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 determined by the cohomology class of its cocycle of transition

functions.

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

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

±

lation on G commute, this is well de¬ned. We write r(u, g) = u.g when the meaning

is clear. The principal right action is obviously 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 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.

37.9. Lemma. Let p : P ’ M be a surjective smooth mapping admitting local

smooth sections near each point in M , and let G be a Lie group which acts freely

on P such that the orbits of the action are exactly the ¬bers p’1 (x) of p. If the

unique mapping „ : P —M P ’ G satisfying ux .„ (ux , vx ) = vx is smooth, then

(p : P ’ M, G) is a principal ¬ber bundle.

Proof. Let the action be a right action by using the group inversion if neces-

sary. 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, with smooth inverse •± (ux ) = (x, „ (s± (x), ux )), a ¬ber re-

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

atlas. Clearly, we have „ (ux , ux .g) = „ (ux , ux ).g and •± (ux ) = (x, „ (s± (x), ux )),

so •± •’1 (x, g) = •± (sβ (x).g) = (x, „ (s± (x), sβ (x).g)) = (x, „ (s± (x), sβ (x)).g), and

β

(U± , •± ) is a principal bundle atlas.

37.9

37.12 37. Bundles and connections 381

37.10. Remarks. In the proof of Lemma (37.9) 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 (37.9) could serve as an equivalent de¬nition for a principal

bundle. From the lemma follows, that the pullback f — P over a smooth mapping

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

37.11. 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

w

M 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.

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

ing. 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. Then χ is ¬ber respecting, so diagram

(a) again makes sense, but it is not a pullback diagram in general.

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 composition 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

& u

(b) p

wM.

χ

¯

M

37.12. 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 ¬nal smooth

mapping.

37.12

382 Chapter VIII. In¬nite dimensional di¬erential geometry 37.12

(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 is also a cocycle 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 „ = „ S : P —M

¯

’1

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 „

considered in (37.8).

Proof. In the setting of diagram (a) 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

¯

’1

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 respecting. For

’1

¯ ±

’1

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 if (ux , s ) is the orbit,

± ±

since the principal right action is free. Thus, ψ± (x, ) : S ’ p’1 (x) is bijective.

’1

¯

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). Therefore, (U± , ψ± ) is a G-atlas for P —G S and

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

that q is smooth and admits local smooth sections, so it is ¬nal, consequently the

smooth structure on P —G S is uniquely de¬ned, and p is smooth. 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

¯ ±

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

±

involving lemma (36.12) shows that γ± ∈ „¦1 (P |U± , g) satis¬es the requirements of

lemma (37.19) and thus is a principal connection on P |U± . Now let (f± ) be a

smooth partition of unity on M which is subordinated to the open cover (U± ), and

let ω := ± (f± —¦ p)γ± . Since both requirements of lemma (37.19) are invariant

under convex linear combinations, ω is a principal connection on P .

37.22. Local descriptions of principal connections. We consider a principal

¬ber bundle (p : P ’ M, G) with some principal ¬ber bundle atlas (U± , •± :

P |U± ’ U± — G) and corresponding cocycle (•±β : U±β ’ G) of transition

functions. We consider the sections s± ∈ C ∞ (U± ← P |U± ) which are given by

•± (s± (x)) = (x, e) and satisfy s± .•±β = sβ , since we have in turn:

•± (sβ (x)) = •± •’1 (x, e) = (x, •±β (x))

β

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

± ±

Let ¦ = ζ —¦ ω ∈ „¦1 (P ; V P ) be a principal connection with connection form ω ∈

„¦1 (P, g). We may associate the following local data to the connection:

(1) ω± := s± — ω ∈ „¦1 (U± , g), the physicists™ version of the connection.

(2) The Christo¬el forms “± ∈ „¦1 (U± ; X(G)) from (37.5), which are given by

(0x , “± (ξx , g)) = ’T (•± ).¦.T (•± )’1 (ξx , 0g ).

Lemma. These local data have the following properties and are related by the fol-

lowing formulas.

(3) The forms ω± ∈ „¦1 (U± , g) satisfy the transition formulas

ω± = Ad(•’1 )ωβ + (•β± )— κl ,

β±

37.22

390 Chapter VIII. In¬nite dimensional di¬erential geometry 37.22

where κl ∈ „¦1 (G, g) is the left Maurer-Cartan form from (36.10). Any set

of such forms with this transition behavior determines a unique principal

connection.

(4) The local expression of ω is given by

(•’1 )— ω(ξx , T µg .X) = (•’1 )— ω(ξx , 0g ) + X = Ad(g ’1 )ω± (ξx ) + X.

± ±

(5) The Christo¬el form “± and ω± are related by

“± (ξx , g) = ’Te (µg ).Ad(g ’1 )ω± (ξx ) = ’Te (µg )ω± (ξx ),

thus the Christo¬el form is right invariant: “± (ξx ) = ’Rω± (ξx ) ∈ X(G).

(6) The local expression of ¦ is given by

(((•± )’1 )— ¦)(ξx , ·g ) = ’“± (ξx , g) + ·g = Te (µg ).ω± (ξx ) + ·g

= Rω± (ξx ) (g) + ·g

for ξx ∈ Tx U± and ·g ∈ Tg G.

(7) The local expression of the curvature R is given by

((•± )’1 )— R = ’R 1

dω± + 2 [ω± ,ω± ]§

g

so that R and „¦ are indeed ˜tensorial™ 2-forms.

Proof. We start with (4).

