0¤i¤k

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

+

i<j

(’1)i q (

= Yi (¦(Y0 , . . . , Yi , . . . , Yk )))

0¤i¤k

(’1)i+j q (¦([Yi , Yj ], Y0 , . . . Yi . . . Yj . . . , Yk ))

+

i<j

= q (d ¦(Y0 , . . . , Yk )) = (q d ¦)(CY0 , . . . , CYk ).

37.32. Corollary. In the situation of theorem (37.31), for the Lie algebra valued

curvature form „¦ ∈ „¦2 (P, g) and the curvature RP [W ] ∈ „¦2 (M ; L(P [W ], P [W ]))

hor

we have the relation

qL(P [W ],P [W ]) RP [W ] = ρ —¦ „¦,

where ρ = Te ρ : g ’ L(W, W ) is the derivative of the representation ρ.

37.32

404 Chapter VIII. In¬nite dimensional di¬erential geometry 38.1

Proof. We use the notation of the proof of theorem (37.31), by which we have for

X, Y ∈ Tu P

(dω dω qP [W ] s)u (X, Y ) = (q d d s)u (X, Y )

= (q RP [W ] s)u (X, Y )

= „ W (u, RP [W ] (Tu p.X, Tu p.Y )s(p(u)))

= (qL(P [W ],P [W ]) RP [W ] )u (X, Y )(qP [W ] s)(u).

‚

On the other hand, let g(t) be a smooth curve in G with g(0) = e and g(t) =

‚t 0

„¦u (X, Y ) ∈ g. Then we have by theorem (37.23.8)

(dω dω q s)u (X, Y ) = (χ— iR dq s)u (X, Y )

since R is horizontal

= (dq s)u (R(X, Y ))

= (dq s)(’ζ„¦(X,Y ) (u)) by (37.20)

’1

(q s)(rg(t) (u))

‚

= ‚t 0

„ W u.g(t)’1 , s(p(u.g(t)’1 ))

‚

= ‚t 0

„ W (u.g(t)’1 , s(p(u)))

‚

= ‚t 0

ρ(g(t))„ W (u, s(p(u)))

‚

= by (37.12)

‚t 0

= ρ („¦u (X, Y ))(q s)(u).

38. Regular Lie Groups

38.1. The right and left logarithmic derivatives. Let M be a manifold, and

let f : M ’ G be a smooth mapping into a Lie group G with Lie algebra g. We

de¬ne the mapping δ r f : T M ’ g by the formula

’1

δ r f (ξx ) := Tf (x) (µf (x) ).Tx f.ξx = (f — κr )(ξx ) for ξx ∈ Tx M,

where κr is the right Maurer-Cartan form from (36.10). Then δ r f is a g-valued

1-form on M , δ r f ∈ „¦1 (M, g). We call δ r f the right logarithmic derivative of f ,

since for f : R ’ (R+ , ·) we have δ r f (x).1 = f (x) = (log —¦f ) (x).

(x)

f

Similarly, the left logarithmic derivative δ l f ∈ „¦1 (M, g) of a smooth mapping f :

M ’ G is given by

δ l f.ξx := Tf (x) (µf (x)’1 ).Tx f.ξx = (F — κl )(ξx ).

Lemma. Let f, g : M ’ G be smooth. Then we have

δ r (f.g)(x) = δ r f (x) + Ad(f (x)).δ r g(x).

38.1

38.1 38. Regular Lie groups 405

Moreover, the di¬erential form δ r f ∈ „¦1 (M, g) satis¬es the ˜left Maurer-Cartan

equation™ (left because it stems from the left action of G on itself )

dδ r f (ξ, ·) ’ [δ r f (ξ), δ r f (·)]g = 0

1

or dδ r f ’ [δ r f, δ r f ]g = 0, §

2

]g was de¬ned in

where ξ, · ∈ Tx M , and where the graded Lie bracket [ , §

(37.20.1).

The left logarithmic derivative also satis¬es a ˜Leibniz rule™ and the ˜right Maurer

Cartan equation™:

δ l (f g)(x) = δ l g(x) + Ad(g(x)’1 ).δ l f (x),

1

dδ l f + [δ l f, δ l f ]g = 0.

§

2

For ˜regular Lie groups™ we will prove a converse to this statement later in (40.2).

Proof. We treat only the right logarithmic derivative, the proof for the left one is

similar.

’1

.f (x)’1

δ r (f.g)(x) = T (µg(x) ).Tx (f.g)

’1 ’1

= T (µf (x) ).T (µg(x) ).T(f (x),g(x)) µ.(Tx f, Tx g)

’1 ’1

= T (µf (x) ).T (µg(x) ). T (µg(x) ).Tx f + T (µf (x) ).Tx g

= δ r f (x) + Ad(f (x)).δ r g(x).

We shall now use principal bundle geometry from section (37). We consider the

trivial principal bundle pr1 : M — G ’ M with right principal action. Then

the submanifolds {(x, f (x).g) : x ∈ M } for g ∈ G form a foliation of M — G,

whose tangent distribution is complementary to the vertical bundle M — T G ‚

T (M —G) and is invariant under the principal right G-action. So it is the horizontal

distribution of a principal connection on M — G ’ M . For a tangent vector

(ξx , Yg ) ∈ Tx M — Tg G the horizontal part is the right translate to the foot point

(x, g) of (ξx , Tx f.ξx ). The decomposition in horizontal and vertical parts according

to this distribution is

’1 ’1

(ξx , Yg ) = (ξx , T (µg ).T (µf (x) ).Tx f.ξx ) + (0x , Yg ’ T (µg ).T (µf (x) ).Tx f.ξx ).

