t

t

Claim 5. Let •t be a curve of local di¬eomorphisms through IdM with ¬rst

m

non-vanishing derivative m!X = ‚t |0 •t , and let ψt be a curve of local di¬eo-

n

morphisms through IdM with ¬rst non-vanishing derivative n!Y = ‚t |0 ψt .

Then the curve of local di¬eomorphisms [•t , ψt ] = ψt —¦ •’1 —¦ ψt —¦ •t has ¬rst

’1

t

non-vanishing derivative

m+n

|0 [•t , ψt ].

(m + n)![X, Y ] = ‚t

From this claim the theorem follows.

By the multinomial version of claim 3 we have

AN f : = ‚t |0 (ψt —¦ •’1 —¦ ψt —¦ •t )— f

’1

N

t

N! j

(‚t |0 •— )(‚t |0 ψt )(‚t |0 (•’1 )— )(‚t |0 (ψt )— )f.

’1

—

i k

= t

t

i!j!k! !

i+j+k+ =N

Let us suppose that 1 ¤ n ¤ m, the case m ¤ n is similar. If N < n all

summands are 0. If N = n we have by claim 4

AN f = (‚t |0 •— )f + (‚t |0 ψt )f + (‚t |0 (•’1 )— )f + (‚t |0 (ψt )— )f = 0.

’1

—

n n n n

t

t

If n < N ¤ m we have, using again claim 4:

N! j

(‚t |0 ψt )(‚t |0 (ψt )— )f + δN (‚t |0 •— )f + (‚t |0 (•’1 )— )f

’1

— m m m

AN f = t

t

j! !

j+ =N

’1

= (‚t |0 (ψt —¦ ψt )— )f + 0 = 0.

N

Now we come to the di¬cult case m, n < N ¤ m + n.

AN f = ‚t |0 (ψt —¦ •’1 —¦ ψt )— f +

’1

(‚t |0 •— )(‚t ’m |0 (ψt —¦ •’1 —¦ ψt )— )f

’1

N N

N m

t t

t

m

+ (‚t |0 •— )f,

N

(1) t

by claim 3, since all other terms vanish, see (3) below. By claim 3 again we get:

N! j

‚t |0 (ψt —¦ •’1 —¦ ψt )— f =

’1

(‚t |0 ψt )(‚t |0 (•’1 )— )(‚t |0 (ψt )— )f

’1

—

N k

t t

j!k! !

j+k+ =N

j ’1

(‚t ’m |0 ψt )(‚t |0 (•’1 )— )f

(‚t |0 ψt )(‚t |0 (ψt )— )f +

— —

N N N m

(2) = t

j m

j+ =N

(‚t |0 (•’1 )— )(‚t ’m |0 (ψt )— )f + ‚t |0 (•’1 )— f

’1

N N

m N

+ t t

m

(‚t ’m |0 ψt )m!L’X f + m!L’X (‚t ’m |0 (ψt )— )f

’1

—

N N

N N

=0+ m m

+ ‚t |0 (•’1 )— f

N

t

= δm+n (m + n)!(LX LY ’ LY LX )f + ‚t |0 (•’1 )— f

N N

t

= δm+n (m + n)!L[X,Y ] f + ‚t |0 (•’1 )— f

N N

t

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24 Chapter I. Manifolds and Lie groups

From the second expression in (2) one can also read o¬ that

‚t ’m |0 (ψt —¦ •’1 —¦ ψt )— f = ‚t ’m |0 (•’1 )— f.

’1

N N

(3) t t

If we put (2) and (3) into (1) we get, using claims 3 and 4 again, the ¬nal result

which proves claim 5 and the theorem:

AN f = δm+n (m + n)!L[X,Y ] f + ‚t |0 (•’1 )— f

N N

t

(‚t |0 •— )(‚t ’m |0 (•’1 )— )f + (‚t |0 •— )f

N N

m N

+ t

t t

m

= δm+n (m + n)!L[X,Y ] f + ‚t |0 (•’1 —¦ •t )— f

N N

t

N

= δm+n (m + n)!L[X,Y ] f + 0.

