ψ±

p’1 (U± ) —S

¯ ±

37.12

37.14 37. Bundles and connections 383

commutes; since its horizontal arrows 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 (37.9). We give below an explicit chart construction.

We rewrite diagram (b) in the following form:

wq wV

»±

=

p’1 (U± ) — S ’1

—G

(V± ) ±

pr1

u u

q

(c)

wV

=

p’1 (U± )

¯ ±

Here V± := p’1 (U± ) ‚ P —G S, and the di¬eomorphism »± is de¬ned by stipulating

¯

»’1 (ψ± (x, s), g) := (•’1 (x, g), g ’1 .s). Then we have

’1

± ±

»’1 (ψ± (x, s), g) = »’1 (ψβ (x, •β± (x).s), g)

’1

’1

β β

= (•’1 (x, g), g ’1 .•β± (x).s)

β

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

±

= »’1 (ψ± (x, s), •±β (x).g),

’1

±

so »± »’1 (ψ± (x, s), g) = (ψ± (x, s), •±β (x).g), and (q : P — S ’ P —G S, G) is

’1 ’1

β

a principal bundle with structure group G and the same cocycle (•±β ) we started

with.

37.13. Corollary. Let (p : E ’ 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, ].

37.14. 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

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 (37.15) below.

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

homomorphism as in (37.11). Furthermore, we consider a smooth left action :

37.14

384 Chapter VIII. In¬nite dimensional di¬erential geometry 37.15

G — S ’ S. Then χ — IdS : P — S ’ P — S is G-equivariant and induces a

mapping χ —G IdS : P —G S ’ P —G S, which is ¬ber respecting over M , ¬berwise

a di¬eomorphism, and a homomorphism of G-bundles in the sense of de¬nition

(37.15) below.

(3) Now we consider the situations 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 principal ¬ber bundle homomorphism, and let f : S ’ S be a

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 also is a

homomorphism of G-bundles.

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

: G — S ’ S , then the inclusion i : {y} ’ S is G-

a ¬xpoint of the action

equivariant, thus IdP —i induces 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, i.e., g.s = s for all s ∈ S, then the associated

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

bundle is trivial.

37.15. De¬nition. In the situation of (37.14), 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 iso-

morphic to the pullback γ — P , and the local representations of γ in pullback-related

¯

¬ber bundle atlas belonging to the two G-bundles are ¬berwise 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 (37.4), 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) = (¯ (x), γ± (x, s)), and γ± (x, ) : S ’ S is

γ

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

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

wS

„S

P —M P [S]

u u

f

χ —M γ

(a)

wS,

S

„

P —M P [S ]

37.15

37.17 37. Bundles and connections 385

which by the assumptions is well de¬ned in the right column.

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.

37.16. 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 ∞ (M ← P [S]).

=

This result follows from (37.14) and (37.15). Since it is very important we include

a direct proof. That this is in general not an isomorphism of smooth structures will

become clear in the proof of (42.21) below.

Proof. If f ∈ C ∞ (P, S)G we construct sf ∈ C ∞ (M ← P [S]) in the following way.

graph(f ) = (Id, f ) : P ’ P — S is G-equivariant, since we have (Id, f )(u.g) =

(u.g, f (u.g)) = (u.g, g ’1 .f (u)) = ((Id, f )(u)).g. So it induces a smooth section

sf ∈ C ∞ (M ← P [S]) as seen from (37.14) and the diagram:

w

(Id, f )

P — {point} ∼ P P —S

=

p q

u u

(a)

w P [S].

sf

M

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

(IdP —M s) : P = P —M M ’ P —M P [S] ’ S. This is G-equivariant since

fs (ux .g) = „ S (ux .g, s(x)) = g ’1 .„ S (ux , s(x)) = g ’1 .fs (ux ) by (37.12). The two

constructions are inverse to each other since we have

fs(f ) (u) = „ S (u, sf (p(u))) = „ S (u, q(u, f (u))) = f (u),

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

37.17. 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 di¬eomorphisms χ : P ’ P which satisfy p —¦ χ = p and

χ(u.g) = χ(u).g.

37.17

386 Chapter VIII. In¬nite dimensional di¬erential geometry 37.18

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

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

=

If (p : P ’ M, G) is a ¬nite dimensional principal bundle then there exists a

structure of a Lie group on Gau(P ) = C ∞ (M ← P [G, conj]), modeled on Cc (M ←

∞

P [g, Ad]). This will be proved in (42.21) below.

Proof. We again use the mapping „ : P —M P ’ G from (37.8). 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 construc-

tions are inverse to each other, namely

χf (ug) = ugf (ug) = ugg ’1 f (u)g = χf (u)g,

fχf (u) = „ (u, χf (u)) = „ (u, u.f (u)) = „ (u, u)f (u) = f (u),

χfχ (u) = ufχ (u) = u„ (u, χ(u)) = χ(u).

37.18. Tangent bundles and vertical bundles. Let (p : E ’ M, S) be a ¬ber

bundle. Recall the vertical subbundle πE : V E = {ξ ∈ T E : T p.ξ = 0} ’ E of T E

from (37.2).

