[LK , iL ] = i([K, L]) ’ (’1)k L(iL K), or

[iL , LK ] = L(iL K) + (’1)k i([L, K]).

This generalizes 7.7.2.

Proof. For f ∈ C ∞ (M, R) we have [iL , LK ]f = iL iK df ’ 0 = iL (df —¦ K) =

df —¦ (iL K) = L(iL K)f . So [iL , LK ] ’ L(iL K) is an algebraic derivation.

[[iL , LK ], d] = [iL , [LK , d]] ’ (’1)k [LK , [iL , d]] =

= 0 ’ (’1)k L([K, L]) = (’1)k [i([L, K]), d].

Since [ , d] kills the L™s and is injective on the i™s, the algebraic part of [iL , LK ]

is (’1)k i([L, K]).

8.7. The space Der „¦(M ) is a graded module over the graded algebra „¦(M )

with the action (ω § D)• = ω § D(•), because „¦(M ) is graded commutative.

Theorem. Let the degree of ω be q, of • be k, and of ψ be . Let the other

degrees be as indicated. Then we have:

[ω § D1 , D2 ] = ω § [D1 , D2 ] ’ (’1)(q+k1 )k2 D2 (ω) § D1 .

(1)

i(ω § L) = ω § i(L)

(2)

ω § LK = L(ω § K) + (’1)q+k’1 i(dω § K).

(3)

[ω § L1 , L2 ]§ = ω § [L1 , L2 ]§ ’

(4)

1 ’1)( 2 ’1)

’ (’1)(q+ i(L2 )ω § L1 .

[ω § K1 , K2 ] = ω § [K1 , K2 ] ’ (’1)(q+k1 )k2 L(K2 )ω § K1

(5)

+ (’1)q+k1 dω § i(K1 )K2 .

[• — X, ψ — Y ] = • § ψ — [X, Y ]

(6)

’ iY d• § ψ — X ’ (’1)k iX dψ § • — Y

’ d(iY • § ψ) — X ’ (’1)k d(iX ψ § •) — Y

= • § ψ — [X, Y ] + • § LX ψ — Y ’ LY • § ψ — X

+ (’1)k (d• § iX ψ — Y + iY • § dψ — X) .

Proof. For (1), (2), (3) write out the de¬nitions. For (4) compute i([ω§L1 , L2 ]§ ).

For (5) compute L([ω § K1 , K2 ]). For (6) use (5) .

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8. Derivations on the algebra of di¬erential forms and the Fr¨licher-Nijenhuis bracket 71

o

8.8. Theorem. For K ∈ „¦k (M ; T M ) and ω ∈ „¦ (M ) the Lie derivative of ω

along K is given by the following formula, where the Xi are vector ¬elds on M .

(LK ω)(X1 , . . . , Xk+ ) =

1

sign σ L(K(Xσ1 , . . . , Xσk ))(ω(Xσ(k+1) , . . . , Xσ(k+ ) ))

= k! !

σ

’1

+ sign σ ω([K(Xσ1 , . . . , Xσk ), Xσ(k+1) ], Xσ(k+2) , . . . )

k! ( ’1)!

σ

k’1

(’1)

+ sign σ ω(K([Xσ1 , Xσ2 ], Xσ3 , . . . ), Xσ(k+2) , . . . ).

(k’1)! ( ’1)! 2!

σ

Proof. It su¬ces to consider K = • — X. Then by 8.7.3 we have L(• — X) =

• § LX ’ (’1)k’1 d• § iX . Now use the global formulas of section 7 to expand

this.

8.9. Theorem. For K ∈ „¦k (M ; T M ) and L ∈ „¦ (M ; T M ) we have for the

Fr¨licher-Nijenhuis bracket [K, L] the following formula, where the Xi are vector

o

¬elds on M .

[K, L](X1 , . . . , Xk+ ) =

1

= sign σ [K(Xσ1 , . . . , Xσk ), L(Xσ(k+1) , . . . , Xσ(k+ ) )]

k! !

σ

’1

+ sign σ L([K(Xσ1 , . . . , Xσk ), Xσ(k+1) ], Xσ(k+2) , . . . )

k! ( ’1)!

σ

k

(’1)

+ sign σ K([L(Xσ1 , . . . , Xσ ), Xσ( +1) ], Xσ( +2) , . . . )

(k’1)! !

σ

k’1

(’1)

+ sign σ L(K([Xσ1 , Xσ2 ], Xσ3 , . . . ), Xσ(k+2) , . . . )

(k’1)! ( ’1)! 2!

σ

(k’1)

(’1)

+ sign σ K(L([Xσ1 , Xσ2 ], Xσ3 , . . . ), Xσ( +2) , . . . ).

(k’1)! ( ’1)! 2!

σ

Proof. It su¬ces to consider K = • — X and L = ψ — Y , then for [• — X, ψ — Y ]

we may use 8.7.6 and evaluate that at (X1 , . . . , Xk+ ). After some combinatorial

computation we get the right hand side of the above formula for K = • — X and

L=ψ—Y.

There are more illuminating ways to prove this formula, see [Michor, 87].

8.10. Local formulas. In a local chart (U, u) on the manifold M we put

Lj dβ — ‚j , and ω | U =

K± d± — ‚i , L | U =

i

ωγ dγ , where

K|U= β

± = (1 ¤ ±1 < ±2 < · · · < ±k ¤ dim M ) is a form index, d± = du±1 § . . . § du±k ,

‚

‚i = ‚ui and so on.

Plugging Xj = ‚ij into the global formulas 8.2, 8.8, and 8.9, we get the

following local formulas:

K±1 ...±k ωi±k+1 ...±k+ ’1 d±

i

iK ω | U =

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

72 Chapter II. Di¬erential forms

K±1 ...±k Lj k+1 ...±k+

[K, L]§ | U = i

i±

j

’1)

’ (’1)(k’1)( Li 1 ...± Ki± +1 ...±k+ d± — ‚j

±

i

LK ω | U = K±1 ...±k ‚i ω±k+1 ...±k+

+ (’1)k (‚±1 K±2 ...±k+1 ) ωi±k+2 ...±k+ d±

i

K±1 ...±k ‚i Lj k+1 ...±k+

i

[K, L] | U = ±

’ (’1)k Li 1 ...± ‚i K± +1 ...±k+

j

±

j

’ kK±1 ...±k’1 i ‚±k Li k+1 ...±k+

±

+ (’1)k Lj 1 ...± ’1 i ‚± K± +1 ...±k+

i

d± — ‚j

±

8.11. Theorem. For Ki ∈ „¦ki (M ; T M ) and Li ∈ „¦ki +1 (M ; T M ) we have

[LK1 + iL1 , LK2 + iL2 ] =

(1)

= L [K1 , K2 ] + iL1 K2 ’ (’1)k1 k2 iL2 K1

+ i [L1 , L2 ]§ + [K1 , L2 ] ’ (’1)k1 k2 [K2 , L1 ] .

Each summand of this formula looks like a semidirect product of graded Lie

algebras, but the mappings

i : „¦(M ; T M ) ’ End(„¦(M ; T M ), [ , ])

]§ )

ad : „¦(M ; T M ) ’ End(„¦(M ; T M ), [ ,

do not take values in the subspaces of graded derivations. We have instead for

K ∈ „¦k (M ; T M ) and L ∈ „¦ +1 (M ; T M ) the following relations:

iL [K1 , K2 ] = [iL K1 , K2 ] + (’1)k1 [K1 , iL K2 ]

(2)

’ (’1)k1 i([K1 , L])K2 ’ (’1)(k1 + )k2

i([K2 , L])K1

[K, [L1 , L2 ]§ ] = [[K, L1 ], L2 ]§ + (’1)kk1 [L1 , [K, L2 ]]§ ’

(3)

’ (’1)kk1 [i(L1 )K, L2 ] ’ (’1)(k+k1 )k2 [i(L2 )K, L1 ]

The algebraic meaning of the relations of this theorem and its consequences in

group theory have been investigated in [Michor, 89]. The corresponding product

of groups is well known to algebraists under the name ˜Zappa-Szep™-product.

Proof. Equation (1) is an immediate consequence of 8.6. Equations (2) and (3)

follow from (1) by writing out the graded Jacobi identity, or as follows: Consider

L(iL [K1 , K2 ]) and use 8.6 repeatedly to obtain L of the right hand side of (2).

Then consider i([K, [L1 , L2 ]§ ]) and use again 8.6 several times to obtain i of the

right hand side of (3).

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

8. Derivations on the algebra of di¬erential forms and the Fr¨licher-Nijenhuis bracket 73

o

8.12. Corollary (of 8.9). For K, L ∈ „¦1 (M ; T M ) we have

[K, L](X, Y ) = [KX, LY ] ’ [KY, LX]

’ L([KX, Y ] ’ [KY, X])

’ K([LX, Y ] ’ [LY, X])

+ (LK + KL)[X, Y ].

8.13. Curvature. Let P ∈ „¦1 (M ; T M ) satisfy P —¦ P = P , i.e. P is a pro-

jection in each ¬ber of T M . This is the most general case of a (¬rst order)

connection. We may call ker P the horizontal space and im P the vertical space

of the connection. If P is of constant rank, then both are sub vector bundles of

T M . If im P is some primarily ¬xed sub vector bundle or (tangent bundle of) a

foliation, P can be called a connection for it. Special cases of this will be treated

extensively later on. The following result is immediate from 8.12.