3.17. Theorem. Let X1 , . . . , Xm be vector ¬elds on M de¬ned in a neighbor-

hood of a point x ∈ M such that X1 (x), . . . , Xm (x) are a basis for Tx M and

[Xi , Xj ] = 0 for all i, j.

‚

Then there is a chart (U, u) of M centered at x such that Xi |U = ‚ui .

Proof. For small t = (t1 , . . . , tm ) ∈ Rm we put

f (t1 , . . . , tm ) = (FlX1 —¦ · · · —¦ FlXm )(x).

tm

t1

By 3.15 we may interchange the order of the ¬‚ows arbitrarily. Therefore

Xi

—¦ FlX1 —¦ · · · )(x) = Xi ((Flx11 —¦ · · · )(x)).

1

, tm ) =

‚ ‚

‚ti f (t , . . . ‚ti (Flti t1 t

So T0 f is invertible, f is a local di¬eomorphism, and its inverse gives a chart

with the desired properties.

3.18. Distributions. Let M be a manifold. Suppose that for each x ∈ M

we are given a sub vector space Ex of Tx M . The disjoint union E = x∈M Ex

is called a distribution on M . We do not suppose, that the dimension of Ex is

locally constant in x.

Let Xloc (M ) denote the set of all locally de¬ned smooth vector ¬elds on M ,

i.e. Xloc (M ) = X(U ), where U runs through all open sets in M . Furthermore

let XE denote the set of all local vector ¬elds X ∈ Xloc (M ) with X(x) ∈ Ex

whenever de¬ned. We say that a subset V ‚ XE spans E, if for each x ∈ M the

vector space Ex is the linear span of the set {X(x) : X ∈ V}. We say that E is a

smooth distribution if XE spans E. Note that every subset W ‚ Xloc (M ) spans

a distribution denoted by E(W), which is obviously smooth (the linear span of

the empty set is the vector space 0). From now on we will consider only smooth

distributions.

An integral manifold of a smooth distribution E is a connected immersed

submanifold (N, i) (see 2.8) such that Tx i(Tx N ) = Ei(x) for all x ∈ N . We

will see in theorem 3.22 below that any integral manifold is in fact an initial

submanifold of M (see 2.14), so that we need not specify the injective immersion

i. An integral manifold of E is called maximal if it is not contained in any strictly

larger integral manifold of E.

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3. Vector ¬elds and ¬‚ows 25

3.19. Lemma. Let E be a smooth distribution on M . Then we have:

1. If (N, i) is an integral manifold of E and X ∈ XE , then i— X makes sense

and is an element of Xloc (N ), which is i|i’1 (UX )-related to X, where UX ‚ M

is the open domain of X.

2. If (Nj , ij ) are integral manifolds of E for j = 1, 2, then i’1 (i1 (N1 ) ©

1

’1

i2 (N2 )) and i2 (i1 (N1 ) © i2 (N2 )) are open subsets in N1 and N2 , respectively;

furthermore i’1 —¦ i1 is a di¬eomorphism between them.

2

3. If x ∈ M is contained in some integral submanifold of E, then it is contained

in a unique maximal one.

Proof. 1. Let UX be the open domain of X ∈ XE . If i(x) ∈ UX for x ∈ N ,

we have X(i(x)) ∈ Ei(x) = Tx i(Tx N ), so i— X(x) := ((Tx i)’1 —¦ X —¦ i)(x) makes

sense. It is clearly de¬ned on an open subset of N and is smooth in x.

2. Let X ∈ XE . Then i— X ∈ Xloc (Nj ) and is ij -related to X. So by lemma

j

3.14 for j = 1, 2 we have

i— X X

ij —¦ Fltj = F lt —¦ ij .