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

action r : P — G ’ P . Let : G — S ’ S be a left action. Then the following

assertions hold:

(1) (T p : T P ’ T M, T G) is a principal ¬ber bundle with principal right action

T r : T P — T G ’ T P , where the structure group T G is the tangent group

of G, see (38.10).

(2) The vertical bundle (π : V P ’ P, g) of the principal bundle is trivial as a

vector bundle over P : V P ∼ P — g.

=

(3) The vertical bundle of the principal bundle as bundle over M is a principal

bundle: (p —¦ π : V P ’ M, T G).

(4) The tangent bundle of the associated bundle P [S, ] is given by

T (P [S, ]) = T P [T S, T ].

(5) The vertical bundle of the associated bundle P [S, ] is given by

V (P [S, ]) = P [T S, T2 ] = P —G T S.

Proof. Let (U± , •± : P |U± ’ U± — G) be a principal ¬ber bundle atlas with

cocycle of transition functions (•±β : U±β ’ G). Since T is a functor which respects

products, (T U± , T •± : T P |T U± ’ T U± — T G) is a principal ¬ber bundle atlas with

cocycle of transition functions (T •±β : T U±β ’ T G), describing the principal ¬ber

bundle (T p : T P ’ T M, T G). The assertion about the principal action is obvious.

So (1) follows. For completeness™ sake, we include here the transition formula for

this atlas in the right trivialization of T G:

T (•± —¦ •’1 )(ξx , Te (µg ).X) = (ξx , Te (µ•±β (x).g ).(δ r •±β (ξx ) + Ad(•±β (x))X)),

β

37.18

37.19 37. Bundles and connections 387

where δ r •±β ∈ „¦1 (U±β , g) is the right logarithmic derivative of •±β , see (38.1)

below.

(2) The mapping (u, X) ’ Te (ru ).X = T(u,e) r.(0u , X) is a vector bundle isomor-

phism P — g ’ V P over P .

(3) Obviously, T r : T P — T G ’ T P is a free right action which acts transitively

on the ¬bers of T p : T P ’ T M . Since V P = (T p)’1 (0M ), the bundle V P ’ M is

isomorphic to T P |0M and T r restricts to a free right action, which is transitive on

the ¬bers, so by lemma (37.9) the result follows.

(4) The transition functions of the ¬ber bundle P [S, ] are given by the expression

—¦ (•±β — IdS ) : U±β — S ’ G — S ’ S. Then the transition functions of T (P [S, ])

are T ( —¦ (•±β — IdS )) = T —¦ (T •±β — IdT S ) : T U±β — T S ’ T G — T S ’ T S, from

which the result follows.

(5) Vertical vectors in T (P [S, ]) have local representations (0x , ·s ) ∈ T U±β —

T S. Under the transition functions of T (P [S, ]) they transform as T ( —¦ (•±β —

IdS )).(0x , ·s ) = T .(0•±β (x) , ·s ) = T ( •±β (x) ).·s = T2 .(•±β (x), ·s ), and this im-

plies the result

37.19. Principal connections. Let (p : P ’ M, G) be a principal ¬ber bundle.

Recall from (37.2) that a (general) connection on P is a ¬ber projection ¦ : T P ’

V P , viewed as a 1-form in „¦1 (P ; V P ) ‚ „¦1 (P ; T P ). Such a connection ¦ is

called a principal connection if it is G-equivariant for the principal right action

r : P — G ’ P , so that T (rg ).¦ = ¦.T (rg ) and ¦ is rg -related to itself, or

(rg )— ¦ = ¦ in the sense of (35.13), for all g ∈ G. By theorem (35.13.7), the

1

curvature R = 2 .[¦, ¦] is then also rg -related to itself for all g ∈ G.

Recall from (37.18.2) that the vertical bundle of P is trivialized as a vector bundle

over P by the principal action. So ω(Xu ) := Te (ru )’1 .¦(Xu ) ∈ g, and in this way

we get a g-valued 1-form ω ∈ „¦1 (P, g), which is called the (Lie algebra valued)

connection form of the connection ¦. Recall from (36.12) the fundamental vector

¬eld mapping ζ : g ’ X(P ) for the principal right action. The de¬ning equation

for ω can be written also as ¦(Xu ) = ζω(Xu ) (u).

Lemma. If ¦ ∈ „¦1 (P ; V P ) is a principal connection on the principal ¬ber bundle

(p : P ’ M, G) then the connection form has the following three properties:

(1) ω reproduces the generators of fundamental vector ¬elds, so that we have

ω(ζX (u)) = X for all X ∈ g.

(2) ω is G-equivariant, ((rg )— ω)(Xu ) = ω(Tu (rg ).Xu ) = Ad(g ’1 ).ω(Xu ) for all

g ∈ G and Xu ∈ Tu P .

(3) We have for the Lie derivative LζX ω = ’ad(X).ω.

Conversely, a 1-form ω ∈ „¦1 (P, g) satisfying (1) de¬nes a connection ¦ on P

by ¦(Xu ) = Te (ru ).ω(Xu ), which is a principal connection if and only if (2) is

satis¬ed.

