mapping T p : T E ’ T M and its kernel ker T p =: V E, which is called the vertical

bundle of E. It is a locally splitting vector subbundle of the tangent bundle T E,

by the following argument: E looks locally like UM — US , where UM is c∞ -open in

a modeling space WM of M and US in a modeling space WS of S. Then T E looks

locally like UM — WM — US — WS , and the mapping T p corresponds to (x, v, y, w) ’

(x, v), so that V E looks locally like UM — 0 — US — WS .

De¬nition. A connection on the ¬ber bundle (p : E ’ M, S) is a vector valued

1-form ¦ ∈ „¦1 (E; V E) with values in the vertical bundle V E such that ¦ —¦ ¦ = ¦

and im¦ = V E; so ¦ is just a projection T E ’ V E.

The kernel ker ¦ is a sub vector bundle of T E, it is called the space of horizontal

vectors or the horizontal bundle, and it is denoted by HE. Clearly, T E = HE •V E

and Tu E = Hu E • Vu E for u ∈ E.

Now we consider the mapping (T p, πE ) : T E ’ T M —M E. Then by de¬nition

(T p, πE )’1 (0p(u) , u) = Vu E, so (T p, πE ) | HE : HE ’ T M —M E is a ¬ber linear

isomorphism, which may be checked in a chart. Its inverse is denoted by

C := ((T p, πE ) | HE)’1 : T M —M E ’ HE ’ T E.

So C : T M —M E ’ T E is ¬ber linear over E and a right inverse for (T p, πE ). C

is called the horizontal lift associated to the connection ¦.

Note the formula ¦(ξu ) = ξu ’ C(T p.ξu , u) for ξu ∈ Tu E. So we can equally well

describe a connection ¦ by specifying C. Then we call ¦ vertical projection and

χ := idT E ’ ¦ = C —¦ (T p, πE ) will be called horizontal projection.

37.2

37.5 37. Bundles and connections 377

37.3. Curvature. If ¦ : T E ’ V E is a connection on the bundle (p : E ’ M, S),

then as in (35.11) the curvature R of ¦ is given by

2R = [¦, ¦] = [Id ’¦, Id ’¦] ∈ „¦2 (E; V E).

¯

The cocurvature R vanishes since the vertical bundle V E is integrable. We have

1

R(X, Y ) = 2 [¦, ¦](X, Y ) = ¦[χX, χY ] by (35.11), so R is an obstruction against

involutivity of the horizontal subbundle in the following sense: If the curvature

R vanishes, then horizontal kinematic vector ¬elds on E also have a horizontal

Lie bracket. Note that for vector ¬elds ξ, · ∈ X(M ) and their horizontal lifts

Cξ, C· ∈ X(E) we have R(Cξ, C·) = [Cξ, C·] ’ C([ξ, ·]). Since the vertical bundle

V E is even integrable, by (35.12) we have the Bianchi identity [¦, R] = 0.

37.4. Pullback. Let (p : E ’ M, S) be a ¬ber bundle, and consider a smooth

mapping f : N ’ M . Let us consider the pullback N —(f,M,p) E := {(n, e) ∈

N — E : f (n) = p(e)}; we will denote it by f — E. The following diagram sets up

w

some further notation for it: p— f

—

fE E

p

f —p

u u

w M.

f

N

Proposition. In the situation above we have:

(1) (f — E, f — p, N, S) is a ¬ber bundle, and p— f is a ¬berwise di¬eomorphism.

(2) If ¦ ∈ „¦1 (E; T E) is a connection on the bundle E, then the vector valued

form f — ¦, given by (f — ¦)u (X) := Tu (p— f )’1 .¦.Tu (p— f ).X for X ∈ Tu E, is

a connection on the bundle f — E. The forms f — ¦ and ¦ are p— f -related in

the sense of (35.13).

(3) The curvatures of f — ¦ and ¦ are also p— f -related.

Proof. (1) If (U± , ψ± ) is a ¬ber bundle atlas of (p : E ’ M, S) in the sense

of (37.1), then (f ’1 (U± ), (f — p, pr2 —¦ψ± —¦ p— f )) is visibly a ¬ber bundle atlas for

the pullback bundle (f — E, f — p, N, S). (2) is obvious. (3) follows from (2) and

(35.13.7).