(•’1 )— ω(ξx , 0g ) = (•’1 )— ω(ξx , Te (µg )0e ) = (ω —¦ T (•± )’1 —¦ T (IdU± —µg ))(ξx , 0e ) =

± ±

= (ω —¦ T (rg —¦ •’1 ))(ξx , 0e ) = Ad(g ’1 )ω(T (•’1 )(ξx , 0e ))

± ±

= Ad(g ’1 )(s± — ω)(ξx ) = Ad(g ’1 )ω± (ξx ).

From this we get

(•’1 )— ω(ξx , T µg .X) = (•’1 )— ω(ξx , 0g ) + (•’1 )— ω(0x , T µg .X)

± ± ±

= Ad(g ’1 )ω± (ξx ) + ω(T (•± )’1 (0x , T µg .X))

= Ad(g ’1 )ω± (ξx ) + ω(ζX (•’1 (x, g)))

±

= Ad(g ’1 )ω± (ξx ) + X.

(5) From the de¬nition of the Christo¬el forms we have

(0x , “± (ξx , g)) = ’T (•± ).¦.T (•± )’1 (ξx , 0g )

= ’T (•± ).Te (r•’1 (x,g) )ω.T (•± )’1 (ξx , 0g )

±

= ’Te (•± —¦ r•’1 (x,g) )ω.T (•± )’1 (ξx , 0g )

±

= ’(0x , Te (µg )ω.T (•± )’1 (ξx , 0g ))

= ’(0x , Te (µg )Ad(g ’1 )ω± (ξx )) = ’(0x , Te (µg )ω± (ξx )).

37.22

37.23 37. Bundles and connections 391

(3) Via (5) the transition formulas for the ω± are easily seen to be equivalent to the

transition formulas for the Christo¬el forms in lemma (37.5). A direct proof goes

as follows: We have s± (x) = sβ (x)•β± (x) = r(sβ (x), •β± (x)) and thus

ω± (ξx ) = ω(Tx (s± ).ξx )

= (ω —¦ T(sβ (x),•β± (x)) r)((Tx sβ .ξx , 0•β± (x) ) ’ (0sβ (x) , Tx •β± .ξx ))

= ω(Tsβ (x) (r•β± (x) ).Tx (sβ ).ξx ) + ω(T•β± (x) (rsβ (x) ).Tx (•β± ).ξx )

= Ad(•β± (x)’1 )ω(Tx (sβ ).ξx )

+ ω(T•β± (x) (rsβ (x) ).T (µ•β± (x) —¦ µ•β± (x)’1 )Tx (•β± ).ξx )

= Ad(•β± (x)’1 )ωβ (ξx ) + ω(Te (rsβ (x)•β± (x) ).(•β± )— κl .ξx )

= Ad(•β± (x)’1 )ωβ (ξx ) + (•β± )— κl (ξx ).

(6) This is clear from the de¬nition of the Christo¬el forms and from (5).

Second proof of (7) First note that the right trivialization or framing (κr , πG ) :

T G ’ g — G induces the isomorphism R : C ∞ (G, g) ’ X(G), given by RX (x) =

Te (µx ).X(x). For the Lie bracket we then have

[RX , RY ] = R’[X,Y ]g +dY.RX ’dX.RY ,

R’1 [RX , RY ] = ’[X, Y ]g + RX (Y ) ’ RY (X).

We write a vector ¬eld on U± — G as (ξ, RX ) where ξ : G ’ X(U± ) and X ∈

C ∞ (U± — G, g). Then the local expression of the curvature is given by

(•± ’1 )— R((ξ, RX ), (·, RY )) = (•’1 )— (R((•± )— (ξ, RX ), (•± )— (·, RY )))

±

= (•’1 )— (¦[(•± )— (ξ, RX ) ’ ¦(•± )— (ξ, RX ), . . . ])

±

= (•’1 )— (¦[(•± )— (ξ, RX ) ’ (•± )— (Rω± (ξ) + RX ), . . . ])

±

= (•’1 )— (¦(•± )— [(ξ, ’Rω± (ξ) ), (·, ’Rω± (·) )])

±

= ((•’1 )— ¦)([ξ, ·]X(U± ) ’ Rω± (ξ) (·) + Rω± (·) (ξ),

±

’ ξ(Rω± (·) ) + ·(Rω± (ξ) ) + R’[ω± (ξ),ω± (·)]+Rω± (ξ) (ω± (·))’Rω± (ξ) (ω± (·)) )

= Rω± ([ξ,·]X(U± ) ’Rω± (ξ) (·)+Rω± (·) (ξ)) ’ Rξ(ω± (·)) + R·(ω± (ξ))

+ R’[ω± (ξ),ω± (·)]+Rω± (ξ) (ω± (·))’Rω± (ξ) (ω± (·))

= ’R .

1

(dω± + 2 [ω± ,ω± ]§ )(ξ,·)

g

37.23. The covariant derivative. Let (p : P ’ M, G) be a principal ¬ber

bundle with principal connection ¦ = ζ —¦ ω. We consider the horizontal projection

χ = IdT P ’¦ : T P ’ HP , cf. (37.2), which satis¬es χ —¦ χ = χ, im χ = HP ,

ker χ = V P , and χ —¦ T (rg ) = T (rg ) —¦ χ for all g ∈ G.

If W is a convenient vector space, we consider the mapping χ— : „¦(P, W ) ’

„¦(P, W ) which is given by

(χ— •)u (X1 , . . . , Xk ) = •u (χ(X1 ), . . . , χ(Xk )).

37.23

392 Chapter VIII. In¬nite dimensional di¬erential geometry 37.23

The mapping χ— is a projection onto the subspace of horizontal di¬erential forms,

i.e. the space „¦hor (P, W ) := {ψ ∈ „¦(P, W ) : iX ψ = 0 for X ∈ V P }. The notion of

horizontal form is independent of the choice of a connection.