Proof. (1) Te (ru ).ω(ζX (u)) = ¦(ζX (u)) = ζX (u) = Te (ru ).X. Since Te (ru ) : g ’

Vu P is an isomorphism, the result follows.

37.19

388 Chapter VIII. In¬nite dimensional di¬erential geometry 37.20

(2) Both directions follow from

Te (rug ).ω(Tu (rg ).Xu ) = ζω(Tu (rg ).Xu ) (ug) = ¦(Tu (rg ).Xu )

Te (rug ).Ad(g ’1 ).ω(Xu ) = ζAd(g’1 ).ω(Xu ) (ug) = Tu (rg ).ζω(Xu ) (u)

= Tu (rg ).¦(Xu ).

‚

(3) Let g(t) be a smooth curve in G with g(0) = e and ‚t |0 g(t) = X. Then

‚

•t (u) = r(u, g(t)) is a smooth curve of di¬eomorphisms on P with ‚t |0 •t = ζX ,

and by the ¬rst claim of lemma (33.19) we have

|0 (rg(t) )— ω |0 Ad(g(t)’1 )ω

‚ ‚

LζX ω = = ’ad(X)ω.

=

‚t ‚t

37.20. Curvature. Let ¦ be a principal connection on the principal ¬ber bundle

(p : P ’ M, G) with connection form ω ∈ „¦1 (P, g). We have already noted in

(37.19) that also the curvature R = 1 [¦, ¦] is then G-equivariant, (rg )— R = R for

2

all g ∈ G. Since R has vertical values we may de¬ne a g-valued 2-form „¦ ∈ „¦2 (P, g)

by „¦(Xu , Yu ) := ’Te (ru )’1 .R(Xu , Yu ), which is called the (Lie algebra-valued)

curvature form of the connection. We also have R(Xu , Yu ) = ’ζ„¦(Xu ,Yu ) (u). We

take the negative sign here to get in ¬nite dimensions the usual curvature form.

We equip the space „¦(P, g) of all g-valued forms on P in a canonical way with the

structure of a graded Lie algebra by

[Ψ, ˜]g (X1 , . . . , Xp+q ) =

(1) §

1

= signσ [Ψ(Xσ1 , . . . , Xσp ), ˜(Xσ(p+1) , . . . , Xσ(p+q) )]g

p! q!

σ

or equivalently by [ψ —X, θ—Y ]§ := ψ §θ—[X, Y ]g . From the latter description it is

clear that d[Ψ, ˜]§ = [dΨ, ˜]§ + (’1)deg Ψ [Ψ, d˜]§ . In particular, for ω ∈ „¦1 (P, g)

we have [ω, ω]§ (X, Y ) = 2[ω(X), ω(Y )]g .

Theorem. The curvature form „¦ of a principal connection with connection form

ω has the following properties:

(2) „¦ is horizontal, i.e., it kills vertical vectors.

(3) „¦ is G-equivariant in the following sense: (rg )— „¦ = Ad(g ’1 ).„¦. Conse-

quently, LζX „¦ = ’ad(X).„¦.

(4) The Maurer-Cartan formula holds: „¦ = dω + 1 [ω, ω]§ .

2

Proof. (2) is true for R by (37.3). For (3) we compute as follows:

Te (rug ).((rg )— „¦)(Xu , Yu ) = Te (rug ).„¦(Tu (rg ).Xu , Tu (rg ).Yu ) =

= ’Rug (Tu (rg ).Xu , Tu (rg ).Yu ) = ’Tu (rg ).((rg )— R)(Xu , Yu ) =

= ’Tu (rg ).R(Xu , Yu ) = Tu (rg ).ζ„¦(Xu ,Yu ) (u) =

= ζAd(g’1 ).„¦(Xu ,Yu ) (ug) = Te (rug ).Ad(g ’1 ).„¦(Xu , Yu ), by (36.12.2).

37.20

37.22 37. Bundles and connections 389

Proof of (4) For X ∈ g we have iζX R = 0 by (2), and using (37.19.3) we get

1 1 1

iζX (dω + [ω, ω]§ ) = iζX dω + [iζX ω, ω]§ ’ [ω, iζX ω]§ =

2 2 2

= LζX ω + [X, ω]§ = ’ad(X)ω + ad(X)ω = 0.

So the formula holds if one vector is vertical, and for horizontal vector ¬elds X, Y ∈

C ∞ (P ← H(P )) we have

R(X, Y ) = ¦[X ’ ¦X, Y ’ ¦Y ] = ¦[X, Y ] = ζω([X,Y ]) ,

1

(dω + [ω, ω]§ )(X, Y ) = Xω(Y ) ’ Y ω(X) ’ ω([X, Y ]) + 0 = ’ω([X, Y ]).

2

37.21. Lemma. Any principal ¬ber bundle (p : P ’ M, G) with smoothly para-

compact basis M admits principal connections.

Proof. Let (U± , •± : P |U± ’ U± — G)± be a principal ¬ber bundle atlas. Let us

de¬ne γ± (T •’1 (ξx , Te µg .X)) := X for ξx ∈ Tx U± and X ∈ g. An easy computation

±