37.5. Local description. Let ¦ be a connection on (p : E ’ M, S). Let us ¬x

a ¬ber bundle atlas (U± ) with transition functions (ψ±β ), and let us consider the

connection ((ψ± )’1 )— ¦ ∈ „¦1 (U± — S; U± — T S), which may be written in the form

(((ψ± )’1 )— ¦)(ξx , ·y ) =: ’“± (ξx , y) + ·y for ξx ∈ Tx U± and ·y ∈ Ty S,

since it reproduces vertical vectors. The “± are given by

(0x , “± (ξx , y)) := ’T (ψ± ).¦.T (ψ± )’1 .(ξx , 0y ).

We consider “± as an element of the space „¦1 (U± ; X(S)), a 1-form on U ± with

values in the Lie algebra X(S) of all kinematic vector ¬elds on the standard ¬ber.

The “± are called the Christo¬el forms of the connection ¦ with respect to the

bundle atlas (U± , ψ± ).

37.5

378 Chapter VIII. In¬nite dimensional di¬erential geometry 37.6

Lemma. The transformation law for the Christo¬el forms is

)).“β (ξx , y) = “± (ξx , ψ±β (x, y)) ’ Tx (ψ±β (

Ty (ψ±β (x, , y)).ξx .

The curvature R of ¦ satis¬es

(ψ± )— R = d“± + 1 [“± , “± ]§ .

’1

X(S)

2

Here d“± is the exterior derivative of the 1-form “± ∈ „¦1 (U± , X(S)) with values in

the convenient vector space X(S).

The formula for the curvature is the Maurer-Cartan formula which in this general

setting appears only on the level of local description.

Proof. From (ψ± —¦ (ψβ )’1 )(x, y) = (x, ψ±β (x, y)) we get that

T (ψ± —¦ (ψβ )’1 ).(ξx , ·y ) = (ξx , T(x,y) (ψ±β ).(ξx , ·y )), and thus

’1 ’1

T (ψβ ).(0x , “β (ξx , y)) = ’¦(T (ψβ )(ξx , 0y )) =

’1

’1

= ’¦(T (ψ± ).T (ψ± —¦ ψβ ).(ξx , 0y )) =

’1

= ’¦(T (ψ± )(ξx , T(x,y) (ψ±β )(ξx , 0y ))) =

’1 ’1

= ’¦(T (ψ± )(ξx , 0ψ±β (x,y) )) ’ ¦(T (ψ± )(0x , T(x,y) ψ±β (ξx , 0y ))) =

’1 ’1

= T (ψ± ).(0x , “± (ξx , ψ±β (x, y))) ’ T (ψ± )(0x , Tx (ψ±β ( , y)).ξx ).

This implies the transformation law.

For the curvature R of ¦ we have by (37.3) and (37.4.3)

(ψ± )— R ((ξ 1 , · 1 ), (ξ 2 , · 2 )) =

’1

= (ψ± )— ¦ [(Id ’(ψ± )— ¦)(ξ 1 , · 1 ), (Id ’(ψ± )— ¦)(ξ 2 , · 2 )] =

’1 ’1 ’1

= (ψ± )— ¦[(ξ 1 , “± (ξ 1 )), (ξ 2 , “± (ξ 2 ))] =

’1

= (ψ± )— ¦ [ξ 1 , ξ 2 ], ξ 1 “± (ξ 2 ) ’ ξ 2 “± (ξ 1 ) + [“± (ξ 1 ), “± (ξ 2 )] =

’1

= ’“± ([ξ 1 , ξ 2 ]) + ξ 1 “± (ξ 2 ) ’ ξ 2 “± (ξ 1 ) + [“± (ξ 1 ), “± (ξ 2 )] =

= d“± (ξ 1 , ξ 2 ) + [“± (ξ 1 ), “± (ξ 2 )]X(S) .

