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 )) =

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

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

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

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

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

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