C ∞ (R, g) =

t

(X, ·) ’ t ’ · + X(s)ds

0

(Y , Y (0)) ← Y,

where on the left hand side the Lie bracket is given by

[(X1 , ·1 ), (X2 , ·2 )] =

t t

= t’[ X2 (s) ds]g + [·1 , X2 (t)]g ’ [·2 , X1 (t)]g ,

X1 (s) ds, X2 (t)]g + [X1 (t),

0 0

[·1 , ·2 ]g ,

and where on the right hand side the bracket is given by

[X, Y ](t) = [X(t), Y (t)]g .

On the right hand sides the evolution operator is

Evolr ∞ (R,G) = C ∞ (R, Evolr ).

C G

38.14. Remarks. Let G be a connected regular Lie group. The smooth homo-

morphism evolr : C ∞ (R, g) ’ G admits local smooth sections. Namely, using a

G

smooth chart near e of G we can choose a smooth curve cg : R ’ G with cg (0) = e

and cg (1) = g, depending smoothly on g for g near e. Then s(g) := δ r cg is a local

smooth section. We have an extension of groups

evolr

∞ G

0 ’ K ’ C (R, g) ’ ’ G ’ {e}

’’

where K = ker(evolr ) is isomorphic to the smooth group {f ∈ C ∞ (R, G) : f (0) =

G

e, f (1) = e} via the mapping Evolr . We do not know whether K is a submanifold.

G

Next we consider the smooth group C ∞ ((S 1 , 1), (G, e)) of all smooth mappings f :

S 1 ’ G with f (1) = e. With pointwise multiplication this is a splitting closed nor-

mal subgroup of the regular Lie group C ∞ (S 1 , G) with the manifold structure given

in (42.21). Moreover, C ∞ (S 1 , G) is the semidirect product C ∞ ((S 1 , 1), (G, e)) G,

where G acts by conjugation on C ∞ ((S 1 , 1), (G, e)). So by theorem (38.6) the

subgroup C ∞ ((S 1 , 1), (G, e)) is also regular.

The right logarithmic derivative for smooth loops δ r : C ∞ (S 1 , G) ’ C ∞ (S 1 , g)

restricts to a di¬eomorphism C ∞ ((S 1 , 1), (G, e)) ’ ker(evolG ) ‚ C ∞ (S 1 , g), thus

the kernel ker(evolG : C ∞ (S 1 , g) ’ G) is a regular Lie group which is isomorphic

to C ∞ ((S 1 , 1), (G, e)). It is also a subgroup (via pullback by the covering mapping

e2πit : R ’ S 1 ) of the regular Lie group (C ∞ (R, g), —). Note that C ∞ (S 1 , g) is not

a subgroup, since it is not closed under the product — if G is not abelian.

38.14

422 Chapter VIII. In¬nite dimensional di¬erential geometry 39.1

39. Bundles with Regular Structure Groups

39.1. Theorem. Let (p : P ’ M, G) be a smooth (locally trivial) principal bun-

dle with a regular Lie group as structure group. Let ω ∈ „¦1 (P, g) be a principal

connection form.

Then the parallel transport for the principal connection exists, is globally de¬ned,

and is G-equivariant. In detail: For each smooth curve c : R ’ M there is a unique

smooth mapping Ptc : R — Pc(0) ’ P such that the following holds:

d

(1) Pt(c, t, u) ∈ Pc(t) , Pt(c, 0) = IdPc(0) , and ω( dt Pt(c, t, u)) = 0.

It has the following further properties:

(2) Pt(c, t) : Pc(0) ’ Pc(t) is G-equivariant, i.e. Pt(c, t, u.g) = Pt(c, t, u).g holds

for all g ∈ G and u ∈ P . Moreover, we have Pt(c, t)— (ζX |Pc(t) ) = ζX |Pc(0)

for all X ∈ g.

(3) For any smooth function f : R ’ R we have

Pt(c, f (t), u) = Pt(c —¦ f, t, Pt(c, f (0), u)).