Now choose xj ∈ Nj such that i1 (x1 ) = i2 (x2 ) = x0 ∈ M and choose vector

¬elds X1 , . . . , Xn ∈ XE such that (X1 (x0 ), . . . , Xn (x0 )) is a basis of Ex0 . Then

i— X1 i— Xn

fj (t1 , . . . , tn ) := (Fltj —¦ · · · —¦ Fltj )(xj )

n

1

‚

is a smooth mapping de¬ned near zero Rn ’ Nj . Since obviously ‚tk |0 fj =

i— Xk (xj ) for j = 1, 2, we see that fj is a di¬eomorphism near 0. Finally we have

j

i— X1 i— Xn

(i’1 —¦ i1 —¦ f1 )(t1 , . . . , tn ) = (i’1 —¦ i1 —¦ Flt1 —¦ · · · —¦ Flt1 )(x1 )

n

2 2 1

= (i’1 —¦ FlX1 —¦ · · · —¦ FlXn —¦i1 )(x1 )

tn

2 t1

i— X1 i— Xn

—¦i’1 —¦ i1 )(x1 )

—¦ · · · —¦ Flt2

= (Flt2 n 2

1

= f2 (t1 , . . . , tn ).

So i’1 —¦ i1 is a di¬eomorphism, as required.

2

3. Let N be the union of all integral manifolds containing x. Choose the union

of all the atlases of these integral manifolds as atlas for N , which is a smooth

atlas for N by 2. Note that a connected immersed submanifold of a separable

manifold is automatically separable (since it carries a Riemannian metric).

3.20. Integrable distributions and foliations.

A smooth distribution E on a manifold M is called integrable, if each point

of M is contained in some integral manifold of E. By 3.19.3 each point is

then contained in a unique maximal integral manifold, so the maximal integral

manifolds form a partition of M . This partition is called the foliation of M

induced by the integrable distribution E, and each maximal integral manifold

is called a leaf of this foliation. If X ∈ XE then by 3.19.1 the integral curve

t ’ FlX (t, x) of X through x ∈ M stays in the leaf through x.

Note, however, that usually a foliation is supposed to have constant dimen-

sions of the leafs, so our notion here is sometimes called a singular foliation.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

26 Chapter I. Manifolds and Lie groups

Let us now consider an arbitrary subset V ‚ Xloc (M ). We say that V is

stable if for all X, Y ∈ V and for all t for which it is de¬ned the local vector ¬eld

(FlX )— Y is again an element of V.

t

If W ‚ Xloc (M ) is an arbitrary subset, we call S(W) the set of all local vector

¬elds of the form (FlX1 —¦ · · · —¦ FlXk )— Y for Xi , Y ∈ W. By lemma 3.14 the ¬‚ow

tk

t1

of this vector ¬eld is

Fl((FlX1 —¦ · · · —¦ FlXk )— Y, t) = FlXkk —¦ · · · —¦ FlX11 —¦ FlY —¦ FlX1 —¦ · · · —¦ FlXk ,

’t

’t

tk tk

t1 t t1

so S(W) is the minimal stable set of local vector ¬elds which contains W.

Now let F be an arbitrary distribution. A local vector ¬eld X ∈ Xloc (M ) is

called an in¬nitesimal automorphism of F , if Tx (FlX )(Fx ) ‚ FFlX (t,x) whenever

t

de¬ned. We denote by aut(F ) the set of all in¬nitesimal automorphisms of F .

By arguments given just above, aut(F ) is stable.

3.21. Lemma. Let E be a smooth distribution on a manifold M . Then the

following conditions are equivalent:

(1) E is integrable.

(2) XE is stable.

(3) There exists a subset W ‚ Xloc (M ) such that S(W) spans E.

(4) aut(E) © XE spans E.

Proof. (1) =’ (2). Let X ∈ XE and let L be the leaf through x ∈ M , with

—

i : L ’ M the inclusion. Then FlX —¦i = i —¦ Fli X by lemma 3.14, so we have

’t ’t

Tx (FlX )(Ex ) = T (FlX ).Tx i.Tx L = T (FlX —¦i).Tx L

’t ’t ’t

—

= T i.Tx (Fli X ).Tx L

’t

= T i.TF li— X (’t,x) L = EF lX (’t,x) .

This implies that (FlX )— Y ∈ XE for any Y ∈ XE .

t

(2) =’ (4). In fact (2) says that XE ‚ aut(E).

