(

(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.

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

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

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

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± .

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

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

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

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

a section g ∈ C ∞ (S 2 E — ) such that gx is a positive de¬nite inner product on

Ex for each x ∈ M . Any vector bundle admits Riemannian metrics: local

existence is clear and we may glue with the help of a partition of unity on

M , since the positive de¬nite sections form an open convex subset. Now let

s = (s1 , . . . , sn ) ∈ C ∞ (GL(Rn , E)|U ) be a local frame ¬eld of the bundle E

over U ‚ M . Now we may apply the Gram-Schmidt orthonormalization pro-

cedure to the basis (s1 (x), . . . , sn (x)) of Ex for each x ∈ U . Since this proce-

dure is smooth (even real analytic), we obtain a frame ¬eld s = (s1 , . . . , sn )

of E over U which is orthonormal with respect to g. We call it an orthonor-

mal frame ¬eld. Now let (U± ) be an open cover of M with orthonormal frame

¬elds s± = (s± , . . . , s± ), where s± is de¬ned on U± . We consider the vector

n

1

bundle charts (U± , ψ± : E|U± ’ U± — Rn ) given by the orthonormal frame

’1

¬elds: ψ± (x, v 1 , . . . , v n ) = s± (x).v i =: s± (x).v. For x ∈ U±β we have

i

sj (x).gβ± i (x) for C -functions g±β j : U±β ’ R. Since s± (x) and

β j ∞

±

si (x) = i

s (x) are both orthonormal bases of Ex , the matrix g±β (x) = (g±β j (x)) is an

β

i

’1

± β

element of O(n). We write s = s .gβ± for short. Then we have ψβ (x, v) =

’1

’1

sβ (x).v = s± (x).g±β (x).v = ψ± (x, g±β (x).v) and consequently ψ± ψβ (x, v) =

(x, g±β (x).v). So the (g±β : U±β ’ O(n)) are the cocycle of transition functions

for the vector bundle atlas (U± , ψ± ). So we have constructed an O(n)-structure

on E. The corresponding principal ¬ber bundle will be denoted by O(Rn , (E, g));

it is usually called the orthonormal frame bundle of E. It is derived from the

linear frame bundle GL(Rn , E) by reduction of the structure group from GL(n)

to O(n). The phenomenon discussed here plays a prominent role in the theory

of classifying spaces.

10.12. Sections of associated bundles. Let (P, p, M, G) be a principal ¬ber

bundle and : G — S ’ S a left action. Let C ∞ (P, S)G denote the space

of all smooth mappings f : P ’ S which are G-equivariant in the sense that

f (u.g) = g ’1 .f (u) holds for g ∈ G and u ∈ P .

Theorem. The sections of the associated bundle P [S, ] correspond exactly

to the G-equivariant mappings P ’ S; we have a bijection C ∞ (P, S)G ∼ =

∞

C (P [S]).

Proof. If f ∈ C ∞ (P, S)G we get sf ∈ C ∞ (P [S]) by the following diagram:

w

(Id, f )

P —S

P

p

u u

q

(a)

w P [S]

sf

M

which exists by 10.9 since graph(f ) = (Id, f ) : P ’ P — S is G-equivariant:

(Id, f )(u.g) = (u.g, f (u.g)) = (u.g, g ’1 .f (u)) = ((Id, f )(u)).g.

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

10. Principal ¬ber bundles and G-bundles 95

If conversely s ∈ C ∞ (P [S]) we de¬ne fs ∈ C ∞ (P, S)G by fs := „ —¦(IdP —M s) :

P = P —M M ’ P —M P [S] ’ S. This is G-equivariant since fs (ux .g) =

„ (ux .g, s(x)) = g ’1 .„ (ux , s(x)) = g ’1 .fs (ux ) by 10.7. The two constructions are

inverse to each other since we have fs(f ) (u) = „ (u, sf (p(u))) = „ (u, q(u, f (u))) =

f (u) and sf (s) (p(u)) = q(u, fs (u)) = q(u, „ (u, s(p(u))) = s(p(u)).

The G-mapping fs : P ’ S determined by a section s of P [S] will be called

the frame form of the section s.

10.13. Theorem. Consider a principal ¬ber bundle (P, p, M, G) and a closed

subgroup K of G. Then the reductions of structure group from G to K corre-

¯

spond bijectively to the global sections of the associated bundle P [G/K, »] in a

¯

canonical way, where » : G—G/K ’ G/K is the left action on the homogeneous

space from 5.11.

