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

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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

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

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&