(4) =’ (3). We can choose W = aut(E) © XE : for X, Y ∈ W we have

(FlX )— Y ∈ XE ; so W ‚ S(W) ‚ XE and E is spanned by W.

t

(3) =’ (1). We have to show that each point x ∈ M is contained in some

integral submanifold for the distribution E. Since S(W) spans E and is stable

we have

T (FlX ).Ex = EFlX (t,x)

(5) t

for each X ∈ S(W). Let dim Ex = n. There are X1 , . . . , Xn ∈ S(W) such that

X1 (x), . . . , Xn (x) is a basis of Ex , since E is smooth. As in the proof of 3.19.2

we consider the mapping

f (t1 , . . . , tn ) := (FlX1 —¦ · · · —¦ FlXn )(x),

tn

t1

de¬ned and smooth near 0 in Rn . Since the rank of f at 0 is n, the image

under f of a small open neighborhood of 0 is a submanifold N of M . We claim

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

3. Vector ¬elds and ¬‚ows 27

that N is an integral manifold of E. The tangent space Tf (t1 ,... ,tn ) N is linearly

generated by

X

—¦ · · · —¦ FlXn )(x) = T (FlX1 —¦ · · · —¦ Fltk’1 )Xk ((FlXk —¦ · · · —¦ FlXn )(x))

(FlX1

‚ k’1

tn tn

t1 t1

‚tk tk

X

= ((FlX11 )— · · · (Fl’tk’1 )— Xk )(f (t1 , . . . , tn )).

k’1

’t

Since S(W) is stable, these vectors lie in Ef (t) . From the form of f and from (5)

we see that dim Ef (t) = dim Ex , so these vectors even span Ef (t) and we have

Tf (t) N = Ef (t) as required.

3.22. Theorem (local structure of foliations). Let E be an integrable

distribution of a manifold M . Then for each x ∈ M there exists a chart (U, u)

with u(U ) = {y ∈ Rm : |y i | < µ for all i} for some µ > 0, and an at most

countable subset A ‚ Rm’n , such that for the leaf L through x we have

u(U © L) = {y ∈ u(U ) : (y n+1 , . . . , y m ) ∈ A}.

Each leaf is an initial submanifold.

If furthermore the distribution E has locally constant rank, this property

holds for each leaf meeting U with the same n.

This chart (U, u) is called a distinguished chart for the distribution or the

foliation. A connected component of U © L is called a plaque.

Proof. Let L be the leaf through x, dim L = n. Let X1 , . . . , Xn ∈ XE be local

vector ¬elds such that X1 (x), . . . , Xn (x) is a basis of Ex . We choose a chart

(V, v) centered at x on M such that the vectors

X1 (x), . . . , Xn (x), ‚v‚ |x , . . . , ‚vm |x

‚

n+1

form a basis of Tx M . Then

f (t1 , . . . , tm ) = (FlX1 —¦ · · · —¦ FlXn )(v ’1 (0, . . . , 0, tn+1 , . . . , tm ))

tn

t1

is a di¬eomorphism from a neighborhood of 0 in Rm onto a neighborhood of x

in M . Let (U, u) be the chart given by f ’1 , suitably restricted. We have

y ∈ L ⇐’ (FlX1 —¦ · · · —¦ FlXn )(y) ∈ L

tn

t1

for all y and all t1 , . . . , tn for which both expressions make sense. So we have

f (t1 , . . . , tm ) ∈ L ⇐’ f (0, . . . , 0, tn+1 , . . . , tm ) ∈ L,

and consequently L © U is the disjoint union of connected sets of the form

{y ∈ U : (un+1 (y), . . . , um (y)) = constant}. Since L is a connected immersed

submanifold of M , it is second countable and only a countable set of constants

can appear in the description of u(L©U ) given above. From this description it is

clear that L is an initial submanifold (2.14) since u(Cx (L©U )) = u(U )©(Rn —0).

The argument given above is valid for any leaf of dimension n meeting U , so

also the assertion for an integrable distribution of constant rank follows.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

28 Chapter I. Manifolds and Lie groups

3.23. Involutive distributions. A subset V ‚ Xloc (M ) is called involutive if

[X, Y ] ∈ V for all X, Y ∈ V. Here [X, Y ] is de¬ned on the intersection of the

domains of X and Y .