The projection χ— has the following properties: χ— (• § ψ) = χ— • § χ— ψ if one of

the two forms has real values, χ— —¦ χ— = χ— , χ— —¦ (rg )— = (rg )— —¦ χ— for all g ∈ G,

χ— ω = 0, and χ— —¦ L(ζX ) = L(ζX ) —¦ χ— . All but the last easily follow from the

corresponding properties of χ. The last property uses that for a smooth curve g(t)

‚ ‚

in G with g(0) = e and ‚t 0 g(t) = X by (33.19) we have LζX = ‚t 0 rg(t) .

We de¬ne the covariant exterior derivative dω : „¦k (P, W ) ’ „¦k+1 (P, W ) by the

prescription dω := χ— —¦ d.

Theorem. The covariant exterior derivative dω has the following properties.

dω (• § ψ) = dω (•) § χ— ψ + (’1)deg • χ— • § dω (ψ) if • or ψ is real valued.

(1)

L(ζX ) —¦ dω = dω —¦ L(ζX ) for each X ∈ g.

(2)

(rg )— —¦ dω = dω —¦ (rg )— for each g ∈ G.

(3)

dω —¦ p— = d —¦ p— = p— —¦ d : „¦(M, W ) ’ „¦hor (P, W ).

(4)

(5)

dω ω = „¦, the curvature form.

(6)

dω „¦ = 0, the Bianchi identity.

dω —¦ χ— ’ dω = χ— —¦ i(R), where R is the curvature.

(7)

dω —¦ dω = χ— —¦ i(R) —¦ d.

(8)

Let „¦hor (P, g)G be the algebra of all horizontal G-equivariant g-valued forms,

(9)

i.e., (rg )— ψ = Ad(g ’1 )ψ. Then for any ψ ∈ „¦hor (P, g)G we have dω ψ =

dψ + [ω, ψ]§ .

(10) The mapping ψ ’ ζψ , where ζψ (X1 , . . . , Xk )(u) = ζψ(X1 ,...,Xk )(u) (u), is

an isomorphism between „¦hor (P, g)G and the algebra „¦hor (P, V P )G of all

horizontal G-equivariant forms with values in the vertical bundle V P . Then

we have ζdω ψ = ’[¦, ζψ ].

Proof. (1) through (4) follow from the properties of χ— .

(5) We have

(dω ω)(ξ, ·) = (χ— dω)(ξ, ·) = dω(χξ, χ·)

= (χξ)ω(χ·) ’ (χ·)ω(χξ) ’ ω([χξ, χ·])

= ’ω([χξ, χ·]) and

’ζ(„¦(ξ, ·)) = R(ξ, ·) = ¦[χξ, χ·] = ζω([χξ,χ·]) .

(6) Using (37.20) we have

dω „¦ = dω (dω + 1 [ω, ω]§ )

2

= χ— ddω + 1 χ— d[ω, ω]§

2

= 1 χ— ([dω, ω]§ ’ [ω, dω]§ ) = χ— [dω, ω]§

2

= [χ— dω, χ— ω]§ = 0, since χ— ω = 0.

37.23

37.23 37. Bundles and connections 393

(7) For • ∈ „¦(P, W ) we have

(dω χ— •)(X0 , . . . , Xk ) = (dχ— •)(χ(X0 ), . . . , χ(Xk ))

(’1)i χ(Xi )((χ— •)(χ(X0 ), . . . , χ(Xi ), . . . , χ(Xk )))

=

0¤i¤k

(’1)i+j (χ— •)([χ(Xi ), χ(Xj )],

+

i<j

χ(X0 ), . . . , χ(Xi ), . . . , χ(Xj ), . . . , χ(Xk ))

(’1)i χ(Xi )(•(χ(X0 ), . . . , χ(Xi ), . . . , χ(Xk )))

=

0¤i¤k

(’1)i+j •([χ(Xi ), χ(Xj )] ’ ¦[χ(Xi ), χ(Xj )],

+

i<j

χ(X0 ), . . . , χ(Xi ), . . . , χ(Xj ), . . . , χ(Xk ))

= (d•)(χ(X0 ), . . . , χ(Xk )) + (iR •)(χ(X0 ), . . . , χ(Xk ))

= (dω + χ— iR )(•)(X0 , . . . , Xk ).

(8) dω dω = dω χ— d = (χ— iR + χ— d)d = χ— iR d holds by (7).

(9) If we insert one vertical vector ¬eld, say ζX for X ∈ g, into dω ψ, we get 0

by de¬nition. For the right hand side we use iζX ψ = 0, and that by (33.19) for

‚

a smooth curve g(t) in G with g(0) = e and ‚t 0 g(t) = X we have LζX ψ =

g(t) —

) ψ = ‚t 0 Ad(g(t)’1 )ψ = ’ad(X)ψ in the computation

‚ ‚

‚t 0 (r

iζX (dψ + [ω, ψ]§ ) = iζX dψ + diζX ψ + [iζX ω, ψ] ’ [ω, iζX ψ]

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

Let now all vector ¬elds ξi be horizontal. Then we get

(dω ψ)(ξ0 , . . . , ξk ) = (χ— dψ)(ξ0 , . . . , ξk ) = dψ(ξ0 , . . . , ξk ),

(dψ + [ω, ψ]§ )(ξ0 , . . . , ξk ) = dψ(ξ0 , . . . , ξk ).

(10) We proceed in a similar manner. Let Ψ be in the space „¦hor (P, V P )G of all

horizontal G-equivariant forms with vertical values. Then for each X ∈ g we have

iζX Ψ = 0. Furthermore, the G-equivariance (rg )— Ψ = Ψ implies that LζX Ψ =

[ζX , Ψ] = 0 by (35.14.5). Using formula (35.9.2) we have

iζX [¦, Ψ] = [iζX ¦, Ψ] ’ [¦, iζX Ψ] + i([¦, ζX ])Ψ + i([Ψ, ζX ])¦

= [ζX , Ψ] ’ 0 + 0 + 0 = 0.