Lemma. We have

¯

[P, P ] = 2R + 2R,

¯

where R, R ∈ „¦2 (M ; T M ) are given by R(X, Y ) = P [(Id ’P )X, (Id ’P )Y ] and

¯

R(X, Y ) = (Id ’P )[P X, P Y ].

If P has constant rank, then R is the obstruction against integrability of the

¯

horizontal bundle ker P , and R is the obstruction against integrability of the

¯

vertical bundle im P . Thus we call R the curvature and R the cocurvature of the

connection P . We will see later, that for a principal ¬ber bundle R is just the

negative of the usual curvature.

8.14. Lemma (Bianchi identity). If P ∈ „¦1 (M ; T M ) is a connection (¬ber

¯

projection) with curvature R and cocurvature R, then we have

¯

[P, R + R] = 0

¯

[R, P ] = iR R + iR R.

¯

¯

Proof. We have [P, P ] = 2R + 2R by 8.13 and [P, [P, P ]] = 0 by the graded

Jacobi identity. So the ¬rst formula follows. We have 2R = P —¦ [P, P ] = i[P,P ] P .

By 8.11.2 we get i[P,P ] [P, P ] = 2[i[P,P ] P, P ] ’ 0 = 4[R, P ]. Therefore [R, P ] =

¯ ¯ ¯

1

4 i[P,P ] [P, P ] = i(R + R)(R + R) = iR R + iR R since R has vertical values and

¯

¯

kills vertical vectors, so iR R = 0; likewise for R.

8.15. f -relatedness of the Fr¨licher-Nijenhuis bracket. Let f : M ’

o

N be a smooth mapping between manifolds. Two vector valued forms K ∈

„¦k (M ; T M ) and K ∈ „¦k (N ; T N ) are called f -related or f -dependent, if for all

Xi ∈ Tx M we have

Kf (x) (Tx f · X1 , . . . , Tx f · Xk ) = Tx f · Kx (X1 , . . . , Xk ).

(1)

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

74 Chapter II. Di¬erential forms

Theorem.

(2) If K and K as above are f -related then iK —¦ f — = f — —¦ iK : „¦(N ) ’

„¦(M ).

(3) If iK —¦ f — | B 1 (N ) = f — —¦ iK | B 1 (N ), then K and K are f -related,

where B 1 denotes the space of exact 1-forms.

(4) If Kj and Kj are f -related for j = 1, 2, then iK1 K2 and iK1 K2 are

f -related, and also [K1 , K2 ]§ and [K1 , K2 ]§ are f -related.

(5) If K and K are f -related then LK —¦ f — = f — —¦ LK : „¦(N ) ’ „¦(M ).

(6) If LK —¦ f — | „¦0 (N ) = f — —¦ LK | „¦0 (N ), then K and K are f -related.

(7) If Kj and Kj are f -related for j = 1, 2, then their Fr¨licher-Nijenhuis

o

brackets [K1 , K2 ] and [K1 , K2 ] are also f -related.

Proof. (2) By 8.2 we have for ω ∈ „¦q (N ) and Xi ∈ Tx M :

(iK f — ω)x (X1 , . . . , Xq+k’1 ) =

sign σ (f — ω)x (Kx (Xσ1 , . . . , Xσk ), Xσ(k+1) , . . . )

1

= k! (q’1)!

σ

1

sign σ ωf (x) (Tx f · Kx (Xσ1 , . . . ), Tx f · Xσ(k+1) , . . . )

= k! (q’1)!

σ

1

sign σ ωf (x) (Kf (x) (Tx f · Xσ1 , . . . ), Tx f · Xσ(k+1) , . . . )

= k! (q’1)!

σ

—

= (f iK ω)x (X1 , . . . , Xq+k’1 )

(3) follows from this computation, since the df , f ∈ C ∞ (M, R) separate

points.

(4) follows from the same computation for K2 instead of ω, the result for the

bracket then follows 8.2.2.

(5) The algebra homomorphism f — intertwines the operators iK and iK by

(2), and f — commutes with the exterior derivative d. Thus f — intertwines the

commutators [iK , d] = LK and [iK , d] = LK .

(6) For g ∈ „¦0 (N ) we have LK f — g = iK d f — g = iK f — dg and f — LK g =

f — iK dg. By (3) the result follows.

(7) The algebra homomorphism f — intertwines LKj and LKj , thus also their

graded commutators, which are equal to L([K1 , K2 ]) and L([K1 , K2 ]), respec-

tively. Then use (6).

8.16. Let f : M ’ N be a local di¬eomorphism. Then we can consider the

pullback operator f — : „¦(N ; T N ) ’ „¦(M ; T M ), given by

(f — K)x (X1 , . . . , Xk ) = (Tx f )’1 Kf (x) (Tx f · X1 , . . . , Tx f · Xk ).

(1)

Note that this is a special case of the pullback operator for sections of natural

vector bundles in 6.15. Clearly K and f — K are then f -related.

Theorem. In this situation we have:

(2) f — [K, L] = [f — K, f — L].

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

Remarks 75

(3) f — iK L = if — K f — L.

(4) f — [K, L]§ = [f — K, f — L]§ .

(5) For a vector ¬eld X ∈ X(M ) and K ∈ „¦(M ; T M ) by 6.15 the Lie

derivative LX K = ‚t 0 (FlX )— K is de¬ned. Then we have LX K =

‚

t

[X, K], the Fr¨licher-Nijenhuis-bracket.

o

This is sometimes expressed by saying that the Fr¨licher-Nijenhuis bracket,

o

§

[ , ] , etc. are natural bilinear concomitants.

Proof. (2) “ (4) are obvious from 8.15. They also follow directly from the geo-

metrical constructions of the operators in question. (5) Obviously LX is R-linear,

so it su¬ces to check this formula for K = ψ — Y , ψ ∈ „¦(M ) and Y ∈ X(M ).

But then

LX (ψ — Y ) = LX ψ — Y + ψ — LX Y by 6.16

= LX ψ — Y + ψ — [X, Y ]

= [X, ψ — Y ] by 8.7.6.

8.17. Remark. At last we mention the best known application of the Fr¨licher-

o

Nijenhuis bracket, which also led to its discovery. A vector valued 1-form J ∈

„¦1 (M ; T M ) with J —¦ J = ’ Id is called a almost complex structure; if it exists,

√

dim M is even and J can be viewed as a ¬ber multiplication with ’1 on T M .

By 8.12 we have

[J, J](X, Y ) = 2([JX, JY ] ’ [X, Y ] ’ J[X, JY ] ’ J[JX, Y ]).

1

The vector valued form 2 [J, J] is also called the Nijenhuis tensor of J, because

we have the following result:

A manifold M with an almost complex structure J is a complex

manifold (i.e., there exists an atlas for M with holomorphic chart-

change mappings) if and only if [J, J] = 0. See [Newlander-Nirenberg,

57].

Remarks

The material on the Lie derivative on natural vector bundles 6.14“6.20 appears

here for the ¬rst time. Most of section 8 is due to [Fr¨licher-Nijenhuis, 56], the

o

formula in 8.9 was proved by [Mangiarotti-Modugno, 84] and [Michor, 87]. The

Bianchi identity 8.14 is from [Michor, 89a].

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

76

CHAPTER III.

BUNDLES AND CONNECTIONS

We begin our treatment of connections in the general setting of ¬ber bundles

(without structure group). A connection on a ¬ber bundle is just a projection

onto the vertical bundle. Curvature and the Bianchi identity is expressed with

the help of the Fr¨licher-Nijenhuis bracket. The parallel transport for such a

o

general connection is not de¬ned along the whole of the curve in the base in

general - if this is the case for all curves, the connection is called complete. We

show that every ¬ber bundle admits complete connections. For complete con-

nections we treat holonomy groups and the holonomy Lie algebra, a subalgebra

of the Lie algebra of all vector ¬elds on the standard ¬ber.

Then we present principal bundles and associated bundles in detail together

with the most important examples. Finally we investigate principal connections

by requiring equivariance under the structure group. It is remarkable how fast

the usual structure equations can be derived from the basic properties of the

Fr¨licher-Nijenhuis bracket. Induced connections are investigated thoroughly -

o

we describe tools to recognize induced connections among general ones.

If the holonomy Lie algebra of a connection on a ¬ber bundle is ¬nite dimen-

sional and consists of complete vector ¬elds on the ¬ber, we are able to show,

that in fact the ¬ber bundle is associated to a principal bundle and the connec-

tion is induced from an irreducible principal connection (theorem 9.11). This is

a powerful generalization of the theorem of Ambrose and Singer.

Connections will be treated once again from the point of view of jets, when

we have them at our disposal in chapter IV.

We think that the treatment of connections presented here o¬ers some di-

dactical advantages besides presenting new results: the geometric content of a

connection is treated ¬rst, and the additional requirement of equivariance under

a structure group is seen to be additional and can be dealt with later - so the

reader is not required to grasp all the structures at the same time. Besides that

it gives new results and new insights. There are naturally appearing connec-

tions in di¬erential geometry which are not principal or induced connections:

The universal connection on the bundle J 1 P/G of all connections of a principal

bundle, and also the Cartan connections.