Since the fundamental vector ¬elds for the right action on G are the left invariant

vector ¬elds, the corresponding connection form is given by

’1

ω r (ξx , Yg ) = T (µg’1 ).(Yg ’ T (µg ).T (µf (x) ).Tx f.ξx ),

ω(x,g) = T (µg’1 ) ’ Ad(g ’1 ).δ r fx ,

r

ω r = κl ’ (Ad —¦ ν).δ r f,

(1)

38.1

406 Chapter VIII. In¬nite dimensional di¬erential geometry 38.1

where κl : T G ’ g is the left Maurer-Cartan form on G (the left trivialization),

given by κl = T (µg’1 ). Note that κl is the principal connection form for the

g

(unique) principal connection p : G ’ {point} with right principal action, which is

¬‚at so that the right (from right action) Maurer-Cartan equation holds in the form

dκl + 1 [κl , κl ]§ = 0.

(2) 2

The principal connection ω r is ¬‚at since we got it via the horizontal leaves, so the

principal curvature form vanishes:

0 = dω r + 1 [ω r , ω r ]§

(3) 2

= dκl + 1 [κl , κl ]§ ’ d(Ad —¦ ν) § δ r f ’ (Ad —¦ ν).dδ r f

2

’ [κl , (Ad —¦ ν).δ r f ]§ + 2 [(Ad —¦ ν).δ r f, (Ad —¦ ν).δ r f ]§

1

= ’(Ad —¦ ν).(dδ r f ’ 1 [δ r f, δ r f ]§ ),

2

where we used (2) and the fact that for ξ ∈ g and a smooth curve c : R ’ G with

c(0) = e and c (0) = ξ we have

Ad(c(t)’1 .g ’1 ) = ’ad(ξ)Ad(g ’1 )

‚

d(Ad —¦ ν)(T (µg )ξ) = ‚t 0

= ’ad κl (T (µg )ξ) (Ad —¦ ν)(g),

d(Ad —¦ ν) = ’(ad —¦ κl ).(Ad —¦ ν).

(4)

So we have dδ r f ’ 1 [δ r f, δ r f ]§ as asserted.

2

For the left logarithmic derivative δ l f the proof is similar, and we discuss only the

essential deviations. First note that on the trivial principal bundle pr1 : M —G ’ M

with left principal action of G the fundamental vector ¬elds are the right invariant

vector ¬elds on G, and that for a principal connection form ω l the curvature form

1

is given by dω l ’ 2 [ω l , ω l ]§ . Look at the proof of theorem (37.20) to see this. The

connection form is then given by

ω l = κr ’ Ad.δ l f,

(1™)

’1

where the right Maurer-Cartan form (κr )g = T (µg ) : Tg G ’ g satis¬es the left

Maurer-Cartan equation

1

dκr ’ [κr , κr ]§ = 0.

(2™)

2

Flatness of ω l now leads to the computation

0 = dω l ’ 1 [ω l , ω l ]§

(3™) 2

= dκr ’ 1 [κr , κr ]§ ’ dAd § δ l f ’ Ad.dδ l f

2

+ [κr , Ad.δ l f ]§ ’ 1 [Ad.δ l f, Ad.δ l f ]§

2

= ’Ad.(dδ l f + 1 [δ l f, δ l f ]§ ),

2

where we have used dAd = (ad —¦ κr )Ad from (36.10.3) directly.

38.1

38.2 38. Regular Lie groups 407

Remark. The second half of the proof of lemma (38.1) can be shortened consid-

erably. Namely, as soon as we know that κr satis¬es the Maurer-Cartan equation

dκr ’ 2 [κr , κr ]§ we get it also for the right logarithmic derivative δ r f = f — κr .

1

But the computations in this proof will be used again in the proof of the converse,

theorem (40.2) below.

38.2. Theorem. [Grabowski, 1993] Let G be a Lie group with exponential mapping

exp : g ’ G. Then for all X, Y ∈ g we have

1

TX exp .Y = Te µexp X . Ad(exp(’tX))Y dt

0

1

exp X

= Te µ . Ad(exp(tX))Y dt.

0

Remark. If G is a Banach Lie group then we have from (36.8.4) and (36.9) the

i

∞

series Ad(exp(tX)) = i=0 t ad(X)i , so that we get the usual formula

i!

∞

TX exp = Te µexp X . i

1

(i+1)! ad(X) .

i=0

Proof. We consider the smooth mapping

f : R2 ’ G, f (s, t) := exp(s(X + tY )). exp(’sX).

Then f (s, 0) = e and

‚0 f (s, 0) = sTexp(sX) µexp(’sX) .TsX exp .Y,

TX exp .Y = Te µexp(X) .‚t f (1, 0).

(1)

Moreover we get

’1

δ r f (s, t).‚s = Tf (s,t) µf (s,t) T µexp(’sX) .‚s exp(s(X + tY ))

+ T µexp(s(X+tY )) .‚s exp(’sX)

’1

= Tf (s,t) µf (s,t) T µexp(’sX) .RX+tY (exp(s(X + tY )))

’ T µexp(s(X+tY )) .LX exp(’sX)

= X + tY ’ Ad(f (s, t))X.

‚t |0 δ r f (s, t).‚s = Y ’ ad(‚t f (s, 0)).X

(2)

Now we use (38.1) to get

0 = d(δ r f )(‚s , ‚t ) ’ [δ r f (‚s ), δ r f (‚t )]

= ‚s (δ r f )(‚t ) ’ ‚t (δ r f )(‚s ) ’ (δ r f )([‚s , ‚t ]) ’ [δ r f (‚s ), δ r f (‚t )].

38.2

408 Chapter VIII. In¬nite dimensional di¬erential geometry 38.3

Since (δ r f )(‚s )|t=0 = 0 we get ‚s (δ r f )(s, 0)(‚t ) = ‚t (δ r f )(s, 0)(‚s ), and from (2)

we then conclude that the curve

c(s) = (δ r f )(s, 0)(‚t ) = ‚t |0 f (s, 0) = sδ r exp(sX).Y ∈ g

(3)

is a solution of the ordinary di¬erential equation

(4) c (s) = Y + [X, c(s)] = Y + ad(X).c(s), c(0) = 0.