Let now all vector ¬elds ξi again be horizontal, then from the huge formula (35.5.1)

for the Fr¨licher-Nijenhuis bracket only the following terms in the fourth and ¬fth

o

line survive:

[¦, Ψ](ξ1 , . . . , ξ +1 ) =

(’1)

= sign σ ¦([Ψ(ξσ1 , . . . , ξσ ), ξσ( +1) ])

!

σ

1

+ sign σ ¦(Ψ([ξσ1 , ξσ2 ], ξσ3 , . . . , ξσ( +1) )).

( ’1)! 2!

σ

37.23

394 Chapter VIII. In¬nite dimensional di¬erential geometry 37.24

For f : P ’ g and horizontal ξ we have ¦[ξ, ζf ] = ζξ(f ) = ζdf (ξ) : It is C ∞ (P, R)-

linear in ξ; or imagine it in local coordinates. So the last expression becomes

’ζdψ(ξ0 ,...,ξk ) = ’ζ(dψ+[ω,ψ]§ )(ξ0 ,...,ξk ) ,

as required.

37.24. Inducing principal connections on associated bundles.

Let (p : P ’ M, G) be a principal bundle with principal right action r : P —G ’ P ,

and let : G — S ’ S be a left action of the structure group G on some manifold

S. Then we consider the associated bundle P [S] = P [S, ] = P —G S, constructed

in (37.12). Recall from (37.18) that its tangent and vertical bundles are given by

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

Let ¦ = ζ —¦ ω ∈ „¦1 (P ; T P ) be a principal connection on the principal bundle P .

¯

We construct the induced connection ¦ ∈ „¦1 (P [S]; T (P [S])) by factorizing as in

the following diagram:

w TP — TS w T (P — S)

¦ — Id =

TP — TS

u u u

Tq = q q Tq

w TP — w T (P —

¯

¦ =

T P —T G T S TS S).

TG G

Let us ¬rst check that the top mapping ¦ — Id is T G-equivariant. For g ∈ G and

X ∈ g the inverse of Te (µg )X in the Lie group T G from (38.10) is denoted by

(Te (µg )X)’1 . Furthermore, by (36.12) we have

T r(ξu , Te (µg )X) = Tu (rg )ξu + Tg (ru )(Te (µg )X) = Tu (rg )ξu + ζX (ug).

We compute

(¦ — Id)(T r(ξu , Te (µg )X), T ((Te (µg )X)’1 , ·s ))

= (¦(Tu (rg )ξu + ζX (ug)), T ((Te (µg )X)’1 , ·s ))

= (¦(Tu (rg )ξu ) + ¦(ζX (ug)), T ((Te (µg )X)’1 , ·s ))

= (Tu (rg )¦ξu + ζX (ug), T ((Te (µg )X)’1 , ·s ))

= (T r(¦(ξu ), Te (µg )X), T ((Te (µg )X)’1 , ·s )).

¯

So the mapping ¦ — Id factors to ¦ as indicated in the diagram, and we have

¯¯ ¯ ¯

¦ —¦ ¦ = ¦ from (¦ — Id) —¦ (¦ — Id) = ¦ — Id. The mapping ¦ is ¬berwise linear,

¯

since ¦ — Id and q = T q are. The image of ¦ is

q (V P — T S) = q (ker(T p : T P — T S ’ T M ))

= ker(T p : T P —T G T S ’ T M ) = V (P [S, ]).

¯

Thus, ¦ is a connection on the associated bundle P [S]. We call it the induced

connection.

37.24

37.25 37. Bundles and connections 395

From the diagram also follows that the vector valued forms ¦—Id ∈ „¦1 (P —S; T P —

¯

T S) and ¦ ∈ „¦1 (P [S]; T (P [S])) are (q : P — S ’ P [S])-related. So by (35.13) we

have for the curvatures

R¦—Id = 2 [¦ — Id, ¦ — Id] = 1 [¦, ¦] — 0 = R¦ — 0,

1

2

¯¯

R¦ = 1 [¦, ¦]

¯ 2

that they are also q-related, i.e., T q —¦ (R¦ — 0) = R¦ —¦ (T q —M T q).

¯

37.25. Recognizing induced connections. Let (p : P ’ M, G) be a principal

¬ber bundle, and let : G — S ’ S be a left action. We consider a connection

Ψ ∈ „¦1 (P [S]; T (P [S])) on the associated bundle P [S] = P [S, ]. Then the following

question arises: When is the connection Ψ induced by a principal connection on P ?

If this is the case, we say that Ψ is compatible with the G-structure on P [S]. The

answer is given in the following

Theorem. Let Ψ be a (general) connection on the associated bundle P [S]. Let us

suppose that the action is in¬nitesimally e¬ective, i.e. the fundamental vector

¬eld mapping ζ : g ’ X(S) is injective.

Then the connection Ψ is induced from a principal connection ω on P if and only

if the following condition is satis¬ed:

In some (equivalently any) ¬ber bundle atlas (U± , ψ± ) of P [S] belonging

to the G-structure of the associated bundle the Christo¬el forms “± ∈

„¦1 (U± ; X(S)) have values in the sub Lie algebra Xf und (S) of fundamental

vector ¬elds for the action .

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

by the proof of theorem (37.12) the induced ¬ber bundle atlas (U± , ψ± : P [S]|U± ’

U± — S) is given by

ψ± (x, s) = q(•’1 (x, e), s),

’1

(1) ±

(ψ± —¦ q)(•’1 (x, g), s) = (x, g.s).