9. General ¬ber bundles and connections

9.1. De¬nition. A (¬ber) bundle (E, p, M, S) consists of manifolds E, M , S,

and a smooth mapping p : E ’ M ; furthermore it is required that each x ∈ M

has an open neighborhood U such that E | U := p’1 (U ) is di¬eomorphic to

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

9. General ¬ber bundles and connections 77

U — S via a ¬ber respecting di¬eomorphism:

w U —S

‘

ψ

E|U

‘p“ &

‘ &pr

)

& 1

U

E is called the total space, M is called the base space, p is a surjective submersion,

called the projection, and S is called standard ¬ber. (U, ψ) as above is called a

¬ber chart or a local trivialization of E.

A collection of ¬ber charts (U± , ψ± ), such that (U± ) is an open cover of M ,

is called a (¬ber) bundle atlas. If we ¬x such an atlas, then (ψ± —¦ ψβ ’1 )(x, s) =

(x, ψ±β (x, s)), where ψ±β : (U± © Uβ ) — S ’ S is smooth and ψ±β (x, ) is a

di¬eomorphism of S for each x ∈ U±β := U± © Uβ . We may thus consider

the mappings ψ±β : U±β ’ Di¬(S) with values in the group Di¬(S) of all

di¬eomorphisms of S; their di¬erentiability is a subtle question, which will not

be discussed in this book, but see [Michor, 88]. In either form these mappings

ψ±β are called the transition functions of the bundle. They satisfy the cocycle

condition: ψ±β (x)—¦ψβγ (x) = ψ±γ (x) for x ∈ U±βγ and ψ±± (x) = IdS for x ∈ U± .

Therefore the collection (ψ±β ) is called a cocycle of transition functions.

Given an open cover (U± ) of a manifold M and a cocycle of transition func-

tions (ψ±β ) we may construct a ¬ber bundle (E, p, M, S) similarly as in 6.4.

9.2. Lemma. Let p : N ’ M be a proper surjective submersion (a ¬bered

manifold) which is proper (i.e. compact sets have compact inverse images) and

let M be connected. Then (N, p, M ) is a ¬ber bundle.

Proof. We have to produce a ¬ber chart at each x0 ∈ M . So let (U, u) be

a chart centered at x0 on M such that u(U ) ∼ Rm . For each x ∈ U let

=

’1

ξx (y) := (Ty u) .u(x), then ξx ∈ X(U ), depending smoothly on x ∈ U , such

that u(Flξx u’1 (z)) = z + t.u(x), so each ξx is a complete vector ¬eld on U .

t

Since p is a submersion, with the help of a partition of unity on p’1 (U ) we may

construct vector ¬elds ·x ∈ X(p’1 (U )) which depend smoothly on x ∈ U and are

p-related to ξx : T p.·x = ξx —¦ p. Thus p —¦ Fl·x = Flξx —¦p by 3.14, so Fl·x is ¬ber

t t t

respecting, and since p is proper and ξx is complete, ·x has a global ¬‚ow too.

Denote p’1 (x0 ) by S. Then • : U — S ’ p’1 (U ), de¬ned by •(x, y) = Fl·x (y),

1

is a di¬eomorphism and is ¬ber respecting, so (U, •’1 ) is a ¬ber chart. Since M

is connected, the ¬bers p’1 (x) are all di¬eomorphic.

9.3. Let (E, p, M, S) be a ¬ber bundle; we consider the tangent mapping T p :

T E ’ T M and its kernel ker T p =: V E which is called the vertical bundle of

E. The following is special case of 8.13.

De¬nition. A connection on the ¬ber bundle (E, p, M, S) is a vector valued 1-

form ¦ ∈ „¦1 (E; V E) with values in the vertical bundle V E such that ¦ —¦ ¦ = ¦

and Im¦ = V E; so ¦ is just a projection T E ’ V E.

If we intend to contrast this general concept of connection with some special

cases which will be discussed later, we will say that ¦ is a general connection.

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

78 Chapter III. Bundles and connections

Since ker ¦ is of constant rank, by 6.6 ker ¦ is a sub vector bundle of T E, it is

called the space of horizontal vectors or the horizontal bundle and it is denoted

by HE. Clearly T E = HE • V E and Tu E = Hu E • Vu E for u ∈ E.

Now we consider the mapping (T p, πE ) : T E ’ T M —M E. We have by

de¬nition (T p, πE )’1 (0p(u) , u) = Vu E, so (T p, πE ) | HE : HE ’ T M —M E is

¬ber linear over E and injective, so by reason of dimensions it is a ¬ber linear

isomorphism: Its inverse is denoted by

C := ((T p, πE ) | HE)’1 : T M —M E ’ HE ’ T E.

So C : T M —M E ’ T E is ¬ber linear over E and is a right inverse for (T p, πE ).

C is called the horizontal lift associated to the connection ¦.

Note the formula ¦(ξu ) = ξu ’ C(T p.ξu , u) for ξu ∈ Tu E. So we can equally

well describe a connection ¦ by specifying C. Then we call ¦ the vertical pro-

jection (no confusion with 6.11 will arise) and χ := idT E ’ ¦ = C —¦ (T p, πE ) will

be called the horizontal projection.

9.4. Curvature. Suppose that ¦ : T E ’ V E is a connection on a ¬ber bundle

(E, p, M, S), then as in 8.13 the curvature R of ¦ is given by

2R = [¦, ¦] = [Id ’¦, Id ’¦] = [χ, χ] ∈ „¦2 (E; V E)

¯

(The cocurvature R vanishes since the vertical bundle V E is integrable). We

1

have R(X, Y ) = 2 [¦, ¦](X, Y ) = ¦[χX, χY ], so R is an obstruction against

integrability of the horizontal subbundle. Note that for vector ¬elds ξ, · ∈

X(M ) and their horizontal lifts Cξ, C· ∈ X(E) we have R(Cξ, C·) = [Cξ, C·] ’

C([ξ, ·]).

Since the vertical bundle V E is integrable, by 8.14 we have the Bianchi iden-

tity [¦, R] = 0.

9.5. Pullback. Let (E, p, M, S) be a ¬ber bundle and consider a smooth map-

ping f : N ’ M . Since p is a submersion, f and p are transversal in the sense

of 2.18 and thus the pullback N —(f,M,p) E exists. It will be called the pullback

of the ¬ber bundle E by f and we will denote it by f — E. The following diagram

sets up some further notation for it:

w

p— f

f —E E

p

f —p

u u

w M.

f

N

Proposition. In the situation above we have:

(1) (f — E, f — p, N, S) is again a ¬ber bundle, and p— f is a ¬ber wise di¬eo-

morphism.

(2) If ¦ ∈ „¦1 (E; T E) is a connection on the bundle E, then the vector valued

form f — ¦, given by (f — ¦)u (X) := Tu (p— f )’1 .¦.Tu (p— f ).X for X ∈ Tu E,

is a connection on the bundle f — E. The forms f — ¦ and ¦ are p— f -related

in the sense of 8.15.

(3) The curvatures of f — ¦ and ¦ are also p— f -related.

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

9. General ¬ber bundles and connections 79

Proof. (1) If (U± , ψ± ) is a ¬ber bundle atlas of (E, p, M, S) in the sense of

9.1, then (f ’1 (U± ), (f — p, pr2 —¦ ψ± —¦ p— f )) is visibly a ¬ber bundle atlas for

(f — E, f — p, N, S), by the formal universal properties of a pullback 2.19. (2) is

obvious. (3) follows from (2) and 8.15.7.

9.6. Let us suppose that a connection ¦ on the bundle (E, p, M, S) has zero

curvature. Then by 9.4 the horizontal bundle is integrable and gives rise to the

horizontal foliation by 3.25.2. Each point u ∈ E lies on a unique leaf L(u) such

that Tv L(u) = Hv E for each v ∈ L(u). The restriction p | L(u) is locally a

di¬eomorphism, but in general it is neither surjective nor is it a covering onto

its image. This is seen by devising suitable horizontal foliations on the trivial

bundle pr2 : R — S 1 ’ S 1 .

9.7. Local description. Let ¦ be a connection on (E, p, M, S). Let us ¬x a

¬ber bundle atlas (U± ) with transition functions (ψ±β ), and let us consider the

connection ((ψ± )’1 )— ¦ ∈ „¦1 (U± — S; U± — T S), which may be written in the

form

((ψ± )’1 )— ¦)(ξx , ·y ) =: ’“± (ξx , y) + ·y for ξx ∈ Tx U± and ·y ∈ Ty S,

since it reproduces vertical vectors. The “± are given by

(0x , “± (ξx , y)) := ’T (ψ± ).¦.T (ψ± )’1 .(ξx , 0y ).

We consider “± as an element of the space „¦1 (U± ; X(S)), a 1-form on U ± with

values in the in¬nite dimensional Lie algebra X(S) of all vector ¬elds on the

standard ¬ber. The “± are called the Christo¬el forms of the connection ¦ with

respect to the bundle atlas (U± , ψ± ).