(4) The parallel transport is smooth as a mapping

Pt : C ∞ (R, M ) —(ev0 ,M,p—¦pr2 ) (R — P ) ’ P,

where C ∞ (R, M ) is considered as a smooth space, see (23.1). If M is a

smooth manifold with a local addition (see (42.4) below), then this holds for

C ∞ (R, M ) replaced by the smooth manifold C∞ (R, M ).

Proof. For a principal bundle chart (U± , •± ) we have the data from (37.22)

s± (x) := •’1 (x, e),

±

ω± := s— ω,

±

ω —¦ T (•’1 ) = (•’1 )— ω ∈ „¦1 (U± — G, g),

± ±

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

± ±

For a smooth curve c : R ’ M the horizontal lift Pt(c, , u) through u ∈

Pc(0) is given among all smooth lifts of c by the ordinary di¬erential equation

d

ω( dt Pt(c, t, u)) = 0 with initial condition Pt(c, 0, u) = u. Locally, we have

•± (Pt(c, t, u)) = (c(t), γ(t)),

so that

0 = Ad(γ(t))ω( dt Pt(c, t, u)) = Ad(γ(t))(ω —¦ T (•’1 ))(c (t), γ (t))

d

±

’1

= Ad(γ(t))((•’1 )— ω)(c (t), γ (t)) = ω± (c (t)) + T (µγ(t) )γ (t),

±

i.e., γ (t) = ’T (µγ(t) ).ω± (c (t)). Thus, γ(t) is given by

γ(t) = EvolG (’ω± (c ))(t).γ(0) = evolG (s ’ ’tω± (c (ts))).γ(0).

39.1

39.2 39. Bundles with regular structure groups 423

By lemma (38.3), we may glue the local solutions over di¬erent bundle charts U± ,

so Pt exists globally.

Properties (1) and (3) are now clear, and (2) can be checked as follows: The

condition ω( dt Pt(c, t, u).g) = Ad(g ’1 )ω( dt Pt(c, t, u)) = 0 implies Pt(c, t, u).g =

d d

Pt(c, t, u.g). For the second assertion we compute for u ∈ Pc(0) :

Pt(c, t)— (ζX |Pc(t) )(u) = T Pt(c, t)’1 ζX (Pt(c, t, u))

= T Pt(c, t)’1 ds |0 Pt(c, t, u). exp(sX)

d

= T Pt(c, t)’1 ds |0 Pt(c, t, u. exp(sX))

d

|0 Pt(c, t)’1 Pt(c, t, u. exp(sX))

d

= ds

d

ds |0 u. exp(sX) = ζX (u).

=

Proof of (4) It su¬ces to check that Pt respects smooth curves. So let (f, g) : R ’

C ∞ (R, M ) —M P ‚ C ∞ (R, M ) — P be a smooth curve. By cartesian closedness

(23.2.3), the smooth curve f : R ’ C ∞ (R, M ) corresponds to a smooth map-

ping f § ∈ C ∞ (R2 , M ). For a principal bundle chart (U± , •± ) as above we have

•± (Pt(f (s), t, g(s))) = (f (s)(t), γ(s, t)), where γ is the evolution curve

γ(s, t) = EvolG ’ω± ( ‚t f § (s,

‚

)) (t).•± (g(s)),

which is clearly smooth in (s, t).

If M admits a local addition then C ∞ (R, M ) also carries the structure of a smooth

manifold by (42.4), which is denoted by C∞ (R, M ) there. Since the identity is

smooth C∞ (R, M ) ’ C ∞ (R, M ) by lemma (42.5), the result follows.

39.2. Theorem. Let (p : P ’ M, G) be a smooth principal bundle with a regular

Lie group as structure group. Let ω ∈ „¦1 (P, g) be a principal connection form.

If the connection is ¬‚at, then the horizontal subbundle H ω (P ) = ker(ω) ‚ T P is

integrable and de¬nes a foliation in the sense of (27.16).

If M is connected then each leaf of this horizontal foliation is a covering of M . All

leaves are isomorphic via right translations. The principal bundle P is associated to

the universal covering of M , which is viewed as principal ¬ber bundle with structure

group the (discrete) fundamental group π1 (M ).