The unique solution for the homogeneous equation with c(0) = c0 is

c(s) = Ad(exp(sX)).c0 , since

c (s) = ‚t |t=0 Ad(exp(tX))Ad(exp(sX))c0 = [X, c(s)],

‚s (Ad(exp(’sX))C(s)) = ’Ad(exp(’sX)).ad(X).C(s)+

+ Ad(exp(’sX))[X, C(s)] = 0

for every other solution C(t). Using the variation of constant ansatz we get the

solution s

c(s) = Ad(exp(sX)) Ad(exp(’tX))Y dt

0

of the inhomogeneous equation (4), which is unique for c(0) = 0 since 0 is the

unique solution of the homogeneous equation with initial value 0. Finally, we have

from (1)

TX exp .Y = Te µexp(X) .c(1)

1

exp(X)

= Te µ .Ad(exp(X)) Ad(exp(’tX))Y dt

0

1

= Te µexp(X) . Ad(exp(’tX))Y dt

0

1

exp(X)

TX exp .Y = Te µ .Ad(exp(X)) Ad(exp(’tX))Y dt

0

1

exp(X)

Ad(exp((1 ’ t)X))Y dt

= Te µ

0

1

exp(X)

= Te µ Ad(exp(rX))Y dr.

0

38.3. Let G be a Lie group with Lie algebra g. For a closed interval I ‚ R and for

X ∈ C ∞ (I, g) we consider the ordinary di¬erential equation

g(t0 ) = e

(1) ‚ ‚

= Te (µg(t) )X(t) = RX(t) (g(t)), or κr ( ‚t g(t)) = X(t),

‚t g(t)

for local smooth curves g in G, where t0 ∈ I.

38.3

38.3 38. Regular Lie groups 409

Lemma.

(2) Local solutions g of the di¬erential equation (1) are uniquely determined.

(3) If for ¬xed X the di¬erential equation (1) has a local solution near each

t0 ∈ I, then it has also a global solution g ∈ C ∞ (I, G).

(4) If for all X ∈ C ∞ (I, g) the di¬erential equation (1) has a local solution near

one ¬xed t0 ∈ I, then it has also a global solution g ∈ C ∞ (I, G) for each

X. Moreover, if the local solutions near t0 depend smoothly on the vector

¬elds X (see the proof for the exact formulation), then so does the global

solution.

(5) The curve t ’ g(t)’1 is the unique local smooth curve h in G which satis¬es

h(t0 ) = e

‚ ‚

or κl ( ‚t h(t)) = ’X(t).

‚t h(t) = Te (µh(t) )(’X(t)) = L’X(t) (h(t)),

Proof. (2) Suppose that g(t) and g1 (t) both satisfy (1). Then on the intersection

of their intervals of de¬nition we have

’1

’1

= ’T (µg1 (t) ).T (µg(t)’1 ).T (µg(t) ).T (µg(t) ).X(t)

‚

‚t (g(t) g1 (t))

+ T (µg(t)’1 ).T (µg1 (t) ).X(t) = 0,

so that g = g1 .

Proof of (3) It su¬ces to prove the claim for every compact subinterval of I, so let

I be compact. If g is a local solution of (1) then t ’ g(t).x is a local solution of

the same di¬erential equation with initial value x. By assumption, for each s ∈ I

there is a unique solution gs of the di¬erential equation with gs (s) = e; so there

exists δs > 0 such that gs (s + t) is de¬ned for |t| < δs . Since I is compact there

exist s0 < s1 < · · · < sk such that I = [s0 , sk ] and si+1 ’ si < δsi . Then we put

for s0 ¤ t ¤ s1

gs0 (t)

±

for s1 ¤ t ¤ s2

gs1 (t).gs0 (s1 )

g(t) := . . .

for si ¤ t ¤ si+1

gs (t).gs (si ) . . . gs (s1 )

i i’1 0

...

which is smooth by the ¬rst case and solves the problem.

Proof of (4) Given X : I ’ g we ¬rst extend X to a smooth curve R ’ g, using

(24.10). For t1 ∈ I, by assumption, there exists a local solution g near t0 of the

translated vector ¬eld t ’ X(t1 ’ t0 + t), thus t ’ g(t0 ’ t1 + t) is a solution near

t1 of X. So by (3) the di¬erential equation has a global solution for X on I.

Now we assume that the local solutions near t0 depend smoothly on the vector

¬eld. So for any smooth curve X : R ’ C ∞ (I, g) we have:

For every compact interval K ‚ R there is a neighborhood UX,K of t0

in I and a smooth mapping g : K — UX,K ’ G with

g(k, t0 ) = e

‚

= Te (µg(k,t) ).X(k)(t) for all k ∈ K, t ∈ UX,K .

‚t g(k, t)

38.3

410 Chapter VIII. In¬nite dimensional di¬erential geometry 38.4

Given a smooth curve X : R ’ C ∞ (I, g) we extend (or lift) it smoothly to X : R ’

C ∞ (R, g) by (24.10). Then the smooth parameter k from the compact interval K

passes smoothly through the proofs given above to give a smooth global solution

g : K — I ’ G. So the ˜solving operation™ respects smooth curves and thus is

˜smooth™.

Proof of (5) One can show in a similar way that h is the unique solution of (5) by

di¬erentiating h1 (t).h(t)’1 . Moreover, the curve t ’ g(t)’1 = h(t) satis¬es (5),

since

’1

’1

= ’T (µg(t)’1 ).T (µg(t) ).T (µg(t) ).X(t) = T (µg(t)’1 ).(’X(t)).