Lemma. The transformation law for the Christo¬el forms is

)).“β (ξx , y) = “± (ξx , ψ±β (x, y)) ’ Tx (ψ±β ( , y)).ξx .

Ty (ψ±β (x,

The curvature R of ¦ satis¬es

(ψ± )— R = d“± + 1 [“± , “± ]§ .

’1

X(S)

2

Here d“± is the exterior derivative of the 1-form “± ∈ „¦1 (U± ; X(S)) with

values in the complete locally convex space X(S). We will later also use the

Lie derivative of it and the usual formulas apply: consult [Fr¨licher, Kriegl, 88]

o

± ±

for calculus in in¬nite dimensional spaces. By [“ , “ ]X(S) we just mean the

2-form (ξ, ·) ’ [“± (ξ), “± (·)]X(S) . See 11.2 for the more sophisticated notation

1± ±

2 [“ , “ ]§ for this.

The formula for the curvature is the Maurer-Cartan formula which in this

general setting appears only in the level of local description.

Proof. From (ψ± —¦ (ψβ )’1 )(x, y) = (x, ψ±β (x, y)) we get that

T (ψ± —¦ (ψβ )’1 ).(ξx , ·y ) = (ξx , T(x,y) (ψ±β ).(ξx , ·y )) and thus:

’1 ’1

T (ψβ ).(0x , “β (ξx , y)) = ’¦(T (ψβ )(ξx , 0y )) =

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80 Chapter III. Bundles and connections

’1

’1

= ’¦(T (ψ± ).T (ψ± —¦ ψβ ).(ξx , 0y )) =

’1

= ’¦(T (ψ± )(ξx , T(x,y) (ψ±β )(ξx , 0y ))) =

’1 ’1

= ’¦(T (ψ± )(ξx , 0ψ±β (x,y) )) ’ ¦(T (ψ± )(0x , T(x,y) ψ±β (ξx , 0y ))) =

’1 ’1

= T (ψ± ).(0x , “± (ξx , ψ±β (x, y))) ’ T (ψ± )(0x , Tx (ψ±β ( , y)).ξx ).

This implies the transformation law.

For the curvature R of ¦ we have by 9.4 and 9.5.3

(ψ± )— R ((ξ 1 , · 1 ), (ξ 2 , · 2 )) =

’1

= (ψ± )— ¦ [(Id ’(ψ± )— ¦)(ξ 1 , · 1 ), (Id ’(ψ± )— ¦)(ξ 2 , · 2 )] =

’1 ’1 ’1

= (ψ± )— ¦[(ξ 1 , “± (ξ 1 )), (ξ 2 , “± (ξ 2 ))] =

’1

= (ψ± )— ¦ [ξ 1 , ξ 2 ], ξ 1 “± (ξ 2 ) ’ ξ 2 “± (ξ 1 ) + [“± (ξ 1 ), “± (ξ 2 )] =

’1

= ’“± ([ξ 1 , ξ 2 ]) + ξ 1 “± (ξ 2 ) ’ ξ 2 “± (ξ 1 ) + [“± (ξ 1 ), “± (ξ 2 )] =

= d“± (ξ 1 , ξ 2 ) + [“± (ξ 1 ), “± (ξ 2 )]X(S) .

9.8. Theorem (Parallel transport). Let ¦ be a connection on a bundle

(E, p, M, S) and let c : (a, b) ’ M be a smooth curve with 0 ∈ (a, b), c(0) = x.

Then there is a neighborhood U of Ex — {0} in Ex — (a, b) and a smooth

mapping Ptc : U ’ E such that:

(1) p(Pt(c, ux , t)) = c(t) if de¬ned, and Pt(c, ux , 0) = ux .

d

(2) ¦( dt Pt(c, ux , t)) = 0 if de¬ned.

(3) Reparametrisation invariance: If f : (a , b ) ’ (a, b) is smooth with

0 ∈ (a , b ), then Pt(c, ux , f (t)) = Pt(c —¦ f, Pt(c, ux , f (0)), t) if de¬ned.

(4) U is maximal for properties (1) and (2).

(5) If the curve c depends smoothly on further parameters then Pt(c, ux , t)

depends also smoothly on those parameters.

d

First proof. In local bundle coordinates ¦( dt Pt(c, ux , t)) = 0 is an ordinary

di¬erential equation of ¬rst order, nonlinear, with initial condition Pt(c, ux , 0) =

ux . So there is a maximally de¬ned local solution curve which is unique. All

further properties are consequences of uniqueness.

Second proof. Consider the pullback bundle (c— E, c— p, (a, b), S) and the pullback

connection c— ¦ on it. It has zero curvature, since the horizontal bundle is 1-

dimensional. By 9.6 the horizontal foliation exists and the parallel transport just

follows a leaf and we may map it back to E, in detail: Pt(c, ux , t) = p— c((c— p |

L(ux ))’1 (t)).

Third proof. Consider a ¬ber bundle atlas (U± , ψ± ) as in 9.7. Then we have

’1

ψ± (Pt(c, ψ± (x, y), t)) = (c(t), γ(y, t)), where

0 = (ψ± )— ¦

’1 d d d d

= ’“±

dt c(t), dt γ(y, t) dt c(t), γ(y, t) + dt γ(y, t),

so γ(y, t) is the integral curve (evolution line) through y ∈ S of the time depen-

d

dent vector ¬eld “± dt c(t) on S. This vector ¬eld visibly depends smoothly

on c. Clearly local solutions exist and all properties follow. For (5) we refer to

[Michor, 83].

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

9. General ¬ber bundles and connections 81

9.9. A connection ¦ on (E, p, M, S) is called a complete connection, if the par-

allel transport Ptc along any smooth curve c : (a, b) ’ M is de¬ned on the whole

of Ec(0) — (a, b). The third proof of theorem 9.8 shows that on a ¬ber bundle

with compact standard ¬ber any connection is complete.

The following is a su¬cient condition for a connection ¦ to be complete:

There exists a ¬ber bundle atlas (U± , ψ± ) and complete Riemannian met-

rics g± on the standard ¬ber S such that each Christo¬el form “± ∈

„¦1 (U± , X(S)) takes values in the linear subspace of g± -bounded vector

¬elds on S.

For in the third proof of theorem 9.8 above the time dependent vector ¬eld

d

±

“ ( dt c(t)) on S is g± -bounded for compact time intervals. So by continuation

the solution exists over c’1 (U± ), and thus globally.

A complete connection is called an Ehresmann connection in [Greub, Halperin,

Vanstone I, 72, p. 314], where it is also indicated how to prove the following

result.

Theorem. Each ¬ber bundle admits complete connections.

Proof. Let dim M = m. Let (U± , ψ± ) be a ¬ber bundle atlas as in 9.1. By

topological dimension theory [Nagata, 65] the open cover (U± ) of M admits a

re¬nement such that any m + 2 members have empty intersection, see also 1.1.

Let (U± ) itself have this property. Choose a smooth partition of unity (f± )

1

subordinated to (U± ). Then the sets V± := { x : f± (x) > m+2 } ‚ U± form still

an open cover of M since f± (x) = 1 and at most m + 1 of the f± (x) can be

nonzero. By renaming assume that each V± is connected. Then we choose an

open cover (W± ) of M such that W± ‚ V± .

Now let g1 and g2 be complete Riemannian metrics on M and S, respectively

(see [Nomizu - Ozeki, 61] or [Morrow, 70]). For not connected Riemannian

manifolds complete means that each connected component is complete. Then

g1 |U± — g2 is a Riemannian metric on U± — S and we consider the metric g :=

—

f± ψ± (g1 |U± — g2 ) on E. Obviously p : E ’ M is a Riemannian submersion

for the metrics g and g1 . We choose now the connection ¦ : T E ’ V E as the

orthonormal projection with respect to the Riemannian metric g.

Claim. ¦ is a complete connection on E.

Let c : [0, 1] ’ M be a smooth curve. We choose a partition 0 = t0 <

t1 < · · · < tk = 1 such that c([ti , ti+1 ]) ‚ V±i for suitable ±i . It su¬ces to

show that Pt(c(ti + ), uc(ti ) , t) exists for all 0 ¤ t ¤ ti+1 ’ ti and all uc(ti ) ,

for all i ” then we may piece them together. So we may assume that c :

[0, 1] ’ V± for some ±. Let us now assume that for some (x, y) ∈ V± — S

the parallel transport Pt(c, ψ± (x, y), t) is de¬ned only for t ∈ [0, t ) for some

’1

0 < t < 1. By the third proof of 9.8 we have Pt(c, ψ± (x, y), t) = ψ± (c(t), γ(t)),

where γ : [0, t ) ’ S is the maximally de¬ned integral curve through y ∈ S

of the time dependent vector ¬eld “± ( dt c(t), ) on S. We put g± := (ψ± )— g,

’1

d

then (g± )(x,y) = (g1 )x — ( β fβ (x)ψβ± (x, )— g2 )y . Since pr1 : (V± — S, g± ) ’

(V± , g1 |V± ) is a Riemannian submersion and since the connection (ψ± )— ¦ is also

’1

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82 Chapter III. Bundles and connections

given by orthonormal projection onto the vertical bundle, we get

t

g1 -lengtht d

∞> |(c (t), dt γ(t))|g± dt =

(c) = g± -length(c, γ) =

0

0

t

d d

—

|c (t)|21 + β fβ (c(t))(ψ±β (c(t), ’) g2 )( dt γ(t), dt γ(t)) dt ≥

= g

0

t t

1

d d

dt ≥ √

≥ f± (c(t)) | dt γ(t)|g2 | dt γ(t)|g2 dt.

m+2

0 0

So g2 -lenght(γ) is ¬nite and since the Riemannian metric g2 on S is complete,

limt’t γ(t) =: γ(t ) exists in S and the integral curve γ can be continued.