(2) ±

¯

Let ¦ = ζ —¦ ω be a principal connection on P , and let ¦ be the induced connection

on the associated bundle P [S]. By (37.5), its Christo¬el symbols are given by

¯ ’1

(0x , “± (ξx , s)) = ’(T (ψ± ) —¦ ¦ —¦ T (ψ± ))(ξx , 0s )

¯

¦

¯

= ’(T (ψ± ) —¦ ¦ —¦ T q —¦ (T (•’1 ) — Id))(ξx , 0e , 0s ) by (1)

±

= ’(T (ψ± ) —¦ T q —¦ (¦ — Id))(T (•’1 )(ξx , 0e ), 0s ) by (37.24)

±

= ’(T (ψ± ) —¦ T q)(¦(T (•’1 )(ξx , 0e )), 0s )

±

= (T (ψ± ) —¦ T q)(T (•’1 )(0x , “± (ξx , e)), 0s ) by (37.22.2)

± ¦

= ’T (ψ± —¦ q —¦ (•’1 — Id))(0x , ω± (ξx ), 0s ) by (37.22.5)

±

= ’Te ( s )ω± (ξx ) by (2)

= ’ζω± (ξx ) (s).

37.25

396 Chapter VIII. In¬nite dimensional di¬erential geometry 37.26

So the condition is necessary.

For the converse let us suppose that a connection Ψ on P [S] is given such that the

Christo¬el forms “± with respect to a ¬ber bundle atlas of the G-structure have

Ψ

values in Xfund (S). Then unique g-valued forms ω± ∈ „¦1 (U± , g) are given by the

equation

“± (ξx ) = ζ(ω± (ξx )),

Ψ

since the action is in¬nitesimally e¬ective. From the transition formulas (37.5)

for the “± follow the transition formulas (37.22.3) for the ω ± , so that they they

Ψ

combine to a unique principal connection on P , which by the ¬rst part of the proof

induces the given connection Ψ on P [S].

37.26. Inducing principal connections on associated vector bundles. Let

(p : P ’ M, G) be a principal ¬ber bundle, and let ρ : G ’ GL(W ) be a represen-

tation of the structure group G on a convenient vector space W . See the beginning

of section (49) for a discussion of such representations. We consider the associated

vector bundle (p : E := P [W, ρ] ’ M, W ), see (37.12).

Recall from (29.9) that the tangent bundle T E = T P —T G T W has two vector

bundle structures, with the projections

πE : T E = T P —T G T W ’ P —G W = E,

T p —¦ pr1 : T E = T P —T G T W ’ T M,

respectively. Recall the vertical bundle V E = ker(T p) which is a vector subbundle

of πE : T E ’ E, and recall the vertical lift mapping vlE : E —M E ’ V E, which

is an isomorphism, (pr1 “πE )“¬berwise linear and also (p“T p)“¬berwise linear.

Now let ¦ = ζ —¦ ω ∈ „¦1 (P ; T P ) be a principal connection on P . We consider

¯

the induced connection ¦ ∈ „¦1 (E; T E) on the associated bundle E from (37.24).

A glance at the following diagram shows that the induced connection is linear in

both vector bundle structures. This property is expressed by calling it a linear

connection, see (37.27), on the associated vector bundle.

w TP — TW

&& ¦ — Id

xx

TP — TW TP — W — W

&&&

xxπ

π(

x

P —W

u

q

Tq Tq

x

x

P —G W = E

π

u xu

πE

E

xw T P —

TP — TW &

&& TW TE

x

TG TG

¯

¦

&( xxT p —¦ pr

&

x

T p —¦ pr 1 1

TM

37.26

37.28 37. Bundles and connections 397

¯

Now we de¬ne the connector K of the linear connection ¦ by

¯

K := pr2 —¦ (vlE )’1 —¦ ¦ : T E ’ V E ’ E —M E ’ E.

Lemma. The connector K : T E ’ E is a vector bundle homomorphism for both

vector bundle structures on T E and satis¬es K —¦ vlE = pr2 : E —M E ’ T E ’ E.

So K is πE “p“¬berwise linear and T p“p“¬berwise linear.

Proof. This follows from the ¬berwise linearity of the parts of K and from its

de¬nition.

37.27. Linear connections. If p : E ’ M is a vector bundle, a connection

Ψ ∈ „¦1 (E; T E) such that Ψ : T E ’ V E ’ T E is additionally T p“T p“¬berwise

linear is called a linear connection.

Equivalently, a linear connection may be speci¬ed by a connector K : T E ’ E

with the three properties of lemma (37.26). For then HE := {ξu : K(ξu ) = 0p(u) }

is a complement to V E in T E which is T p“¬berwise linearly chosen.

37.28. Covariant derivative on vector bundles. Let p : E ’ M be a vector

bundle with a linear connection, given by a connector K : T E ’ E with the

properties in lemma (37.26).

For any manifold N , smooth mapping s : N ’ E, and kinematic vector ¬eld

X ∈ X(N ) we de¬ne the covariant derivative of s along X by

:= K —¦ T s —¦ X : N ’ T N ’ T E ’ E.

(1) Xs

∞

If f : N ’ M is a ¬xed smooth mapping, let us denote by Cf (N, E) the vector

space of all smooth mappings s : N ’ E with p —¦ s = f ” they are called sections

of E along f . It follows from the universal property of the pullback that the vector

space Cf (N, E) is canonically linearly isomorphic to the space C ∞ (N ← f — E) of

∞

sections of the pullback bundle. Then the covariant derivative may be viewed as a

bilinear mapping

∞ ∞

: X(N ) — Cf (N, E) ’ Cf (N, E).