A smooth distribution E on M is called involutive if there exists an involutive

subset V ‚ Xloc (M ) spanning E.

For an arbitrary subset W ‚ Xloc (M ) let L(W) be the set consisting of

all local vector ¬elds on M which can be written as ¬nite expressions using

Lie brackets and starting from elements of W. Clearly L(W) is the smallest

involutive subset of Xloc (M ) which contains W.

3.24. Lemma. For each subset W ‚ Xloc (M ) we have

E(W) ‚ E(L(W)) ‚ E(S(W)).

In particular we have E(S(W)) = E(L(S(W))).

Proof. We will show that for X, Y ∈ W we have [X, Y ] ∈ XE(S(W)) , for then by

induction we get L(W) ‚ XE(S(W)) and E(L(W)) ‚ E(S(W)).

Let x ∈ M ; since by 3.21 E(S(W)) is integrable, we can choose the leaf L

through x, with the inclusion i. Then i— X is i-related to X, i— Y is i-related to

Y , thus by 3.10 the local vector ¬eld [i— X, i— Y ] ∈ Xloc (L) is i-related to [X, Y ],

and [X, Y ](x) ∈ E(S(W))x , as required.

3.25. Theorem. Let V ‚ Xloc (M ) be an involutive subset. Then the distribu-

tion E(V) spanned by V is integrable under each of the following conditions.

(1) M is real analytic and V consists of real analytic vector ¬elds.

(2) The dimension of E(V) is constant along all ¬‚ow lines of vector ¬elds in

V.

Proof. (1) For X, Y ∈ V we have dt (FlX )— Y = (FlX )— LX Y , consequently

d

t t

dk X— X— k

(Flt ) Y = (Flt ) (LX ) Y , and since everything is real analytic we get for

dtk

x ∈ M and small t

tk d k tk

(FlX )— Y | (FlX )— Y (x) = (LX )k Y (x).

(x) = k0

t t

k! dt k!

k≥0 k≥0

Since V is involutive, all (LX )k Y ∈ V. Therefore we get (FlX )— Y (x) ∈ E(V)x

t

for small t. By the ¬‚ow property of FlX the set of all t satisfying (FlX )— Y (x) ∈

t

E(V)x is open and closed, so it follows that 3.21.2 is satis¬ed and thus E(V) is

integrable.

(2) We choose X1 , . . . , Xn ∈ V such that X1 (x), . . . , Xn (x) is a basis of

E(V)x . For X ∈ V, by hypothesis, E(V)FlX (t,x) has also dimension n and ad-

mits X1 (FlX (t, x)), . . . , Xn (FlX (t, x)) as basis for small t. So there are smooth

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3. Vector ¬elds and ¬‚ows 29

functions fij (t) such that

n

X

fij (t)Xj (FlX (t, x)).

[X, Xi ](Fl (t, x)) =

j=1

X X

= T (FlX ).[X, Xi ](FlX (t, x)) =

d

dt T (Fl’t ).Xi (Fl (t, x)) ’t

n

fij (t)T (FlX ).Xj (FlX (t, x)).

= ’t

j=1

So the Tx M -valued functions gi (t) = T (FlX ).Xi (FlX (t, x)) satisfy the linear

’t

n

d

ordinary di¬erential equation dt gi (t) = j=1 fij (t)gj (t) and have initial values

in the linear subspace E(V)x , so they have values in it for all small t. There-

fore T (FlX )E(V)FlX (t,x) ‚ E(V)x for small t. Using compact time intervals

’t

and the ¬‚ow property one sees that condition 3.21.2 is satis¬ed and E(V) is

integrable.

Example. The distribution spanned by W ‚ Xloc (R2 ) is involutive, but not

integrable, where W consists of all global vector ¬elds with support in R2 \ {0}

‚

and the ¬eld ‚x1 ; the leaf through 0 should have dimension 1 at 0 and dimension

2 elsewhere.

3.26. By a time dependent vector ¬eld on a manifold M we mean a smooth

mapping X : J — M ’ T M with πM —¦ X = pr2 , where J is an open interval.