9.10. Holonomy groups and Lie algebras. Let (E, p, M, S) be a ¬ber bun-

dle with a complete connection ¦, and let us assume that M is connected. We

choose a ¬xed base point x0 ∈ M and we identify Ex0 with the standard ¬ber S.

For each closed piecewise smooth curve c : [0, 1] ’ M through x0 the parallel

transport Pt(c, , 1) =: Pt(c, 1) (pieced together over the smooth parts of c)

is a di¬eomorphism of S. All these di¬eomorphisms form together the group

Hol(¦, x0 ), the holonomy group of ¦ at x0 , a subgroup of the di¬eomorphism

group Di¬(S). If we consider only those piecewise smooth curves which are ho-

motopic to zero, we get a subgroup Hol0 (¦, x0 ), called the restricted holonomy

group of the connection ¦ at x0 .

Now let C : T M —M E ’ T E be the horizontal lifting as in 9.3, and let R

be the curvature (9.4) of the connection ¦. For any x ∈ M and Xx ∈ Tx M

the horizontal lift C(Xx ) := C(Xx , ) : Ex ’ T E is a vector ¬eld along Ex .

For Xx and Yx ∈ Tx M we consider R(CXx , CYx ) ∈ X(Ex ). Now we choose

any piecewise smooth curve c from x0 to x and consider the di¬eomorphism

Pt(c, t) : S = Ex0 ’ Ex and the pullback Pt(c, 1)— R(CXx , CYx ) ∈ X(S). Let

us denote by hol(¦, x0 ) the closed linear subspace, generated by all these vector

¬elds (for all x ∈ M , Xx , Yx ∈ Tx M and curves c from x0 to x) in X(S) with

respect to the compact C ∞ -topology (see [Hirsch, 76]), and let us call it the

holonomy Lie algebra of ¦ at x0 .

Lemma. hol(¦, x0 ) is a Lie subalgebra of X(S).

Proof. For X ∈ X(M ) we consider the local ¬‚ow FlCX of the horizontal lift of

t

X. It restricts to parallel transport along any of the ¬‚ow lines of X in M . Then

for vector ¬elds X, Y, U, V on M the expression

CX — CY — CX — CZ —

d

dt |0 (Fls ) (Flt ) (Fl’s ) (Flz ) R(CU, CV )|Ex0

= (FlCX )— [CY, (FlCX )— (FlCZ )— R(CU, CV )]|Ex0

’s

s z

CX — CZ —

= [(Fls ) CY, (Flz ) R(CU, CV )]|Ex0

is in hol(¦, x0 ), since it is closed in the compact C ∞ -topology and the derivative

can be written as a limit. Thus

[(FlCX )— [CY1 , CY2 ], (FlCZ )— R(CU, CV )]|Ex0 ∈ hol(¦, x0 )

s z

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9. General ¬ber bundles and connections 83

by the Jacobi identity and

[(FlCX )— C[Y1 , Y2 ], (FlCZ )— R(CU, CV )]|Ex0 ∈ hol(¦, x0 ),

s z

so also their di¬erence

[(FlCX )— R(CY1 , CY2 ), (FlCZ )— R(CU, CV )]|Ex0

s z

is in hol(¦, x0 ).

9.11. The following theorem is a generalization of the theorem of Ambrose

and Singer on principal connections. The reader who does not know principal

connections is advised to read parts of sections 10 and 11 ¬rst. We include this

result here in order not to disturb the development in section 11 later.

Theorem. Let ¦ be a complete connection on the ¬bre bundle (E, p, M, S) and

let M be connected. Suppose that for some (hence any) x0 ∈ M the holonomy

Lie algebra hol(¦, x0 ) is ¬nite dimensional and consists of complete vector ¬elds

on the ¬ber Ex0

Then there is a principal bundle (P, p, M, G) with ¬nite dimensional structure

group G, an irreducible connection ω on it and a smooth action of G on S such

that the Lie algebra g of G equals the holonomy Lie algebra hol(¦, x0 ), the ¬bre

bundle E is isomorphic to the associated bundle P [S], and ¦ is the connection

induced by ω. The structure group G equals the holonomy group Hol(¦, x0 ). P

and ω are unique up to isomorphism.

By a theorem of [Palais, 57] a ¬nite dimensional Lie subalgebra of X(Ex0 )

like hol(¦, x0 ) consists of complete vector ¬elds if and only if it is generated by

complete vector ¬elds as a Lie algebra.

Proof. Let us again identify Ex0 and S. Then g := hol(¦, x0 ) is a ¬nite dimen-

sional Lie subalgebra of X(S), and since each vector ¬eld in it is complete, there

is a ¬nite dimensional connected Lie group G0 of di¬eomorphisms of S with Lie

algebra g, see [Palais, 57].

Claim 1. G0 contains Hol0 (¦, x0 ), the restricted holonomy group.

Let f ∈ Hol0 (¦, x0 ), then f = Pt(c, 1) for a piecewise smooth closed curve c

through x0 , which is nullhomotopic. Since the parallel transport is essentially

invariant under reparametrisation, 9.8, we can replace c by c —¦ g, where g is

smooth and ¬‚at at each corner of c. So we may assume that c itself is smooth.

Since c is homotopic to zero, by approximation we may assume that there is a

smooth homotopy H : R2 ’ M with H1 |[0, 1] = c and H0 |[0, 1] = x0 . Then

ft := Pt(Ht , 1) is a curve in Hol0 (¦, x0 ) which is smooth as a mapping R—S ’ S.

The rest of the proof of claim 1 will follow.

Claim 2. ( dt ft ) —¦ ft’1 =: Zt is in g for all t.

d

To prove claim 2 we consider the pullback bundle H — E ’ R2 with the induced

connection H — ¦. It is su¬cient to prove claim 2 there. Let X = ds and Y = dt

d d

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

84 Chapter III. Bundles and connections

be the constant vector ¬elds on R2 , so [X, Y ] = 0. Then Pt(c, s) = FlCX |S and

s

so on. We put

ft,s = FlCX —¦ FlCY —¦ FlCX —¦ FlCY : S ’ S,

’s ’t s t

so ft,1 = ft . Then we have in the vector space X(S)

’1

( dt ft,s ) —¦ ft,s = ’(FlCX )— CY + (FlCX )— (FlCY )— (FlCX )— CY,

d

’s

s s t

1

’1 ’1

d d d

—¦ ( dt ft,s ) —¦ ft,s ds

( dt ft,1 ) ft,1 = ds

0

1

’(FlCX )— [CX, CY ] + (FlCX )— [CX, (FlCY )— (FlCX )— CY ]

= ’s

s s t

0

’(FlCX )— (FlCY )— (FlCX )— [CX, CY ] ds.

’s

s t

Since [X, Y ] = 0 we have [CX, CY ] = ¦[CX, CY ] = R(CX, CY ) and

(FlCX )— CY = C (FlX )— Y + ¦ (FlCX )— CY

t t t

t

CX —

d

= CY + dt ¦(Flt ) CY dt

0

t

¦(FlCX )— [CX, CY ] dt

= CY + t

0

t

¦(FlCX )— R(CX, CY ) dt

= CY + t

0

t

(FlCX )— R(CX, CY ) dt.

= CY + t

0

The ¬‚ows (FlC Xs )— and its derivative at 0 LCX = [CX, ] do not lead out of

’1

d

g, thus all parts of the integrand above are in g. So ( dt ft,1 ) —¦ ft,1 is in g for all

t and claim 2 follows.

Now claim 1 can be shown as follows. There is a unique smooth curve g(t)

d

in G0 satisfying Te (ρg(t) )Zt = Zt .g(t) = dt g(t) and g(0) = e; via the action of

G0 on S the curve g(t) is a curve of di¬eomorphisms on S, generated by the

time dependent vector ¬eld Zt , so g(t) = ft and f = f1 is in G0 . So we get

Hol0 (¦, x0 ) ⊆ G0 .

Claim 3. Hol0 (¦, x0 ) equals G0 .

In the proof of claim 1 we have seen that Hol0 (¦, x0 ) is a smoothly arcwise

connected subgroup of G0 , so it is a connected Lie subgroup by the results cited

in 5.6. It su¬ces thus to show that the Lie algebra g of G0 is contained in the

Lie algebra of Hol0 (¦, x0 ), and for that it is enough to show, that for each ξ in a

linearly spanning subset of g there is a smooth mapping f : [’1, 1] — S ’ S such

ˇ ˇ ˇ

that the associated curve f lies in Hol0 (¦, x0 ) with f (0) = 0 and f (0) = ξ.