‚

‚t (g(t) )

38.4. De¬nition. Regular Lie groups. If for each X ∈ C ∞ (R, g) there exists

g ∈ C ∞ (R, G) satisfying

±

g(0) = e

‚ g(t)

‚t g(t) = Te (µ )X(t) = RX(t) (g(t)),

(1)

‚

or κr ( ‚t g(t)) = δ r g(‚t ) = X(t)

then we write

evolr (X) = evolG (X) := g(1),

G

Evolr (X)(t) := evolG (s ’ tX(ts)) = g(t),

G

and call them the right evolution of the curve X in G. By lemma (38.3), the solution

of the di¬erential equation (1) is unique, and for global existence it is su¬cient that

it has a local solution. Then

Evolr : C ∞ (R, g) ’ {g ∈ C ∞ (R, G) : g(0) = e}

G

is bijective with inverse δ r . The Lie group G is called a regular Lie group if evolr :

C ∞ (R, g) ’ G exists and is smooth. We also write

evoll (X) = evolG (X) := h(1),

G

Evoll (X)(t) := evoll (s ’ tX(ts)) = h(t),

G G

if h is the (unique) solution of

±

h(0) = e

‚

‚t h(t) = Te (µh(t) )(X(t)) = LX(t) (h(t)),

(2)

‚

or κl ( ‚t h(t)) = δ l h(‚t ) = X(t).

Clearly, evoll : C ∞ (R, g) ’ G exists and is also smooth if evolr does, since we have

evoll (X) = evolr (’X)’1 by lemma (38.3).

38.4

38.5 38. Regular Lie groups 411

If f ∈

Let us collect some easily seen properties of the evolution mappings.

C ∞ (R, R) then we have

Evolr (X)(f (t)) = Evolr (f .(X —¦ f ))(t).Evolr (X)(f (0)),

Evoll (X)(f (t)) = Evoll (X)(f (0)).Evoll (f .(X —¦ f ))(t).

If • : G ’ H is a smooth homomorphism between regular Lie groups then the

diagrams

wC wC

•— •—

C ∞ (R, g) ∞

C ∞ (R, g) ∞

(R, h) (R, h)

u u u u

(3) evolG evolH EvolG EvolH

wH wC

• •—

C ∞ (R, G) ∞

G (R, H)

‚

commutes, since ‚t •(g(t)) = T •.T (µg(t) ).X(t) = T (µ•(g(t)) ).• .X(t). Note that

each regular Lie group admits an exponential mapping, namely the restriction of

evolr to the constant curves R ’ g. A Lie group is regular if and only if its universal

covering group is regular.

This notion of regularity is a weakening of the same notion of [Omori et al., 1982,

1983, etc.], who considered a sort of product integration property on a smooth Lie

group modeled on Fr´chet spaces. Our notion here is due to [Milnor, 1984]. Up to

e

now the following statement holds:

All known Lie groups are regular.

Any Banach Lie group is regular since we may consider the time dependent right

invariant vector ¬eld RX(t) on G and its integral curve g(t) starting at e, which

exists and depends smoothly on (a further parameter in) X. In particular, ¬nite

dimensional Lie groups are regular. For di¬eomorphism groups the evolution oper-

ator is just integration of time dependent vector ¬elds with compact support, see

section (43) below.

38.5. Some abelian regular Lie groups. For (E, +), where E is a convenient

1

vector space, we have evol(X) = 0 X(t)dt, so convenient vector spaces are regular

abelian Lie groups. We shall need ˜discrete™ subgroups, which is not an obvious

notion since (E, +) is not a topological group: the addition is continuous only as a

mapping c∞ (E —E) ’ c∞ E and not for the cartesian product of the c∞ -topologies.

Next let Z be a ˜discrete™ subgroup of a convenient vector space E in the sense that

there exists a c∞ -open neighborhood U of zero in E such that U © (z + U ) = … for

all 0 = z ∈ Z (equivalently (U ’ U ) © (Z \ 0) = …). For that it su¬ces e.g., that Z

is discrete in the bornological topology on E. Then E/Z is an abelian but possibly

non Hausdor¬ Lie group. It does not su¬ce to take Z discrete in the c∞ -topology:

Take as Z the subgroup generated by A in RN—c0 in the proof of (4.26).(iv).

Let us assume that Z ful¬lls the stronger condition: there exists a symmetric c∞ -

open neighborhood W of 0 such that (W + W ) © (z + W + W ) = … for all 0 = z ∈ Z

(equivalently (W + W + W + W ) © (Z \ 0) = …). Then E/Z is Hausdor¬ and thus an

38.5

412 Chapter VIII. In¬nite dimensional di¬erential geometry 38.6

abelian regular Lie group, since its universal cover E is regular. Namely, for x ∈ Z,

/

we have to ¬nd neighborhoods U and V of 0 such that (Z + U ) © (x + Z + V ) = ….

There are two cases. If x ∈ Z + W + W then there is a unique z ∈ Z with

x ∈ z + W + W , and we may choose U, V ‚ W such that (z + U ) © (x + V ) = …;

then (Z + U ) © (x + Z + V ) = …. In the other case, if x ∈ Z + W + W , then we

/

have (Z + W ) © (x + Z + W ) = ….

Notice that the two conditions above and their consequences also hold for gen-

eral (non-abelian) (regular) Lie groups instead of E and their ˜discrete™ normal

subgroups (which turn out to be central if G is connected).

It would be nice if any regular abelian Lie group would be of the form E/Z described

above. A ¬rst result in this direction is that for an abelian Lie group G with Lie

algebra g which admits a smooth exponential mapping exp : g ’ G one can easily

‚

check by using (38.2) that ‚t (exp(’tX). exp(tX + Y )) = 0, so that exp is a smooth

homomorphism of Lie groups.

Let us consider some examples. More examples can be found in [Banaszczyk, 1984,

1986, 1991]. For the ¬rst one we consider a discrete subgroup Z ‚ RN . There

exists a neighborhood of 0, without loss of generality of the form U — RN\n for

U ‚ Rn , with U © (Z \ 0) = …. Then we consider the following diagram of Lie group

homomorphisms

wR wR

N\n N\n

0

u u u

wR w R /Z (S 1 )k — RN\(n’k)

N N

Z

∼ π

u u u u

=

wR w R /π(Z)

n n

(S 1 )k — Rn’k

π(Z)

which has exact lines and columns. For the right hand column we use a diagram

chase to see this. Choose a global linear section of π inverting π|Z. This factors to

a global homomorphism of the right hand side column.