37.6. Parallel transport. Let ¦ be a connection on a bundle (p : E ’ M, S),

and let c : (a, b) ’ M be a smooth curve with 0 ∈ (a, b), c(0) = x. The parallel

transport along c is a smooth mapping Ptc : U ’ E, where U is a neighborhood

of Diag(a, b) —(c—¦pr2 ,M,p) E in (a, b) — (a, b) —(c—¦pr2 ,M,p) E, such that the following

properties hold:

(1) U © ((a, b) — {s} — {uc(s) }) is connected for each s ∈ (a, b) and each uc(s) ∈

Ec(s) .

(2) p(Pt(c, t, s, uc(s) )) = c(t) if de¬ned, and Pt(c, s, s, uc(s) ) = uc(s) .

d

(3) ¦( dt Pt(c, t, s, uc(s) )) = 0 if de¬ned.

37.6

37.7 37. Bundles and connections 379

(4) If Pt(c, t, s, uc(s) ) exists then Pt(c, r, s, uc(s) ) = Pt(c, r, t, Pt(c, t, s, uc(s) )) in

the sense that existence of both sides is equivalent and we have equality.

(5) U is maximal for properties (1) to (4).

(6) Pt also depends smoothly on c in the Fr¨licher space C ∞ ((a, b), M ), see

o

(23.1), in the following sense: For any smooth mapping c : R — (a, b) ’ M

we have: For each s ∈ (a, b), each r ∈ R, and each u ∈ Ec(r,s) there are

a neighborhood Uc,r,s,u of (r, s, s, u) in R — (a, b) — (a, b) —(c—¦pr1,3 ,M,p) E ‚

R — (a, b) — (a, b) — E such that Uc,r,s,u (r , t, s , u ) ’ Pt(c(r , ), t, s , u )

is de¬ned and smooth.

(7) Reparameterization invariance: If f : (a , b ) ’ (a, b) is smooth, then we

have Pt(c, f (t), f (s), uc(f (s)) ) = Pt(c —¦ f, t, s, uc(f (s)) )

Requirements (1) “ (5) are essential. (6) is a further requirement which is not nec-

essary for the uniqueness result below, and (7) is a consequence of this uniqueness

result.

Proposition. The parallel transport along c is unique if it exists.

Proof. Consider the pullback bundle (c— E, c— p, (a, b), S) and the pullback con-

nection c— ¦ on it. We shall need the horizontal lift C : T (a, b) —(a,b) c— E ’

T (c— E) = (T c)— (T E) associated to c— ¦, from (37.2). Consider the constant vec-

tor ¬eld ‚ ∈ X((a, b)) and its horizontal lift C(‚) ∈ X(c— E) which is given by

C(‚)(us ) = C(‚|s , us ) ∈ Tus (c— E). Now from the properties of the parallel trans-

port we see that t ’ Pt(c(s+ ), t, s, us ) is a ¬‚ow line of the horizontal vector

¬eld C(‚) with initial value us = (s, uc(s) ) ∈ (c— E)s ∼ {s} — Ec(s) . (3) says that

=

it has the ¬‚ow property, so that by uniqueness of the ¬‚ow (32.16) we see that

C(‚)

Pt(c, t) = Flt is unique if it exists.

At this place one could consider complete connections (those whose parallel trans-

port exists globally), which then give rise to holonomy groups, even for ¬ber bundles

without structure groups. In ¬nite dimensions some deep results are available, see

[Kol´ˇ, Michor, Slov´k, 1993, pp81].

ar a

37.7. De¬nition. Let G be a Lie group, and let (p : E ’ M, S) be a ¬ber bundle

as in (37.1). 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.

37.7

380 Chapter VIII. In¬nite dimensional di¬erential geometry 37.9

To be more precise, two G-atlas are said to be equivalent (to describe the same

G-bundle), if their union is also a G-atlas. This translates to the two cocycles of

transition functions as follows, where we assume that the two coverings of M are

the same (by passing to the common re¬nement, if necessary): (•±β ) 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 .

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. 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

(p : E ’ M, S, G) for a ¬ber bundle with speci¬ed G-bundle structure.

37.8. 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 (37.7) 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 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)