By de¬nition we may assume ξ = Pt(c, 1)— R(CXx , CYx ) for Xx , Yx ∈ Tx M

and a smooth curve c in M from x0 to x. We extend Xx and Yx to vector ¬elds

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9. General ¬ber bundles and connections 85

X and Y ∈ X(M ) with [X, Y ] = 0 near x. We may also suppose that Z ∈ X(M )

is a vector ¬eld which extends c (t) along c(t): if c is simple we approximate it

by an embedding and can consequently extend c (t) to such a vector ¬eld. If c

is not simple we do this for each simple piece of c and have then several vector

¬elds Z instead of one below. So we have

ξ = (FlCZ )— R(CX, CY ) = (FlCZ )— [CX, CY ] since [X, Y ](x) = 0

1 1

2

= (FlCZ )— 1 dt2 |t=0 (FlCY —¦ FlCX —¦ FlCY —¦ FlCX ) by 3.16

d

’t ’t

1 t t

2

1 d2 CZ

—¦ FlCY —¦ FlCX —¦ FlCY —¦ FlCX —¦ FlCZ ),

2 dt2 |t=0 (Fl’1

= ’t ’t t t 1

where the parallel transport in the last equation ¬rst follows c from x0 to x, then

follows a small closed parallelogram near x in M (since [X, Y ] = 0 near x) and

then follows c back to x0 . This curve is clearly nullhomotopic.

Step 4. Now we make Hol(¦, x0 ) into a Lie group which we call G, by taking

Hol0 (¦, x0 ) = G0 as its connected component of the identity. Then the quotient

group Hol(¦, x0 )/ Hol0 (¦, x0 ) is countable, since the fundamental group π1 (M )

is countable (by Morse theory M is homotopy equivalent to a countable CW-

complex).

Step 5. Construction of a cocycle of transition functions with values in G. Let

(U± , u± : U± ’ Rm ) be a locally ¬nite smooth atlas for M such that each

u± : U± ’ Rm ) is surjective. Put x± := u’1 (0) and choose smooth curves c± :

±

[0, 1] ’ M with c± (0) = x0 and c± (1) = x± . For each x ∈ U± let cx : [0, 1] ’ M

±

be the smooth curve t ’ u’1 (t.u± (x)), then cx connects x± and x and the

± ±

mapping (x, t) ’ cx (t) is smooth U± — [0, 1] ’ M . Now we de¬ne a ¬bre bundle

±

’1

atlas (U± , ψ± : E|U± ’ U± — S) by ψ± (x, s) = Pt(cx , 1) Pt(c± , 1) s. Then ψ± is

±

CXx

x

smooth since Pt(c± , 1) = Fl1 for a local vector ¬eld Xx depending smoothly

on x. Let us investigate the transition functions.

’1

ψ± ψβ (x, s) = x, Pt(c± , 1)’1 Pt(cx , 1)’1 Pt(cx , 1) Pt(cβ , 1) s

± β

= x, Pt(cβ .cx .(cx )’1 .(c± )’1 , 4) s

β ±

=: (x, ψ±β (x) s), where ψ±β : U±β ’ G.

Clearly ψβ± : Uβ± — S ’ S is smooth which implies that ψβ± : Uβ± ’ G is

also smooth. (ψ±β ) is a cocycle of transition functions and we use it to glue

a principal bundle with structure group G over M which we call (P, p, M, G).

From its construction it is clear that the associated bundle P [S] = P —G S equals

(E, p, M, S).

Step 6. Lifting the connection ¦ to P .

For this we have to compute the Christo¬el symbols of ¦ with respect to the

atlas of step 5. To do this directly is quite di¬cult since we have to di¬erentiate

the parallel transport with respect to the curve. Fortunately there is another

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86 Chapter III. Bundles and connections

way. Let c : [0, 1] ’ U± be a smooth curve. Then we have

’1

ψ± (Pt(c, t)ψ± (c(0), s)) =

= c(t), Pt((c± )’1 , 1) Pt((cc(0) )’1 , 1) Pt(c, t) Pt(cc(0) , 1) Pt(c± , 1)s

± ±

= (c(t), γ(t).s),

where γ(t) is a smooth curve in the holonomy group G. Let “± ∈ „¦1 (U± , X(S))

be the Christo¬el symbol of the connection ¦ with respect to the chart (U± , ψ± ).

From the third proof of theorem 9.8 we have

’1

ψ± (Pt(c, t)ψ± (c(0), s)) = (c(t), γ (t, s)),

¯

where γ (t, s) is the integral curve through s of the time dependent vector ¬eld

¯

±d

“ ( dt c(t)) on S. But then we get

“± ( dt c(t))(¯ (t, s)) =

d d d d

γ dt γ (t, s) = dt (γ(t).s) =

¯ ( dt γ(t)).s,

( dt γ(t)) —¦ γ(t)’1 ∈ g.

“± ( dt c(t)) =

d d

So “± takes values in the Lie sub algebra of fundamental vector ¬elds for the

action of G on S. By theorem 11.9 below the connection ¦ is thus induced by a

principal connection ω on P . Since by 11.8 the principal connection ω has the

˜same™ holonomy group as ¦ and since this is also the structure group of P , the

principal connection ω is irreducible, see 11.7.

10. Principal ¬ber bundles and G-bundles

10.1. De¬nition. Let G be a Lie group and let (E, p, M, S) be a ¬ber bundle

as in 9.1. A G-bundle structure on the ¬ber bundle consists of the following

data:

(1) A left action : G — S ’ S of the Lie group on the standard ¬ber.

(2) A ¬ber bundle atlas (U± , ψ± ) whose transition functions (ψ±β ) act on S

via the G-action: There is a family of smooth mappings (•±β : U±β ’ G)

which satis¬es the cocycle condition •±β (x)•βγ (x) = •±γ (x) for x ∈

U±βγ and •±± (x) = e, the unit in the group, such that ψ±β (x, s) =

(•±β (x), s) = •±β (x).s.

A ¬ber bundle with a G-bundle structure is called a G-bundle. A ¬ber bundle

atlas as in (2) is called a G-atlas and the family (•±β ) is also called a cocycle of

transition functions, but now for the G-bundle. G is called the structure group

of the bundle.

To be more precise, two G-atlases are said to be equivalent (to describe the

same G-bundle), if their union is also a G-atlas. This translates as follows to

the two cocycles of transition functions, where we assume that the two coverings

of M are the same (by passing to the common re¬nement, if necessary): (•±β )

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10. Principal ¬ber bundles and G-bundles 87

and (•±β ) are called cohomologous if there is a family („± : U± ’ G) such that

•±β (x) = „± (x)’1 .•±β (x).„β (x) holds for all x ∈ U±β , compare with 6.4.

In (2) one should specify only an equivalence class of G-bundle structures

or only a cohomology class of cocycles of G-valued transition functions. The

proof of 6.4 now shows that from any open cover (U± ) of M , some cocycle of

transition functions (•±β : U±β ’ G) for it, and a left G-action on a manifold

S, we may construct a G-bundle, which depends only on the cohomology class

of the cocycle. By some abuse of notation we write (E, p, M, S, G) for a ¬ber

bundle with speci¬ed G-bundle structure.

Examples. The tangent bundle of a manifold M is a ¬ber bundle with structure

group GL(m). More general a vector bundle (E, p, M, V ) as in 6.1 is a ¬ber

bundle with standard ¬ber the vector space V and with GL(V )-structure.

10.2. De¬nition. A principal (¬ber) bundle (P, p, M, G) is a G-bundle with

typical ¬ber a Lie group G, where the left action of G on G is just the left

translation.

So by 10.1 we are given a bundle atlas (U± , •± : P |U± ’ U± — G) such

that we have •± •’1 (x, a) = (x, •±β (x).a) for the cocycle of transition functions

β

(•±β : U±β ’ G). This is now called a principal bundle atlas. Clearly the

principal bundle is uniquely speci¬ed by the cohomology class of its cocycle of

transition functions.

Each principal bundle admits a unique right action r : P — G ’ P , called the

principal right action, given by •± (r(•’1 (x, a), g)) = (x, ag). Since left and right

±

translation on G commute, this is well de¬ned. As in 5.10 we write r(u, g) = u.g

when the meaning is clear. The principal right action is visibly free and for any

ux ∈ Px the partial mapping rux = r(ux , ) : G ’ Px is a di¬eomorphism onto

the ¬ber through ux , whose inverse is denoted by „ux : Px ’ G. These inverses

together give a smooth mapping „ : P —M P ’ G, whose local expression is

„ (•’1 (x, a), •’1 (x, b)) = a’1 .b. This mapping is also uniquely determined by

± ±

the implicit equation r(ux , „ (ux , vx )) = vx , thus we also have „ (ux .g, ux .g ) =

g ’1 .„ (ux , ux ).g and „ (ux , ux ) = e.

When considering principal bundles the reader should think of frame bundles

as the foremost examples for this book. They will be treated in 10.11 below.

10.3. Lemma. Let p : P ’ M be a surjective submersion (a ¬bered manifold),

and let G be a Lie group which acts freely on P from the right such that the

orbits of the action are exactly the ¬bers p’1 (x) of p. Then (P, p, M, G) is a

principal ¬ber bundle.