As next example we consider Z(N) ‚ R(N) . Then, obviously, R(N) /Z(N) = (S 1 )(N) ,

which is a real analytic manifold modeled on R(N) , similar to the ones which are

treated in section (47). The reader may convince himself that any Lie group covered

by R(N) is isomorphic to (S 1 )(A) — R(N\A) for A ⊆ N.

As another example, one may check easily that ∞ /(ZN © ∞ ) = (S 1 )N , equipped

with the ˜uniform box topology™; compare with the remark at the end of (27.3).

38.6. Extensions of Lie groups. Let H and K be Lie groups. A Lie group G

is called a smooth extension of H with kernel K if we have a short exact sequence

of groups

p

i

{e} ’ K ’ G ’ H ’ {e},

’’

(1)

38.6

38.6 38. Regular Lie groups 413

such that i and p are smooth, and one of the following two equivalent conditions is

satis¬ed:

(2) p admits a local smooth section s near e (equivalently near any point), and

i is initial (27.11).

(3) i admits a local smooth retraction r near e (equivalently near any point),

and p is ¬nal (27.15).

Of course, by s(p(x))i(r(x)) = x the two conditions are equivalent, and then G is

locally di¬eomorphic to K — H via (r, p) with local inverse (i —¦ pr1 ).(s —¦ pr2 ).

Not every smooth exact sequence of Lie groups admits local sections as required in

(2). Let, for example, K be a closed linear subspace in a convenient vector space

G which is not a direct summand, and let H be G/K. Then the tangent mapping

at 0 of a local smooth splitting would make K a direct summand.

p

i

Theorem. Let {e} ’ K ’ G ’ H ’ {e} be a smooth extension of Lie groups.

’ ’

Then G is regular if and only if both K and H are regular.

Proof. Clearly, the induced sequence of Lie algebras also is exact,

p

i

0 ’ k ’ g ’ h ’ 0,

’’

with a bounded linear section Te s of p . Thus, g is isomorphic to k — h as convenient

vector space.

Let us suppose that K and H are regular. Given X ∈ C ∞ (R, g), we consider

‚

Y (t) := p (X(t)) ∈ h with evolution curve h satisfying ‚t h(t) = T (µh(t) ).Y (t) and

h(0) = e. By lemma (38.3) it su¬ces to ¬nd smooth local solutions g near 0 of

‚ g(t)

‚t g(t) = T (µ ).X(t) with g(0) = e, depending smoothly on X. We look for

solutions of the form g(t) = s(h(t)).i(k(t)), where k is a local evolution curve in K

‚

of a suitable curve t ’ Z(t) in k, i.e., ‚t k(t) = T (µk(t) ).Z(t), and k(0) = e. For

this ansatz we have

s(h(t)).i(k(t)) = T (µs(h(t)) ).T i. ‚t k(t) + T (µi(k(t)) ).T s. ‚t h(t)

‚ ‚ ‚ ‚

‚t g(t) = ‚t

= T (µs(h(t)) ).T i.T (µk(t) ).Z(t) + T (µi(k(t)) ).T s.T (µh(t) ).Y (t),

and we want this to be

T (µg(t) ).X(t) = T (µs(h(t)).i(k(t)) ).X(t) = T (µi(k(t)) ).T (µs(h(t)) ).X(t).

Using i —¦ µk = µi(k) —¦ i, one quickly sees that

’1

i .Z(t) := Ad s(h(t))’1 . X(t) ’ T (µs(h(t)) ).T s.T (µh(t) ).Y (t) ∈ ker p

solves the problem, so G is regular.

Let now G be regular. If Y ∈ C ∞ (R, h), then p —¦ Evolr (s —¦ Y ) = EvolH (Y ), by

G

∞

the diagram in (38.4.3). If U ∈ C (R, k) then p —¦ EvolG (i —¦ U ) = EvolH (0) = e, so

that EvolG (i —¦ U )(t) ∈ i(K) for all t and thus equals i(EvolK (U )(t)).

38.6

414 Chapter VIII. In¬nite dimensional di¬erential geometry 38.8

38.7. Subgroups of regular Lie groups. Let G and K be Lie groups, let G

be regular, and let i : K ’ G be a smooth homomorphism which is initial (27.11)

with Te i = i : k ’ g injective. We suspect that K is then regular, but we are only

able to prove this under the following assumption.

There are an open neighborhood U ‚ G of e and a smooth mapping

p : U ’ E into a convenient vector space E such that p’1 (0) = K © U

and p is constant on left cosets Kg © U .

Proof. For Z ∈ C ∞ (R, k) we consider g(t) = EvolG (i —¦ Z)(t) ∈ G. Then we have

‚ g(t)

‚t (p(g(t))) = T p.T (µ ).i (Z(t)) = 0 by the assumption, so p(g(t)) is constant

p(e) = 0, thus g(t) = i(h(t)) for a smooth curve h in H, since i is initial. Then

h = EvolH (Y ) since Te i is injective, and h depends smoothly on Z since i is

initial.

38.8. Abelian and central extensions. From theorem (38.6), it is clear that

any smooth extension G of a regular Lie group H with an abelian regular Lie group

(K, +) is regular. We shall describe EvolG in terms of EvolG , EvolK , and in terms

of the action of H on K and the cocycle c : H — H ’ K if the latter exists.

Let us ¬rst recall these notions. If we have a smooth extension with abelian normal

subgroup K,

p

i

{e} ’ K ’ G ’ H ’ {e},

’’

then a unique smooth action ± : H — K ’ K by automorphisms is given by

i(±h (k)) = s(h)i(k)s(h)’1 , where s is any smooth local section of p de¬ned near h.