Proof. Let (U± , u± : U± ’ u± (U± ) ‚ E± ) be a smooth chart of the manifold

M and let x± ∈ U± be such that u± (x± ) = 0 and the c∞ -open subset u± (U± ) is

star-shaped in E± . Let us also suppose that we have a principal ¬ber bundle chart

(U± , •± : P |U± ’ U± — G). We may cover M by such U± .

We shall now construct for each w± ∈ Px± a smooth section ψ± : U± ’ P whose

image is an integral submanifold for the horizontal subbundle ker(ω). Namely, for

x ∈ U± let cx (t) := u’1 (tu± (x)) for t ∈ [0, 1]. Then we put

±

ψ± (x) := Pt(cx , 1, w± ).

39.2

424 Chapter VIII. In¬nite dimensional di¬erential geometry 39.2

We have to show that the image of T ψ± is contained in the horizontal bundle

ker(ω). Then we get Tx ψ± = T p|H ω (p)’1(x) . This is a consequence of the following

ψ±

notationally more suitable claim.

Let h : R2 ’ U± be smooth with h(0, s) = x± for all s.

‚

Claim: ‚s Pt(h( , s), 1, w± ) is horizontal.

Let •± (w± ) = (x± , g± ) ∈ U± — G. Then from the proof of theorem (39.1) we know

that

•± Pt(h( , s), 1, w± ) = (h(1, s), γ(1, s)), where

γ(t, s) = γ (t, s).g±

˜

‚

γ (t, s) = evolG u ’ ’tω± ( ‚t h(tu, s))

˜

= EvolG ’(h— ω± )(‚t ( , s)) (t),

ω± = s— ω, s± (x) = •’1 (x, e).

± ±

1

Since the curvature „¦ = dω + 2 [ω, ω]§ = 0 we have

‚s (h— ω± )(‚t ) = ‚t (h— ω± )(‚s ) ’ d(h— ω± )(‚t , ‚s ) ’ (h— ω± )([‚t , ‚s ])

= ‚t (h— ω± )(‚s ) + [(h— ω± )(‚t ), (h— ω± )(‚s )]g ’ 0.

Using this and the expression for T evolG from (38.10), we have

’‚s (h— ω± )(‚t )( , s)

‚

‚s γ (1, s)

˜ = T’(h— ω± )(‚t )( ,s) evolG .

1

Ad(˜ (t, s)’1 )‚s (h— ω± )(‚t ) dt

= ’T (µγ (1,s) ). γ

˜

0

1

Ad(˜ (t, s)’1 )‚t (h— ω± )(‚s ) dt+

= ’T (µγ (1,s) ). γ

˜

0

1

Ad(˜ (t, s)’1 ).ad((h— ω± )(‚t )).(h— ω± )(‚s ) dt .

+ γ

0

Next we integrate by parts, use (36.10.3), and use κl (‚t γ (t, s)’1 ) = (h— ω± )(‚t )(t, s)

˜

from (38.3):

1

Ad(˜ (t, s)’1 )‚t (h— ω± )(‚s ) dt =

γ

0

t=1

1

’1 — ’1 —

=’ ‚t Ad(˜ (t, s)

γ ) (h ω± )(‚s ) dt + Ad(˜ (t, s)

γ )(h ω± )(‚s )

0 t=0

1

Ad(˜ (t, s)’1 ).ad κl ‚t (˜ (t, s)’1 ) .(h— ω± )(‚s ) dt

=’ γ γ

0

+ Ad(˜ (1, s)’1 )(h— ω± )(‚s )(1, s) ’ 0

γ

1

Ad(˜ (t, s)’1 ).ad (h— ω± )(‚t ) .(h— ω± )(‚s ) dt

=’ γ

0

+ Ad(˜ (1, s)’1 )(h— ω± )(‚s )(1, s),

γ

39.2

39.3 39. Bundles with regular structure groups 425

so that ¬nally

= ’T (µγ (1,s) ).Ad(˜ (1, s)’1 )(h— ω± )(‚s )(1, s)