If the action is a left one we may turn it into a right one by using the group

inversion if necessary.

Proof. Let s± : U± ’ P be local sections (right inverses) for p : P ’ M such that

(U± ) is an open cover of M . Let •’1 : U± — G ’ P |U± be given by •’1 (x, a) =

± ±

s± (x).a, which is obviously injective with invertible tangent mapping, so its

inverse •± : P |U± ’ U± — G is a ¬ber respecting di¬eomorphism. So (U± , •± )

is already a ¬ber bundle atlas. Let „ : P —M P ’ G be given by the implicit

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88 Chapter III. Bundles and connections

equation r(ux , „ (ux , ux )) = ux , where r is the right G-action. „ is smooth

by the implicit function theorem and clearly we have „ (ux , ux .g) = „ (ux , ux ).g

and •± (ux ) = (x, „ (s± (x), ux )). Thus we have •± •’1 (x, g) = •± (sβ (x).g) =

β

(x, „ (s± (x), sβ (x).g)) = (x, „ (s± (x), sβ (x)).g) and (U± , •± ) is a principal bundle

atlas.

10.4. Remarks. In the proof of lemma 10.3 we have seen, that a principal

bundle atlas of a principal ¬ber bundle (P, p, M, G) is already determined if we

specify a family of smooth sections of P , whose domains of de¬nition cover the

base M .

Lemma 10.3 can serve as an equivalent de¬nition for a principal bundle. But

this is true only if an implicit function theorem is available, so in topology

or in in¬nite dimensional di¬erential geometry one should stick to our original

de¬nition.

From the lemma itself it follows, that the pullback f — P over a smooth mapping

f : M ’ M is again a principal ¬ber bundle.

10.5. Homogeneous spaces. Let G be a Lie group with Lie algebra g. Let K

be a closed subgroup of G, then by theorem 5.5 K is a closed Lie subgroup whose

Lie algebra will be denoted by k. By theorem 5.11 there is a unique structure

of a smooth manifold on the quotient space G/K such that the projection p :

G ’ G/K is a submersion, so by the implicit function theorem p admits local

sections.

Theorem. (G, p, G/K, K) is a principal ¬ber bundle.

Proof. The group multiplication of G restricts to a free right action µ : G — K ’

G, whose orbits are exactly the ¬bers of p. By lemma 10.3 the result follows.

For the convenience of the reader we discuss now the best known homogeneous

spaces.

The group SO(n) acts transitively on S n’1 ‚ Rn . The isotropy group of the

˜north pole™ (1, 0, . . . , 0) is the subgroup

1 0

0 SO(n ’ 1)

which we identify with SO(n ’ 1). So S n’1 = SO(n)/SO(n ’ 1) and we have a

principal ¬ber bundle (SO(n), p, S n’1 , SO(n ’ 1)). Likewise

(O(n), p, S n’1 , O(n ’ 1)),

(SU (n), p, S 2n’1 , SU (n ’ 1)),

(U (n), p, S 2n’1 , U (n ’ 1)), and

(Sp(n), p, S 4n’1 , Sp(n ’ 1)) are principal ¬ber bundles.

The Grassmann manifold G(k, n; R) is the space of all k-planes containing 0

in Rn . The group O(n) acts transitively on it and the isotropy group of the

k-plane Rk — {0} is the subgroup

O(k) 0

,

O(n ’ k)

0

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10. Principal ¬ber bundles and G-bundles 89

therefore G(k, n; R) = O(n)/O(k) — O(n ’ k) is a compact manifold and we get

the principal ¬ber bundle (O(n), p, G(k, n; R), O(k) — O(n ’ k)). Likewise

˜

(SO(n), p, G(k, n; R), SO(k) — SO(n ’ k)),

(U (n), p, G(k, n; C), U (k) — U (n ’ k)), and

(Sp(n), p, G(k, n; H), Sp(k) — Sp(n ’ k)) are principal ¬ber bundles.

The Stiefel manifold V (k, n; R) is the space of all orthonormal k-frames in

n

R . Clearly the group O(n) acts transitively on V (k, n; R) and the isotropy

subgroup of (e1 , . . . , ek ) is Ik — O(n ’ k), so V (k, n; R) = O(n)/O(n ’ k) is a

compact manifold and (O(n), p, V (k, n; R), O(n ’ k)) is a principal ¬ber bundle.

But O(k) also acts from the right on V (k, n; R), its orbits are exactly the ¬bers

of the projection p : V (k, n; R) ’ G(k, n; R). So by lemma 10.3 we get a prin-

cipal ¬ber bundle (V (k, n, R), p, G(k, n; R), O(k)). Indeed we have the following

diagram where all arrows are projections of principal ¬ber bundles, and where

the respective structure groups are written on the arrows:

w V (k, n; R)

O(n ’ k)

O(n)

u u

(a) O(k) O(k)

w G(k, n; R)

V (n ’ k, n; R)

O(n ’ k)

It is easy to see that V (k, n) is also di¬eomorphic to the space { A ∈ L(Rk , Rn ) :

At .A = Ik }, i.e. the space of all linear isometries Rk ’ Rn . There are further-

more complex and quaternionic versions of the Stiefel manifolds.

Further examples will be given by means of jets in section 12.

10.6. Homomorphisms. Let χ : (P, p, M, G) ’ (P , p , M , G) be a principal

¬ber bundle homomorphism, i.e. a smooth G-equivariant mapping χ : P ’ P .

Then obviously the diagram

wP

χ

P

u u

p

(a) p

wM

M χ

commutes for a uniquely determined smooth mapping χ : M ’ M . For each

x ∈ M the mapping χx := χ|Px : Px ’ Pχ(x) is G-equivariant and therefore a

di¬eomorphism, so diagram (a) is a pullback diagram. We denote by PB(G) the

category of principal G-bundles and their homomorphisms.

But the most general notion of a homomorphism of principal bundles is the

following. Let ¦ : G ’ G be a homomorphism of Lie groups. χ : (P, p, M, G) ’

(P , p , M , G ) is called a homomorphism over ¦ of principal bundles, if χ : P ’

P is smooth and χ(u.g) = χ(u).¦(g) holds for all u ∈ P and g ∈ G. Then χ is

¬ber respecting, so diagram (a) makes again sense, but it is no longer a pullback

diagram in general. Thus we obtain the category PB of principal bundles and

their homomorphisms.

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90 Chapter III. Bundles and connections

If χ covers the identity on the base, it is called a reduction of the structure

group G to G for the principal bundle (P , p , M , G ) ” the name comes from

the case, when ¦ is the embedding of a subgroup.

By the universal property of the pullback any general homomorphism χ of

principal ¬ber bundles over a group homomorphism can be written as the com-

position of a reduction of structure groups and a pullback homomorphism as

follows, where we also indicate the structure groups:

&& w (χ P , G ) w (P , G )

—

(P, G)

&p&

&u

(

(b)

u

p

wM.

χ

M

10.7. Associated bundles. Let (P, p, M, G) be a principal bundle and let

: G — S ’ S be a left action of the structure group G on a manifold S. We

consider the right action R : (P — S) — G ’ P — S, given by R((u, s), g) =

(u.g, g ’1 .s).

Theorem. In this situation we have:

(1) The space P —G S of orbits of the action R carries a unique smooth

manifold structure such that the quotient map q : P — S ’ P —G S is a

submersion.

(2) (P —G S, p, M, S, G) is a G-bundle in a canonical way, where p : P —G S ’

¯ ¯

M is given by

wP—

q

P —S S

G

pr1

u u

(a) p

¯

w M.

p

P

In this diagram qu : {u} — S ’ (P —G S)p(u) is a di¬eomorphism for each

u ∈ P.

(3) (P — S, q, P —G S, G) is a principal ¬ber bundle with principal action R.

(4) If (U± , •± : P |U± ’ U± — G) is a principal bundle atlas with cocycle

of transition functions (•±β : U±β ’ G), then together with the left

action : G — S ’ S this cocycle is also one for the G-bundle (P —G

S, p, M, S, G).

¯

Notation. (P —G S, p, M, S, G) is called the associated bundle for the action

¯

: G — S ’ S. We will also denote it by P [S, ] or simply P [S] and we will

write p for p if no confusion is possible. We also de¬ne the smooth mapping

¯

’1

S

„ = „ : P —M P [S, ] ’ S by „ (ux , vx ) := qux (vx ). It satis¬es „ (u, q(u, s)) = s,

q(ux , „ (ux , vx )) = vx , and „ (ux .g, vx ) = g ’1 .„ (ux , vx ). In the special situation,

where S = G and the action is left translation, so that P [G] = P , this mapping

coincides with „ = „ G considered in 10.2. We denote by {u, s} ∈ P —G S the

G-orbit through (u, s) ∈ P — S.