(2)

In particular, for f = IdM we have

: X(M ) — C ∞ (M ← E) ’ C ∞ (M ← E).

Lemma. This covariant derivative has the following properties:

is C ∞ (N, R)-linear in X ∈ X(N ). Moreover, for a tangent vector

(3) Xs

∞

Xx ∈ Tx N the mapping Xx : Cf (N, E) ’ Ef (x) makes sense, and we

have ( X s)(x) = X(x) s. Thus, s ∈ „¦1 (N ; f — E).

∞

(4) X s is R-linear in s ∈ Cf (N, E).

(5) X (h.s) = dh(X).s + h. X s for h ∈ C ∞ (N, R), the derivation property of

X.

(6) For any manifold Q, smooth mapping g : Q ’ N , and Yy ∈ Ty Q we have

Yy (s —¦ g). If Y ∈ X(Q) and X ∈ X(N ) are g-related, then we

T g.Yy s =

have Y (s —¦ g) = ( X s) —¦ g.

37.28

398 Chapter VIII. In¬nite dimensional di¬erential geometry 37.29

Proof. All these properties follow easily from de¬nition (1).

For vector ¬elds X, Y ∈ X(M ) and a section s ∈ C ∞ (M ← E) an easy computation

shows that

RE (X, Y )s := s’ ’

Xs [X,Y ] s

X Y Y

]’

= ([ X, [X,Y ] )s

Y

is C ∞ (M, R)-linear in X, Y , and s. By the method of (14.3), it follows that RE is a

2-form on M with values in the vector bundle L(E, E), i.e. RE ∈ „¦2 (M ; L(E, E)).

It is called the curvature of the covariant derivative.

∞

For f : N ’ M , vector ¬elds X, Y ∈ X(N ), and a section s ∈ Cf (N, E) along f

one can prove that

= (f — RE )(X, Y )s := RE (T f.X, T f.Y )s.

s’ ’

Xs [X,Y ] s

X Y Y

37.29. Covariant exterior derivative. Let p : E ’ M be a vector bundle with

a linear connection, given by a connector K : T E ’ E.

For a smooth mapping f : N ’ M let „¦(N ; f — E) be the vector space of all forms

on N with values in the vector bundle f — E. We can also view them as forms on N

with values along f in E, but we do not introduce an extra notation for this.

As in (32.1), (33.2), and (35.1) we have to assume the

Convention. We consider each derivation and homomorphism to be a sheaf mor-

phism (compare (32.1) and the de¬nition of modular 1-forms in (33.2)), or we

assume that all manifolds in question are smoothly regular.

The graded space „¦(N ; f — E) is a graded „¦(N )-module via

(• § ¦)(X1 , . . . , Xp+q ) =

1

= sign(σ) •(Xσ1 , . . . , Xσp )¦(Xσ(p+1) , . . . , Xσ(p+q) ).

p! q!

σ

Any A ∈ „¦p (N ; f — L(E, E)) de¬nes a graded module homomorphism

µ(A) : „¦(N ; f — E) ’ „¦(N ; f — E),

(1)

(µ(A)¦)(X1 , . . . , Xp+q ) =

1

= sign(σ) A(Xσ1 , . . . , Xσp )(¦(Xσ(p+1) , . . . , Xσ(p+q) )),

p! q!

σ

deg A. deg •

µ(A)(• § ¦) = (’1) • § µ(A)(¦).

But in general not all graded module homomorphisms are of this form, recall the

distinction between modular di¬erential forms and di¬erential forms in (33.2). This

is only true if the modeling spaces of N have the bornological approximation prop-

erty; the proof is as in (33.5).

37.29

37.29 37. Bundles and connections 399

The covariant exterior derivative is given by

d : „¦p (N ; f — E) ’ „¦p+1 (N ; f — E)

(2)

p

(’1)i

(d ¦)(X0 , . . . , Xp ) = Xi ¦(X0 , . . . , Xi , . . . , Xp )+

i=0

(’1)i+j ¦([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xp ),

+

0¤i<j¤p

where the Xi are vector ¬elds on N . It will be shown below that it is indeed well

de¬ned, i.e. that d ¦ ∈ „¦p+1 (N ; f — E). Now we only see that it is a modular

di¬erential form.

The covariant Lie derivative along a vector ¬eld X ∈ X(N ) is given by

LX : „¦p (N ; f — E) ’ „¦p (N ; f — E)

(3)

(LX ¦)(X1 , . . . , Xp ) = X (¦(X1 , . . . , Xp ))’

’ ¦(X1 , . . . , [X, Xi ], . . . , Xp ).

i

Again we will show below that it is well de¬ned. Finally we recall the insertion

operator

iX : „¦p (N ; f — E) ’ „¦p’1 (N ; f — E)

(4)

(LX ¦)(X1 , . . . , Xp’1 ) = ¦(X, X1 , . . . , Xp’1 )

Theorem. The covariant exterior derivative d , and the covariant Lie derivative

are well de¬ned and have the following properties.

For s ∈ C ∞ (N ← f — E) = „¦0 (N ; f — E) we have (d s)(X) = X s.

(5)

d (• § ¦) = d• § ¦ + (’1)deg • • § d ¦.

(6)

For smooth g : Q ’ N and ¦ ∈ „¦(N ; f — E) we have d (g — ¦) = g — (d ¦).

(7)

d d ¦ = µ(f — RE )¦.

(8)

iX (• § ¦) = iX • § ¦ + (’1)deg • • § iX ¦.

(9)

LX (• § ¦) = LX • § ¦ + • § LX ¦.

(10)

[LX , iY ] = LX —¦ iY ’ iY —¦ LX = i[X,Y ] .

(11)

[iX , d ] = iX —¦ d + d —¦ iX = LX .