If moreover p admits a global smooth section s : H ’ G, which we assume without

loss of generality to satisfy s(e) = e, then we consider the smooth mapping c :

H — H ’ K given by ic(h1 , h2 ) := s(h1 ).s(h2 ).s(h1 .h2 )’1 . Via the di¬eomorphism

K — H ’ G given by (k, h) ’ i(k).s(h) the identity corresponds to (0, e), the

multiplication and the inverse in G look as follows:

(1) (k1 , h1 ).(k2 , h2 ) = (k1 + ±h1 k2 + c(h1 , h2 ), h1 h2 ),

(k, h)’1 = (’±h’1 (k) ’ c(h’1 , h), h’1 ).

Associativity and (0, e)2 = (0, e) correspond to the fact that c satis¬es the following

cocycle condition and normalization

±h1 (c(h2 , h3 )) ’ c(h1 h2 , h3 ) + c(h1 , h2 h3 ) ’ c(h1 , h2 ) = 0,

(2)

c(e, e) = 0.

These imply that c(e, h) = 0 = c(h, e) and ±h (c(h’1 , h)) = c(h, h’1 ). For a central

extension the action is trivial, ±h = IdK for all h ∈ H.

If conversely H acts smoothly by automorphisms on an abelian Lie group K and

if c : H — H ’ K satis¬es (2), then (1) describes a smooth Lie group structure on

K — H, which is a smooth extension of H over K with a global smooth section.

38.8

38.10 38. Regular Lie groups 415

For later purposes, let us compute

(0, h1 ).(0, h2 )’1 = (’±h1 (c(h’1 , h2 )) + c(h1 , h’1 ), h1 h’1 ),

2 2 2

’1 ’1

’1

).Yh1 + T (c( , h’1 )).Yh1 , T (µh2 ).Yh1 ).

T(0,h1 ) (µ(0,h2 ) ).(0, Yh1 ) = (’T (±c(h2 ,h2 )

2

Let us now assume that K and H are moreover regular Lie groups. We consider

a curve t ’ X(t) = (U (t), Y (t)) in the Lie algebra g which as convenient vector

space equals k — h. From the proof of (38.6) we get that

g(t) := EvolG (U, Y )(t) = (0, h(t)).(k(t), e) = (±h(t) (k(t)), h(t)), where

h(t) := EvolH (Y )(t) ∈ H,

’1

(Z(t), 0) := AdG (0, h(t))’1 (U (t), Y (t)) ’ T µ(0,h(t)) .(0, ‚t h(t)) ,

‚

’1

) ’ T (c( , h(t)’1 )) . ‚t h(t) ,

Z(t) = T0 (±h(t)’1 ). U (t) + T (±c(h(t) ,h(t)) ‚

k(t) := EvolK (Z)(t) ∈ K.

38.9. Semidirect products. From theorem (38.6) we see immediately that the

semidirect product of regular Lie groups is regular. Since we shall need explicit

formulas later we specialize the proof of (38.6) to this case.

Let H and K be regular Lie groups with Lie algebras h and k, respectively. Let

± : H — K ’ K be smooth such that ±∨ : H ’ Aut(K) is a group homomorphism.

Then the semidirect product K H is the Lie group K — H with multiplication

(k, h).(k , h ) = (k.±h (k ), h.h ) and inverse (k, h)’1 = (±h’1 (k)’1 , h’1 ). We have

then T(e,e) (µ(k ,h ) ).(U, Y ) = (T (µk ).U + T (±k ).Y, T (µh ).Y ).

Now we consider a curve t ’ X(t) = (U (t), Y (t)) in the Lie algebra k h. Since

s : h ’ (e, h) is a smooth homomorphism of Lie groups, from the proof of (38.6)

we get that

g(t) := EvolK H (U, Y )(t) = (e, h(t)).(k(t), e) = (±h(t) (k(t)), h(t)), where

h(t) := EvolH (Y )(t) ∈ H,

’1

(Z(t), 0) := AdK H (e, h(t) )(U (t), 0) = (Te (±h(t)’1 ).U (t), 0),

k(t) := EvolK (Z)(t) ∈ K.

38.10. Corollary. Let G be a Lie group. Then via right trivialization (κr , πG ) :

T G ’ g — G the tangent group T G is isomorphic to the semidirect product g G,

where G acts by Ad : G ’ Aut(g).

Therefore, if G is a regular Lie group, then T G ∼ g G also is regular, and T evolr

= G

evolr G . ∞ ∞

In particular, for (Y, X) ∈ C (R, g — g) = T C (R, g),

corresponds to T

where X is the footpoint, we have

1

Ad(Evolr (X)(s)’1 ).Y (s) ds, evolr (X)

evolr G (Y, X) Ad(evolr (X))

= G G G

g

0

1

Ad(Evolr (X)(s)’1 ).Y (s) ds,

TX evolr .Y = T (µevolr (X) ).

G G

G

0

t

Ad(Evolr (X)(s)’1 ).Y (s) ds.

TX (Evolr ( )(t)).Y = T (µEvolr (X)(t) ).

G G

G

0

38.10

416 Chapter VIII. In¬nite dimensional di¬erential geometry 38.10

The expression in (38.2) for the derivative of the exponential mapping is a special

case of the expression for T evolG , for constant curves in g. Note that in the semidi-

rect product representation T G ∼ g G the footpoint appears in the right factor

=

G, contrary to our usual convention. We followed this also in T g = g g.

Proof. Via right trivialization the tangent group T G is the semidirect product

g G, where G acts on the Lie algebra g by Ad : G ’ Aut(g), because by (36.2)

we have for g, h ∈ G and X, Y ∈ g, where µ = µG is the multiplication on G:

T(g,h) µ.(RX (g), RY (h)) = T (µh ).RX (g) + T (µg ).RY (h)

= T (µh ).T (µg ).X + T (µg ).T (µh ).Y

= RX (gh) + RAd(g)Y (gh),

’1

Tg ν.RX (g) = ’T (µg ).T (µg’1 ).T (µg ).X

= ’RAd(g’1 )X (g ’1 ),

so that µT G and νT G are given by

(1) µg G ((X, g), (Y, h)) = (X + Ad(g)Y, gh)

= (’Ad(g ’1 )X, g ’1 ).