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10. Principal ¬ber bundles and G-bundles 91

Proof. In the setting of the diagram in (2) the mapping p —¦ pr1 is constant on

the R-orbits, so p exists as a mapping. Let (U± , •± : P |U± ’ U± — G) be a

¯

principal bundle atlas with transition functions (•±β : U±β ’ G). We de¬ne

ψ± : U± — S ’ p’1 (U± ) ‚ P —G S by ψ± (x, s) = q(•’1 (x, e), s), which is ¬ber

’1 ’1

¯ ±

’1

respecting. For each orbit in p (x) ‚ P —G S there is exactly one s ∈ S such

¯

that this orbit passes through (•’1 (x, e), s), namely s = „ G (ux , •’1 (x, e))’1 .s

± ±

’1

if (ux , s ) is the orbit, since the principal right action is free. Thus ψ± (x, ) :

S ’ p’1 (x) is bijective. Furthermore

¯

ψβ (x, s) = q(•’1 (x, e), s)

’1

β

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

± ±

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

’1

±

’1

so ψ± ψβ (x, s) = (x, •±β (x).s) So (U± , ψ± ) is a G-atlas for P —G S and makes

it into a smooth manifold and a G-bundle. The de¬ning equation for ψ± shows

that q is smooth and a submersion and consequently the smooth structure on

P —G S is uniquely de¬ned, and p is smooth by the universal properties of a

¯

submersion.

By the de¬nition of ψ± the diagram

wU

•± — Id

p’1 (U± ) — S —G—S

±

u u

q Id —

(b)

wU

ψ±

p’1 (U± ) —S

¯ ±

commutes; since its lines are di¬eomorphisms we conclude that qu : {u} — S ’

p’1 (p(u)) is a di¬eomorphism. So (1), (2), and (4) are checked.

¯

(3) follows directly from lemma 10.3.

10.8. Corollary. Let (E, p, M, S, G) be a G-bundle, speci¬ed by a cocycle of

transition functions (•±β ) with values in G and a left action of G on S. Then

from the cocycle of transition functions we may glue a unique principal bundle

(P, p, M, G) such that E = P [S, ].

This is the usual way a di¬erential geometer thinks of an associated bundle.

He is given a bundle E, a principal bundle P , and the G-bundle structure then

is described with the help of the mappings „ and q. We remark that in standard

di¬erential geometric situations, the elements of the principal ¬ber bundle P play

the role of certain frames for the individual ¬bers of each associated ¬ber bundle

E = P [S, ]. Every frame u ∈ Px is interpreted as the above di¬eomorphism

q u : S ’ Ex .

10.9. Equivariant mappings and associated bundles.

1. Let (P, p, M, G) be a principal ¬ber bundle and consider two left actions

of G, : G — S ’ S and : G — S ’ S . Let furthermore f : S ’ S be

a G-equivariant smooth mapping, so f (g.s) = g.f (s) or f —¦ g = g —¦ f . Then

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92 Chapter III. Bundles and connections

IdP —f : P — S ’ P — S is equivariant for the actions R : (P — S) — G ’ P — S

and R : (P —S )—G ’ P —S and is thus a homomorphism of principal bundles,

so there is an induced mapping

w P —S

Id —f

P —S

u u

q q

(a)

wP—

Id —G f

P —G S S,

G

which is ¬ber respecting over M , and a homomorphism of G-bundles in the sense

of the de¬nition 10.10 below.

2. Let χ : (P, p, M, G) ’ (P , p , M , G) be a homomorphism of principal ¬ber

bundles as in 10.6. Furthermore we consider a smooth left action : G — S ’ S.

Then χ — IdS : P — S ’ P — S is G-equivariant (a homomorphism of principal

¬ber bundles) and induces a mapping χ—G IdS : P —G S ’ P —G S, which is ¬ber

respecting over M , ¬ber wise a di¬eomorphism, and again a homomorphism of

G-bundles in the sense of de¬nition 10.10 below.

3. Now we consider the situation of 1 and 2 at the same time. We have two

associated bundles P [S, ] and P [S , ]. Let χ : (P, p, M, G) ’ (P , p , M , G) be

a homomorphism principal ¬ber bundles and let f : S ’ S be an G-equivariant

mapping. Then χ — f : P — S ’ P — S is clearly G-equivariant and therefore

induces a mapping χ —G f : P [S, ] ’ P [S , ] which again is a homomorphism

of G-bundles.

4. Let S be a point. Then P [S] = P —G S = M . Furthermore let y ∈ S be

a ¬xed point of the action : G — S ’ S , then the inclusion i : {y} ’ S is

G-equivariant, thus IdP —i induces the mapping IdP —G i : M = P [{y}] ’ P [S ],

which is a global section of the associated bundle P [S ].

If the action of G on S is trivial, so g.s = s for all s ∈ S, then the associ-

ated bundle is trivial: P [S] = M — S. For a trivial principal ¬ber bundle any

associated bundle is trivial.

10.10. De¬nition. In the situation of 10.9, a smooth ¬ber respecting mapping

γ : P [S, ] ’ P [S , ] covering a smooth mapping γ : M ’ M of the bases is

called a homomorphism of G-bundles, if the following conditions are satis¬ed:

P is isomorphic to the pullback γ — P , and the local representations of γ in

pullback-related ¬ber bundle atlases belonging to the two G-bundles are ¬ber

wise G-equivariant.

Let us describe this in more detail now. Let (U± , ψ± ) be a G-atlas for P [S , ]

with cocycle of transition functions (•±β ), belonging to the principal ¬ber bundle

atlas (U± , •± ) of (P , p , M , G). Then the pullback-related principal ¬ber bundle

atlas (U± = γ ’1 (U± ), •± ) for P = γ — P as described in the proof of 9.5 has the

cocycle of transition functions (•±β = •±β —¦ γ); it induces the G-atlas (U± , ψ± )

’1

for P [S, ]. Then (ψ± —¦ γ —¦ ψ± )(x, s) = (γ(x), γ± (x, s)) and γ± (x, ) : S ’ S

is required to be G-equivariant for all ± and all x ∈ U± .

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10. Principal ¬ber bundles and G-bundles 93

Lemma. Let γ : P [S, ] ’ P [S , ] be a homomorphism of G-bundles as de-

¬ned above. Then there is a homomorphism χ : (P, p, M, G) ’ (P , p , M , G)

of principal bundles and a G-equivariant mapping f : S ’ S such that γ =

χ —G f : P [S, ] ’ P [S , ].

Proof. The homomorphism χ : (P, p, M, G) ’ (P , p , M , G) of principal ¬ber

bundles is already determined by the requirement that P = γ — P , and we have

γ = χ. The G-equivariant mapping f : S ’ S can be read o¬ the following

diagram which by the assumptions is seen to be well de¬ned in the right column:

wS

„S

P —M P [S]

u u

f

χ —M γ

(a)

wS

„S

P —M P [S ]

So a homomorphism of G-bundles is described by the whole triple (χ : P ’

P , f : S ’ S (G-equivariant), γ : P [S] ’ P [S ]), such that diagram (a)

commutes.

10.11. Associated vector bundles. Let (P, p, M, G) be a principal ¬ber bun-

dle, and consider a representation ρ : G ’ GL(V ) of G on a ¬nite dimensional

vector space V . Then P [V, ρ] is an associated ¬ber bundle with structure group

G, but also with structure group GL(V ), for in the canonically associated ¬ber

bundle atlas the transition functions have also values in GL(V ). So by section 6

P [V, ρ] is a vector bundle.

Now let F be a covariant smooth functor from the category of ¬nite dimen-

sional vector spaces and linear mappings into itself, as considered in section

6.7. Then clearly F —¦ ρ : G ’ GL(V ) ’ GL(F(V )) is another representa-

tion of G and the associated bundle P [F(V ), F —¦ ρ] coincides with the vector

bundle F(P [V, ρ]) constructed with the method of 6.7, but now it has an ex-

tra G-bundle structure. For contravariant functors F we have to consider the

representation F —¦ ρ —¦ ν, similarly for bifunctors. In particular the bifunctor

L(V, W ) may be applied to two di¬erent representations of two structure groups

of two principal bundles over the same base M to construct a vector bundle

L(P [V, ρ], P [V , ρ ]) = (P —M P )[L(V, V ), L —¦ ((ρ —¦ ν) — ρ )].

If (E, p, M ) is a vector bundle with n-dimensional ¬bers we may consider

the open subset GL(Rn , E) ‚ L(M — Rn , E), a ¬ber bundle over the base M ,

whose ¬ber over x ∈ M is the space GL(Rn , Ex ) of all invertible linear map-

pings. Composition from the right by elements of GL(n) gives a free right

action on GL(Rn , E) whose orbits are exactly the ¬bers, so by lemma 10.3 we

have a principal ¬ber bundle (GL(Rn , E), p, M, GL(n)). The associated bundle

GL(Rn , E)[Rn ] for the standard representation of GL(n) on Rn is isomorphic

to the vector bundle (E, p, M ) we started with, for the evaluation mapping

ev : GL(Rn , E) — Rn ’ E is invariant under the right action R of GL(n), and

locally in the image there are smooth sections to it, so it factors to a ¬ber linear

di¬eomorphism GL(Rn , E)[Rn ] = GL(Rn , E) —GL(n) Rn ’ E. The principal

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94 Chapter III. Bundles and connections

bundle GL(Rn , E) is called the linear frame bundle of E. Note that local sec-

tions of GL(Rn , E) are exactly the local frame ¬elds of the vector bundle E as

discussed in 6.5.

To illustrate the notion of reduction of structure group, we consider now

a vector bundle (E, p, M, Rn ) equipped with a Riemannian metric g, that is