‚

‚s γ (1, s)

˜ γ

˜

= ’T (µγ (1,s) ).(h— ω± )(‚s )(1, s),

˜

= T (µg± ). ‚s γ (1, s)

‚ ‚

‚s γ(1, s) ˜

= ’T (µγ(1,s) ).Ad(γ(1, s)’1 )(h— ω± )(‚s )(1, s)

Pt(h( , s), 1, w± ) = ((•’1 )— ω)

‚ ‚ ‚

ω ‚s h(1, s), ‚s γ(1, s)

±

‚s

= Ad(γ(1, s)’1 )ω± ( ‚s h(1, s)) ’ Ad(γ(1, s)’1 )(h— ω± )(‚s )(1, s) = 0,

‚

where at the end we used (37.22.4). Thus, the claim follows.

By the claim and by uniqueness of the parallel transport (39.1.1) for any smooth

curve c in U± the horizontal curve ψ± (c(t)) coincides with Pt(c, t, ψ± (c(0))).

To ¬nish the proof, we may now glue overlapping right translations of ψ± (U± ) to

maximal integral manifolds of the horizontal subbundle. As subset such an integral

manifold consists of all endpoints of parallel transports of a ¬xed point. These are

di¬eomorphic covering spaces of M . Let us ¬x base points x0 ∈ M and u0 ∈ Px0 .

The parallel transport Pt(c, 1, u0 ) depends only on the homotopy class relative

to the ends of the curve c, by the claim above, so that a group homomorphism

˜

ρ : π1 (M ) ’ G is given by Pt(γ, 1, u0 ) = u0 .ρ([γ]). Now let M ’ M be the

universal cover of M , a principal bundle with discrete structure group π1 (M ),

viewed as the space of homotopy classes relative to the ends of smooth curves

starting from x0 . Then the mapping

˜

M — G ’ P,

([c], g) ’ Pt(c, 1, u0 ).g

˜ ˜

factors to a smooth mapping from the associated bundle M [G] = M —π1 (M ) G to

˜

P which is a di¬eomorphism, since we can ¬nd local smooth sections P ’ M — G

in the following way: For u ∈ P choose a smooth curve cu from x0 to p(u), and

˜

consider ([cu ], „ (Pt(cu , 1, u0 ), u)) ∈ M — G.

It is not clear, however, whether the integral submanifolds of the theorem are initial

submanifolds of P , or whether they intersect each ¬ber in a totally disconnected

subset, since M might have uncountable fundamental group.

39.3. Holonomy groups. Let (p : P ’ M, G) be a principal ¬ber bundle with

regular structure group G so that the parallel transport exists along all curves by

theorem (39.1). Let ¦ = ζ —¦ ω be a principal connection. We assume that M is

connected, and we ¬x x0 ∈ M .

Now let us ¬x u0 ∈ Px0 . Consider the subgroup Hol(ω, u0 ) of the structure group

G which consists of all elements „ (u0 , Pt(c, t, u0 )) ∈ G for c any piecewise smooth

closed loop through x0 . Reparameterizing c by a function which is ¬‚at at each

corner of c we may assume that any such c is smooth. We call Hol(ω, u0 ) the

39.3

426 Chapter VIII. In¬nite dimensional di¬erential geometry 40.2

holonomy group of the connection. If we consider only those curves c which are null-

homotopic, we obtain the restricted holonomy group Hol0 (ω, u0 ), a normal subgroup

in Hol(ω, u0 ).

Theorem. (1) We have Hol(ω, u0 .g) = conj(g ’1 ) Hol(ω, u0 ) and Hol0 (ω, u0 .g) =

conj(g ’1 ) Hol0 (ω, u0 ).

(2) For every curve c in M with c(0) = x0 we have Hol(ω, Pt(c, t, u0 )) =

Hol(ω, u0 ) and Hol0 (ω, Pt(c, t, u0 )) = Hol0 (ω, u0 ).

Proof. (1) This follows from the properties of the mapping „ from (37.8) and from

the G-equivariance of the parallel transport:

„ (u0 .g, Pt(c, 1, u0 .g)) = „ (u0 .g, Pt(c, 1, u0 ).g) = g ’1 .„ (u0 , Pt(c, 1, u0 )).g.