(12)

Proof. By the formula above d ¦ is a priori de¬ned as a modular di¬erential form

and we have to show that it really lies in „¦(M ; f — E). For that let s— ∈ C ∞ (N ←

f — E ) be a local smooth section on U ⊆ N along f |U : U ’ M of the dual vector

bundle E ’ M . Then ¦, s— ∈ „¦k (N ), and for the canonical covariant derivative

on the dual bundle (write down its connector!) we have

d ¦, s— = d ¦, s— + (’1)k ¦, s—

E

§,

which shows that d ¦ ∈ „¦k (N, f — E) since d respects „¦— (N ) by (33.12).

37.29

400 Chapter VIII. In¬nite dimensional di¬erential geometry 37.29

(5) is just (33.11). (7) follows from (37.28.6).

(11) Take the di¬erence of the following two expressions:

(LX iY ¦)(Z1 , . . . , Zk ) = X ((iY ¦)(Z1 , . . . , Zk ))’

’ (iY ¦)(Z1 , . . . , [X, Zi ], . . . , Zk )

i

’

= X (¦(Y, Z1 , . . . , Zk )) ¦(Y, Z1 , . . . , [X, Zi ], . . . , Zk )

i

(iY LX ¦)(Z1 , . . . , Zk ) = LX ¦(Y, Z1 , . . . , Zk )

’ ¦([X, Y ], Z1 , . . . , Zk )’

= X (¦(Y, Z1 , . . . , Zk ))

’ ¦(Y, Z1 , . . . , [X, Zi ], . . . , Zk ).

i

(10) Let • be of degree p and ¦ of degree q. We prove the result by induction on

p + q. Suppose that (5) is true for p + q < k. Then for X we have by (9), by (11),

and by induction

(iY LX )(• § ¦) = (LX iY )(• § ¦) ’ i[X,Y ] (• § ¦)

= LX (iY • § ¦ + (’1)p • § iY ¦) ’ i[X,Y ] • § ¦ ’ (’1)p • § i[X,Y ] ¦

= LX iY • § ¦ + iY • § LX ¦ + (’1)p LX • § iY ¦+

+ (’1)p • § LX iY ¦ ’ i[X,Y ] • § ¦ ’ (’1)p • § i[X,Y ] ¦

iY (LX • § ¦ + • § LX ¦) = iY LX • § ¦ + (’1)p LX • § iY ¦+

+ iY • § LX ¦ + (’1)p • § iY LX ¦.

Using again (11), we get the result since the iY for all local vector ¬elds Y to-

gether act point separating on each space of di¬erential forms, in both cases of the

convention.

(12) We write out all relevant expressions.

(LX0 ¦)(X1 , . . . , Xk ) = X0 (¦(X1 , . . . , Xk ))+

k

(’1)0+j ¦([X0 , Xj ], X1 , . . . , Xj , . . . , Xk )

+

j=1

(iX0 d ¦)(X1 , . . . , Xk ) = d ¦(X0 , . . . , Xk )

k

(’1)i

= Xi (¦(X0 , . . . , Xi , . . . , Xk )) +

i=0

(’1)i+j ¦([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xk ).

+

0¤i<j

k

(’1)i’1

(d iX0 ¦)(X1 , . . . , Xk ) = Xi ((iX0 ¦)(X1 , . . . , Xi , . . . , Xk )) +

i=1

(’1)i+j’2 (iX0 ¦)([Xi , Xj ], X1 , . . . , Xi , . . . , Xj , . . . , Xk )

+

1¤i<j

37.29

37.30 37. Bundles and connections 401

k

(’1)i

=’ ’

Xi (¦(X0 , X1 , . . . , Xi , . . . , Xk ))

i=1

(’1)i+j ¦([Xi , Xj ], X0 , X1 , . . . , Xi , . . . , Xj , . . . , Xk ).

’

1¤i<j

By summing up, the result follows.

(6) We prove the result again by induction on p + q. Suppose that (6) is true for

p + q < k. Then for each local vector ¬eld X we have by (10), (9), (12), and by

induction

iX d (• § ¦) = LX (• § ¦) ’ d iX (• § ¦)

= LX • § ¦ + • § LX ¦ ’ d (iX • § ¦ + (’1)p • § iX ¦)

= iX d• § ¦ + d iX • § ¦ + • § iX d¦ + • § d iX ¦ ’ d iX • § ¦

’ (’1)p’1 iX • § d ¦ ’ (’1)p d • § iX ¦ ’ • § d iX ¦

= iX (d • § ¦ + (’1)p • § d ¦).

Since X is arbitrary, the result follows.

(8) follows from a direct computation. The usual fast proofs are not conclusive

in in¬nite dimensions. The computation is similar to the one for the proof of

(33.18.4), and only the de¬nitions (2) of d and (37.28) of RE , and the Jacobi

identity enter.

37.30. Let (p : P ’ M, G) be a principal ¬ber bundle, and let ρ : G ’ GL(W )

be a representation of the structure group G on a convenient vector space W , as in

(49.1).

Theorem. There is a canonical isomorphism from the space of P [W, ρ]-valued dif-

ferential forms on M onto the space of horizontal G-equivariant W -valued di¬er-

ential forms on P :

q : „¦(M ; P [W, ρ]) ’ „¦hor (P, W )G := {• ∈ „¦(P, W ) : iX • = 0

for all X ∈ V P, (rg )— • = ρ(g ’1 ) —¦ • for all g ∈ G},

In particular, for W = R with trivial representation we see that

p— : „¦(M ) ’ „¦hor (P )G = {• ∈ „¦hor (P ) : (rg )— • = •}

also is an isomorphism. We have q (• § ¦) = p— • § q ¦ for • ∈ „¦(M ) and

¦ ∈ „¦(M ; P [W ]).

