u0 ).

(2) For each curve c in M with c(0) = x0 we have Hol(ω, Pt(c, t, u0 )) =

Hol(ω, u0 ) and Hol0 (ω, Pt(c, t, u0 )) = Hol0 (ω, u0 ).

(3) Hol0 (ω, u0 ) is a connected Lie subgroup of G and the quotient group

Hol(ω, u0 )/ Hol0 (ω, u0 ) is at most countable, so Hol(ω, u0 ) is also a Lie

subgroup of G.

(4) The Lie algebra hol(ω, u0 ) ‚ g of Hol(ω, u0 ) is linearly generated by

{„¦(Xu , Yu ) : Xu , Yu ∈ Tu P }, and it is isomorphic to the holonomy Lie

algebra hol(¦, x0 ) we considered in 9.10.

(5) For u0 ∈ Px0 let P (ω, u0 ) be the set of all Pt(c, t, u0 ) for c any (piecewise)

smooth curve in M with c(0) = x0 and for t ∈ R. Then P (ω, u0 ) is

a sub ¬ber bundle of P which is invariant under the right action of

Hol(ω, u0 ); so it is itself a principal ¬ber bundle over M with structure

group Hol(ω, u0 ) and we have a reduction of structure group, cf. 10.6 and

10.13. The pullback of ω to P (ω, u0 ) is then again a principal connection

form i— ω ∈ „¦1 (P (ω, u0 ); hol(ω, u0 )).

(6) P is foliated by the leaves P (ω, u), u ∈ Px0 .

(7) If the curvature „¦ = 0 then Hol0 (ω, u0 ) = {e} and each P (ω, u) is a

covering of M .

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11. Principal and induced connections 107

(8) If one uses piecewise C k -curves for 1 ¤ k < ∞ in the de¬nition, one gets

the same holonomy groups.

In view of assertion 5 a principal connection ω is called irreducible if Hol(ω, u0 )

equals the structure group G for some (equivalently any) u0 ∈ Px0 .

Proof. 1. This follows from the properties of the mapping „ from 10.2 and from

the G-equivariance of the parallel transport.

The rest of this theorem is a compilation of well known results, and we refer

to [Kobayashi-Nomizu I, 63, p. 83¬] for proofs.

11.8. Inducing 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 10.7.

Recall from 10.18 that its tangent and vertical bundle 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 the following

diagram:

w w

¦ — Id =

TP — TS TP — TS T (P — S)

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 (»g )X in the Lie group T G is denoted by (Te (»g )X)’1 ,

see lemma 10.17. Furthermore by 5.13 we have

T r(ξu , Te (»g )X) = Tu (rg )ξu + T r((0P — LX )(u, g))

= Tu (rg )ξu + Tg (ru )(Te (»g )X)

= Tu (rg )ξu + ζX (ug).

We may compute

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

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

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

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

= (T r(¦(ξu ), Te (»g )X), T ((Te (»g )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, ]).

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108 Chapter III. Bundles and connections

¯

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

connection.

From the diagram it also follows, that the vector valued forms ¦ — Id ∈

¯

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

1

So by 8.15 we have for the curvatures

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

2 2

1¯ ¯

R¦ = [¦, ¦],

¯ 2

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

¯

By uniqueness of the solutions of the de¬ning di¬erential equation we also get

that

Pt¦ (c, t, q(u, s)) = q(Pt¦ (c, t, u), s).

¯

11.9. Recognizing induced connections. We consider again a principal

¬ber bundle (P, p, M, G) and a left action : G — S ’ S. Suppose that Ψ ∈

„¦1 (P [S]; T (P [S])) is a connection on the associated bundle P [S] = P [S, ]. Then

the following question arises: When is the connection Ψ induced from a principal

connection on P ? If this is the case, we say that Ψ is compatible with the G-

bundle 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] belong-

ing to the G-bundle 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 10.7 it is seen that 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 9.7 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 11.8

±

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

±

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11. Principal and induced connections 109

= (T (ψ± ) —¦ T q)(T (•’1 )(0x , “± (ξx , e)), 0s ) by 11.4.(3)

± ¦

= ’T (ψ± —¦ q —¦ (•’1 — Id))(0x , ω± (ξx ), 0s ) by 11.4.(7)

±

= ’Te ( s )ω± (ξx ) by (2)

= ’ζω± (ξx ) (s).

So the condition is necessary. Now let us conversely suppose that a connection

Ψ on P [S] is given such that the Christo¬el forms “± with respect to a ¬ber

Ψ

bundle atlas of the G-structure have values in Xf und (S). Then unique g-valued

forms ω± ∈ „¦1 (U± ; g) are given by the equation “± (ξx ) = ζ(ω± (ξx )), since the

Ψ

action is in¬nitesimally e¬ective. From the transition formulas 9.7 for the “± Ψ

follow the transition formulas 11.4.(5) for the ω ± , so that they give a unique

principal connection on P , which by the ¬rst part of the proof induces the given

connection Ψ on P [S].

11.10. Inducing connections on associated vector bundles.

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 ¬nite dimensional vector space W .

We consider the associated vector bundle (E := P [W, ρ], p, M, W ), from 10.11.

Recall from 6.11 that 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.

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

¯

the induced connection ¦ ∈ „¦1 (E; T (E)) from 11.8. Inserting the projections

of both vector bundle structures on T (E) into the diagram in 11.8 we get the

following diagram

xw

&& ¦ — Id

TP — TW TP — TW TP — W — W

&& x

xxπ

(

&

x

π

P —W

u

q

Tq Tq

x

x

P —G W = E

u xu