νg G (X, g)

Now we shall prove that the following diagram commutes and that the equations

of the corollary follow. The lower triangle commutes by de¬nition.

xw C

∼

=

∞ ∞

T C (R, g) (R, g g)

x

xx

u evol

x u

evolg

T evolG G

TG

wg

TG G

∼

=

For that we choose X, Y ∈ C ∞ (R, g). Let us ¬rst consider the evolution operator

of the tangent group T G in the picture g G. On (g, +) the evolution mapping is

the de¬nite integral, so going through the prescription (38.9) for evolg G we have

the following data:

(2) evolg G (Y, X) = (h(1), g(1)), where

g(t) := EvolG (X)(t) ∈ G,

Z(t) := Ad(g(t)’1 ).Y (t) ∈ g,

t

Ad(g(u)’1 ).Y (u) du ∈ g,

h0 (t) := Evol(g,+) (Z)(t) =

0

t

Ad(g(u)’1 ).Y (u) du ∈ g.

h(t) := Ad(g(t))h0 (t) = Ad(g(t))

0

This shows the ¬rst equation in the corollary. The di¬erential equation for the

curve (h(t), g(t)), which by lemma (38.3) has a unique solution starting at (0, e),

38.10

38.10 38. Regular Lie groups 417

looks as follows, using (1):

(h(t),g(t))

(h (t), h(t)), g (t) = T(0,e) (µg ). (Y (t), 0), X(t)

G

g(t)

= Y (t) + dAd(X(t)).h(t), 0 + Ad(e).h(t) , T (µG ).X(t) ,

(3) h (t) = Y (t) + ad(X(t))h(t),

g(t)

g (t) = T (µG ).X(t).

For the computation of T evolG we let

g(t, s) := evolG u ’ t(X(tu) + sY (tu)) = EvolG (X + sY )(t),

δ r g(‚t (t, s)) = X(t) + sY (t).

satisfying

Then T evolG (Y, X) = ‚s |0 g(1, s), and the derivative ‚s |0 g(t, s) in T G corresponds

to the element

’1

(T (µg(t,0) ).‚s |0 g(t, s), g(t, 0)) = (δ r g(‚s (t, 0)), g(t, 0)) ∈ g G

via right trivialization. For the right hand side we have g(t, 0) = g(t), so it remains

to show that δ r g(‚s (t, 0)) = h(t). We will show that δ r g(‚s (t, 0)) is the unique

solution of the di¬erential equation (3) for h(t). Using the Maurer Cartan equation

1

dδ r g ’ 2 [δ r g, δ r g]§ = 0 from lemma (38.1) we get

‚t δ r g(‚s ) = ‚s δ r g(‚t ) + d(δ r g)(‚t , ‚s ) + δ r g([‚t , ‚s ])

= ‚s δ r g(‚t ) + [δ r g(‚t ), δ r g(‚s )]g + 0

= ‚s (X(t) + sY (t)) + [X(t) + sY (t), δ r g(‚s )]g ,

so that for s = 0 we get

‚t δ r g(‚s (t, 0)) = Y (t) + [X(t), δ r g(‚s (t, 0))]g

= Y (t) + ad(X(t))δ r g(‚s (t, 0)).

Thus, δ r g(‚s (t, 0)) is a solution of the inhomogeneous linear ordinary di¬erential

equation (3), as required.

It remains to check the last formula. Note that X ’ tX(t ) is a bounded linear

operator. So we have

Evolr (X)(t) = evolr (s ’ tX(ts)),

G G

TX (Evolr ( r

)(t)).Y = TtX(t ) evolG .(tY (t ))

G

1

))(s)’1 .tY (ts) ds

AdG Evolr (tX(t

= T (µevolr (tX(t )) ). G

G

0

1

AdG evolr (stX(st ))’1 .tY (ts) ds

= T (µEvolr (X)(t) ). G

G

0

t

AdG Evolr (X)(s)’1 .Y (s) ds.

= T (µEvolr (X)(t) ). G

G

0

38.10

418 Chapter VIII. In¬nite dimensional di¬erential geometry 38.12

38.11. Current groups. We have another stability result: If G is regular and M

is a ¬nite dimensional manifold then also the space of all smooth mappings M ’ G

is a a regular Lie group, denoted by C∞ (M, G), with evolC∞ (M,G) = C∞ (M, evolG ),

see (42.21) below.

38.12. Theorem. For a regular Lie group G we have

evolr (X).evolr (Y ) = evolr t ’ X(t) + AdG (Evolr (X)(t)).Y (t) ,

evolr (X)’1 = evolr t ’ ’AdG (Evolr (X)(t)’1 ).X(t) ,

so that evolr : C ∞ (R, g) ’ G is a surjective smooth homomorphism of Lie groups,

where on C ∞ (R, g) we consider the operations

(X — Y )(t) = X(t) + AdG (Evolr (X)(t)).Y (t),

X ’1 (t) = ’AdG (Evolr (X)(t)’1 ).X(t).

With this operations and with 0 as unit element (C ∞ (R, g), —) becomes a regular

Lie group. Its Lie algebra is C ∞ (R, g) with bracket

t t

[X, Y ]C ∞ (R,g) (t) = X(s) ds, Y (t) + X(t), Y (s) ds

g g

0 0

t t

‚

= X(s) ds, Y (s) ds .

‚t

g

0 0

Its evolution operator is given by

1

AdG (EvolG (Y s )(v)’1 ).X(v)(s) dv,

s

evol(C ∞ (R,g),—) (X) := AdG (evolG (Y )).

0

s

Y s (t) := X(t)(u)du.