The isomorphism

q : „¦0 (M ; P [W ]) = C ∞ (M ← P [W ]) ’ „¦0 (P, W )G = C ∞ (P, W )G

hor

is a special case of the one from (37.16).

37.30

402 Chapter VIII. In¬nite dimensional di¬erential geometry 37.31

Proof. Let • ∈ „¦k (P, W )G , X1 , . . . , Xk ∈ Tu P , and X1 , . . . , Xk ∈ Tu P such

hor

that Tu p.Xi = Tu p.Xi for each i. Then there is a g ∈ G such that ug = u :

q(u, •u (X1 , . . . , Xk )) = q(ug, ρ(g ’1 )•u (X1 , . . . , Xk ))

= q(u , ((rg )— •)u (X1 , . . . , Xk ))

= q(u , •ug (Tu (rg ).X1 , . . . , Tu (rg ).Xk ))

= q(u , •u (X1 , . . . , Xk )), since Tu (rg )Xi ’ Xi ∈ Vu P.

Thus, a vector bundle valued form ¦ ∈ „¦k (M ; P [W ]) is uniquely determined by

¦p(u) (Tu p.X1 , . . . , Tu p.Xk ) := q(u, •u (X1 , . . . , Xk )).

For the converse recall the smooth mapping „ W : P —M P [W, ρ] ’ W from (37.12),

which satis¬es „ W (u, q(u, w)) = w, q(ux , „ W (ux , vx )) = vx , and „ W (ux g, vx ) =

ρ(g ’1 )„ W (ux , vx ).

For ¦ ∈ „¦k (M ; P [W ]) we de¬ne q ¦ ∈ „¦k (P, W ) as follows. For Xi ∈ Tu P we put

(q ¦)u (X1 , . . . , Xk ) := „ W (u, ¦p(u) (Tu p.X1 , . . . , Tu p.Xk )).

Then q ¦ is smooth and horizontal. For g ∈ G we have

((rg )— (q ¦))u (X1 , . . . , Xk ) = (q ¦)ug (Tu (rg ).X1 , . . . , Tu (rg ).Xk )

= „ W (ug, ¦p(ug) (Tug p.Tu (rg ).X1 , . . . , Tug p.Tu (rg ).Xk ))

= ρ(g ’1 )„ W (u, ¦p(u) (Tu p.X1 , . . . , Tu p.Xk ))

= ρ(g ’1 )(q ¦)u (X1 , . . . , Xk ).

Clearly, the two constructions are inverse to each other.

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

¦ = ζ —¦ ω, and let ρ : G ’ GL(W ) be a representation of the structure group G on

a convenient vector space W , as in (49.1). We consider the associated vector bundle

¯

(p : E := P [W, ρ] ’ M, W ), the induced connection ¦ on it, and the corresponding

covariant derivative.

Theorem. The covariant exterior derivative dω from (37.23) on P and the covari-

ant exterior derivative for P [W ]-valued forms on M are connected by the mapping

q from (37.30), as follows:

q —¦ d = dω —¦ q : „¦(M ; P [W ]) ’ „¦hor (P, W )G .

Proof. Let us consider ¬rst f ∈ „¦0 (P, W )G = C ∞ (P, W )G , then f = q s for

hor

∞

s ∈ C (M ← P [W ]), and we have f (u) = „ W (u, s(p(u))) and s(p(u)) = q(u, f (u))

by (37.30) and (37.16). Therefore, we have T s.T p.Xu = T q(Xu , T f.Xu ), where

T f.Xu = (f (u), df (Xu )) ∈ T W = W — W . If χ : T P ’ HP is the horizontal

37.31

37.32 37. Bundles and connections 403

projection as in (37.23), we have T s.T p.Xu = T s.T p.χ.Xu = T q(χ.Xu , T f.χ.Xu ).

So we get

(q d s)(Xu ) = „ W (u, (d s)(T p.Xu ))

= „ W (u, T p.Xu s) by (37.29.5)

= „ W (u, K.T s.T p.Xu ) by (37.28.1)

= „ W (u, K.T q(χ.Xu , T f.χ.Xu )) from above

= „ W (u, pr2 .vl’1 .¦.T q(χ.Xu , T f.χ.Xu ))

¯ by (37.26)

P [W ]

„ W (u, pr2 .vl’1 ] .T q.(¦ — Id)(χ.Xu , T f.χ.Xu ))

= by (37.24)

P [W

„ W (u, pr2 .vl’1 ] .T q(0u , T f.χ.Xu ))

= since ¦.χ = 0

P [W

= „ W (u, q. pr2 .vl’1 .(0u , T f.χ.Xu )) since q is ¬ber linear

P —W

= „ W (u, q(u, df.χ.Xu )) = (χ— df )(Xu )

= (dω q s)(Xu ).

Now we turn to the general case. Let Yi for i = 0, . . . , k be local vector ¬elds on M ,

and let CYi be their horizontal lifts to P . Then T p.CYi = yi —¦ p, so Yi and CYi are

p-related. Since both q d ¦ and dω q are horizontal, it su¬ces to check that they

coincide on all local vector ¬elds of the form CYi . Since C[Yi , Yj ] = χ[CYi , CYj ],

we get from the special case above and the de¬nition of q :

(’1)i (CYi )(q ¦)(CY0 , . . . , CYi , . . . , CYk )

(dω q ¦)(CY0 , . . . , CYk ) =

0¤i¤k

(’1)i+j (q ¦)([CYi , CYj ], CY0 , . . . CYi . . . CYj . . . , CYk )

+

i<j

(’1)i (CYi )(q (¦(Y0 , . . . , Yi , . . . , Yk )