(2) By reparameterizing the curve c we may assume that t = 1, and we put

Pt(c, 1, u0 ) =: u1 . Then by de¬nition for an element g ∈ G we have g ∈ Hol(ω, u1 )

if and only if g = „ (u1 , Pt(e, 1, u1 )) for some closed smooth loop e through x1 :=

c(1) = p(u1 ), that is,

Pt(c, 1)(rg (u0 )) = rg (Pt(c, 1)(u0 )) = u1 g = Pt(e, 1)(Pt(c, 1)(u0 ))

u0 g = Pt(c, 1)’1 Pt(e, 1) Pt(c, 1)(u0 ) = Pt(c.e.c’1 , 3)(u0 ),

where c.e.c’1 is the curve traveling along c(t) for 0 ¤ t ¤ 1, along e(t ’ 1) for

1 ¤ t ¤ 2, and along c(3 ’ t) for 2 ¤ t ¤ 3. This is equivalent to g ∈ Hol(ω, u0 ).

Furthermore, e is null-homotopic if and only if c.e.c’1 is null-homotopic, so we also

have Hol0 (ω, u1 ) = Hol0 (ω, u0 ).

40. Rudiments of Lie Theory for Regular Lie Groups

40.1. From Lie algebras to Lie groups. It is not true in general that every

convenient Lie algebra is the Lie algebra of a convenient Lie group. This is also

wrong for Banach Lie algebras and Banach Lie groups; one of the ¬rst examples is

from [Van Est, Korthagen, 1964], see also [de la Harpe, 1972].

To Lie subalgebras in the Lie algebra of a Lie group, in general, do not correspond

Lie subgroups. We shall give easy examples in (43.6).

In principle, one should be able to tell whether a given convenient Lie algebra is

the Lie algebra of a regular Lie group, but we have no idea how to do that.

40.2. The Cartan developing. Let G be a connected Lie group with Lie algebra

g. For a smooth mapping f : M ’ G we have considered in (38.1) the right

’1

logarithmic derivative δ r f ∈ „¦1 (M, g) which is given by δ r fx := T (µf (x) ) —¦ Tx f :

Tx M ’ Tf (x) G ’ g and which satis¬es the left (from the left action) Maurer-

Cartan equation

1

dδ r f ’ [δ r f, δ r f ]g = 0.

§

2

40.2

40.2 40. Rudiments of Lie theory for regular Lie groups 427

Similarly, the left logarithmic derivative δ l f ∈ „¦1 (M, g) of f ∈ C ∞ (M, G) is given

by δ l fx := T (µf (x)’1 ) —¦ Tx f : Tx M ’ Tf (x) G ’ g and satis¬es the right Maurer

Cartan equation

1

dδ l f + [δ l f, δ l f ]g = 0.

§

2

For regular Lie groups we have the following converse, which for ¬nite dimensional

Lie groups can be found in [Onishchik, 1961, 1964, 1967], or in [Gri¬th, 1974]

(proved with moving frames); see also [Alekseevsky, Michor, 1995b, 5.2].

Theorem. Let G be a connected regular Lie group with Lie algebra g.

If a 1-form • ∈ „¦1 (M, g) satis¬es d• + 2 [•, •]g = 0 then for each simply connected

1

§

subset U ‚ M there exists a smooth mapping f : U ’ G with δ l f = •|U , and f is

uniquely determined up to a left translation in G.

If a 1-form ψ ∈ „¦1 (M, g) satis¬es dψ ’ 1 [ψ, ψ]g = 0 then for each simply connected

§

2

subset U ‚ M there exists a smooth mapping f : U ’ G with δ r f = ψ|U , and f is

uniquely determined up to a right translation in G.

The mapping f is called the left Cartan developing of •, or the right Cartan devel-

oping of ψ, respectively.

Proof. Let us treat the right logarithmic derivative since it leads to a principal

connection for a bundle with right principal action. For the left logarithmic deriva-