0

Proof. For X, Y ∈ C ∞ (R, g) we compute

Evolr (X)(t).Evolr (Y )(t) =

‚

‚t

r r r

= T (µEvol (Y )(t)

).T (µEvol (X)(t)

).X(t) + T (µEvolr (X)(t) ).T (µEvol (Y )(t)

).Y (t)

r

(X)(t).Evolr (Y )(t)

).(X(t) + AdG (Evolr (X)(t))Y (t)),

= T (µEvol

which implies also

Evolr (X)’1 = Evolr (X ’1 ).

Evolr (X).Evolr (Y ) = Evolr (X — Y ),

Thus, Evolr : C ∞ (R, g) ’ C ∞ (R, G) is a group isomorphism onto the subgroup

{c ∈ C ∞ (R, G) : c(0) = e} of C ∞ (R, G) with the pointwise product, which, how-

ever, is only a Fr¨licher space, see (23.1) Nevertheless, it follows that the product

o

38.12

38.12 38. Regular Lie groups 419

on C ∞ (R, g) is associative. It is clear that these operations are smooth, hence the

convenient vector space C ∞ (R, g) becomes a Lie group and C ∞ (R, G) becomes a

manifold.

Now we aim for the Lie bracket. We have

(X — Y —X ’1 )(t) = — ’Ad(Evolr (X)’1 ).X

X + Ad(Evolr (X)).Y (t)

= X(t) + Ad(Evolr (X)(t)).Y (t)’

’ Ad Evolr (X — Y )(t) .Ad Evolr (X)(t)’1 .X(t)

= X(t) + Ad Evolr (X)(t) .Y (t)’

’ Ad Evolr (X)(t) .Ad Evolr (Y )(t) .Ad Evolr (X)(t)’1 .X(t).

We shall need

T0 AdG (Evolr ( )(t)) .Y = Te AdG .T0 (Evolr ( )(t)).Y

t

= adg Y (s) ds , by (38.10).

0

Using this, we can di¬erentiate the conjugation,

) — X ’1 ).Y )(t)

(AdC ∞ (R,g) (X).Y )(t) = (T0 (X — (

= 0 + Ad(Evolr (X)(t)).Y (t)’

)(t))).Y .Ad(Evolr (X)(t)’1 ).X(t)

’ Ad(Evolr (X)(t)). T0 (Ad(Evolr (

= Ad(Evolr (X)(t)).Y (t)’

t

Y (s) ds .Ad(Evolr (X)(t)’1 ).X(t)

r

’ Ad(Evol (X)(t)).adg

0

t

r r

= Ad(Evol (X)(t)).Y (t) ’ adg . Ad(Evol (X)(t)). Y (s) ds .X(t).

0

Now we can compute the Lie bracket

[X,Y ]C ∞ (R,g) (t) = T0 (AdC ∞ (R,g) ( ).Y ).X (t)

t

r r

)(t)) .X .Y (t) ’ 0 ’ Ad(Evol (0)(t)).

= T0 Ad(Evol ( Y (s) ds, X(t)

g

0

t t

’

= X(s) ds, Y (t) Y (s) ds, X(t)

g g

0 0

t t

= X(s) ds, Y (t) + X(t), Y (s) ds

g g

0 0

t t

‚

= X(s) ds, Y (s) ds .

‚t

g

0 0

38.12

420 Chapter VIII. In¬nite dimensional di¬erential geometry 38.13

We show that the Lie group (C ∞ (R, g), —) is regular. Let X ∨ ∈ C ∞ (R, C ∞ (R, g))

correspond to X ∈ C ∞ (R2 , g). We look for g ∈ C ∞ (R2 , g) which satis¬es the

equation (38.4.1):

µg(t, )

(Y )(s) = (Y — g(t, ))(s) = Y (s) + AdG (EvolG (Y )(s)).g(t, s)

= T0 (µg(t, )

‚

‚t g(t, s) ).X(t, ) (s)

= X(t, s) + T0 AdG (EvolG ( )(s)) .X(t, ) .g(t, s)

s

= X(t, s) + adg X(t, u)du .g(t, s)

0

s

= X(t, s) + X(t, u)du, g(t, s) .

g

0

This is the di¬erential equation (38.10.3) for h(t), depending smoothly on a further

parameter s, which has the following unique solution given by (38.10.2)

t

AdG (EvolG (Y s )(v)’1 ).X(v, s) dv

s

g(t, s) := AdG (EvolG (Y )(t)).

0

s

Y s (t) := X(t, u)du.

0

Since this solution is obviously smooth in X, the Lie group C ∞ (R, g) is regular.

For convenience (yours, not ours) we show now (once more) that this, in fact, is a

s

solution. Putting Y s (t) := 0 X(t, u)du we have by (36.10.3)

‚

‚t g(t, s) =

t

Ad(Evol(Y s )(v)’1 ).X(v, s) dv

dAd( ‚t Evol(Y s )(t)).

‚

=

0

+ Ad(Evol(Y s )(t)).Ad(Evol(Y s )(t)’1 ).X(t, s)

t

Evol(Y s )(t)

Ad(Evol(Y s )(v)’1 ).X(v, s) dv

r s

= ((ad —¦ κ ).Ad) T (µ ).Y (t) .

0

+ X(t, s)

t

Ad(Evol(Y s )(v)’1 ).X(v, s) dv + X(t, s)

s s

= ad(Y (t)).Ad(Evol(Y )(t)).

0

s

= X(t, u)du, g(t, s) + X(t, s).

g

0

38.13. Corollary. Let G be a regular Lie group. Then as Fr¨licher spaces and

o

groups we have the following isomorphisms

G ∼ {f ∈ C ∞ (R, G) : f (0) = e} G ∼ C ∞ (R, G),

(C ∞ (R, g), —) = =

where g ∈ G acts on f by (±g (f ))(t) = g.f (t).g ’1 , and on X ∈ C ∞ (R, g) by

±g (X)(t) = AdG (g)(X(t)). The leftmost space is a smooth manifold, thus all spaces

are regular Lie groups.

38.13

38.14 38. Regular Lie groups 421

For the Lie algebras we have an isomorphism

g ∼ C ∞ (R, g),

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 ))