Proof. By theorem 10.12 the section s ∈ C ∞ (P [G/K]) corresponds to fs ∈

¯

C ∞ (P, G/K)G , which is a surjective submersion since the action » : G—G/K ’

’1

G/K is transitive. Thus Ps := fs (¯) is a submanifold of P which is stable under

e

the right action of K on P . Furthermore the K-orbits are exactly the ¬bers of

the mapping p : Ps ’ M , so by lemma 10.3 we get a principal ¬ber bundle

(Ps , p, M, K). The embedding Ps ’ P is then a reduction of structure groups

as required.

If conversely we have a principal ¬ber bundle (P , p , M, K) and a reduction of

structure groups χ : P ’ P , then χ is an embedding covering the identity of M

and is K-equivariant, so we may view P as a sub ¬ber bundle of P which is stable

under the right action of K. Now we consider the mapping „ : P —M P ’ G

from 10.2 and restrict it to P —M P . Since we have „ (ux , vx .k) = „ (ux , vx ).k

for k ∈ K this restriction induces f : P ’ G/K by

wG

„

P —M P

p

u u

w G/K;

f

P = P —M P /K

and from „ (ux .g, vx ) = g ’1 .„ (ux , vx ) it follows that f is G-equivariant as re-

quired. Finally f ’1 (¯) = {u ∈ P : „ (u, Pp(u) ) ⊆ K } = P , so the two construc-

e

tions are inverse to each other.

10.14. The bundle of gauges. If (P, p, M, G) is a principal ¬ber bundle we

denote by Aut(P ) the group of all G-equivariant di¬eomorphisms χ : P ’ P .

Then p —¦ χ = χ —¦ p for a unique di¬eomorphism χ of M , so there is a group

homomorphism from Aut(P ) into the group Di¬(M ) of all di¬eomorphisms of

M . The kernel of this homomorphism is called Gau(P ), the group of gauge

transformations. So Gau(P ) is the space of all χ : P ’ P which satisfy p —¦ χ = p

and χ(u.g) = χ(u).g.

Theorem. The group Gau(P ) of gauge transformations is equal to the space

C ∞ (P, (G, conj))G ∼ C ∞ (P [G, conj]).

=

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

96 Chapter III. Bundles and connections

Proof. We use again the mapping „ : P —M P ’ G from 10.2. For χ ∈

Gau(P ) we de¬ne fχ ∈ C ∞ (P, (G, conj))G by fχ := „ —¦ (Id, χ). Then fχ (u.g) =

„ (u.g, χ(u.g)) = g ’1 .„ (u, χ(u)).g = conjg’1 fχ (u), so fχ is indeed G-equivariant.

If conversely f ∈ C ∞ (P, (G, conj))G is given, we de¬ne χf : P ’ P by

χf (u) := u.f (u). It is easy to check that χf is indeed in Gau(P ) and that the

two constructions are inverse to each other.

10.15. The tangent bundles of homogeneous spaces. Let G be a Lie

group and K a closed subgroup, with Lie algebras g and k, respectively. We

recall the mapping AdG : G ’ AutLie (g) from 4.24 and put AdG,K := AdG |K :

K ’ AutLie (g). For X ∈ k and k ∈ K we have AdG,K (k)X = AdG (k)X =

AdK (k)X ∈ k, so k is an invariant subspace for the representation AdG,K of K

in g, and we have the factor representation Ad⊥ : K ’ GL(g/k). Then

0 ’ k ’ g ’ g/k ’ 0

(a)

is short exact and K-equivariant.

Now we consider the principal ¬ber bundle (G, p, G/K, K) and the associated

vector bundles G[g/k, Ad⊥ ] and G[k, AdK ].

Theorem. In these circumstances we have

T (G/K) = G[g/k, Ad⊥ ] = (G —K g/k, p, G/K, g/k).

¯

The left action g ’ T (»g ) of G on T (G/K) corresponds to the canonical left

action of G on G—K g/k. Furthermore G[g/k, Ad⊥ ]•G[k, AdK ] is a trivial vector

bundle.

Proof. For p : G ’ G/K we consider the tangent mapping Te p : g ’ Te (G/K)¯