An integral curve of X is a smooth curve c : I ’ M with c(t) = X(t, c(t)) for

™

all t ∈ I, where I is a subinterval of J.

¯ ¯

There is an associated vector ¬eld X ∈ X(J — M ), given by X(t, x) =

(1t , X(t, x)) ∈ Tt R — Tx M .

By the evolution operator of X we mean the mapping ¦X : J — J — M ’ M ,

de¬ned in a maximal open neighborhood of the diagonal in M —M and satisfying

the di¬erential equation

dX

= X(t, ¦X (t, s, x))

dt ¦ (t, s, x)

X

¦ (s, s, x) = x.

¯

It is easily seen that (t, ¦X (t, s, x)) = FlX (t ’ s, (s, x)), so the maximally de¬ned

evolution operator exists and is unique, and it satis¬es

¦X = ¦ X —¦ ¦ X

t,s t,r r,s

whenever one side makes sense (with the restrictions of 3.7), where ¦X (x) =

t,s

¦(t, s, x).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

30 Chapter I. Manifolds and Lie groups

4. Lie groups

4.1. De¬nition. A Lie group G is a smooth manifold and a group such that

the multiplication µ : G — G ’ G is smooth. We shall see in a moment, that

then also the inversion ν : G ’ G turns out to be smooth.

We shall use the following notation:

µ : G — G ’ G, multiplication, µ(x, y) = x.y.

»a : G ’ G, left translation, »a (x) = a.x.

ρa : G ’ G, right translation, ρa (x) = x.a.

ν : G ’ G, inversion, ν(x) = x’1 .

e ∈ G, the unit element.

Then we have »a —¦ »b = »a.b , ρa —¦ ρb = ρb.a , »’1 = »a’1 , ρ’1 = ρa’1 , ρa —¦ »b =

a a

»b —¦ ρa . If • : G ’ H is a smooth homomorphism between Lie groups, then we

also have • —¦ »a = »•(a) —¦ •, • —¦ ρa = ρ•(a) —¦ •, thus also T •.T »a = T »•(a) .T •,

etc. So Te • is injective (surjective) if and only if Ta • is injective (surjective) for

all a ∈ G.

4.2. Lemma. T(a,b) µ : Ta G — Tb G ’ Tab G is given by

T(a,b) µ.(Xa , Yb ) = Ta (ρb ).Xa + Tb (»a ).Yb .

Proof. Let ria : G ’ G — G, ria (x) = (a, x) be the right insertion and let

lib : G ’ G — G, lib (x) = (x, b) be the left insertion. Then we have

T(a,b) µ.(Xa , Yb ) = T(a,b) µ.(Ta (lib ).Xa + Tb (ria ).Yb ) =

= Ta (µ —¦ lib ).Xa + Tb (µ —¦ ria ).Yb = Ta (ρb ).Xa + Tb (»a ).Yb .

4.3. Corollary. The inversion ν : G ’ G is smooth and

Ta ν = ’Te (ρa’1 ).Ta (»a’1 ) = ’Te (»a’1 ).Ta (ρa’1 ).

Proof. The equation µ(x, ν(x)) = e determines ν implicitly. Since we have

Te (µ(e, )) = Te (»e ) = Id, the mapping ν is smooth in a neighborhood of e by

the implicit function theorem. From (ν —¦ »a )(x) = x’1 .a’1 = (ρa’1 —¦ ν)(x) we

may conclude that ν is everywhere smooth. Now we di¬erentiate the equation

µ(a, ν(a)) = e; this gives in turn

0e = T(a,a’1 ) µ.(Xa , Ta ν.Xa ) = Ta (ρa’1 ).Xa + Ta’1 (»a ).Ta ν.Xa ,

Ta ν.Xa = ’Te (»a )’1 .Ta (ρa’1 ).Xa = ’Te (»a’1 ).Ta (ρa’1 ).Xa .

4.4. Example. The general linear group GL(n, R) is the group of all invertible

real n — n-matrices. It is an open subset of L(Rn , Rn ), given by det = 0 and a

Lie group.

Similarly GL(n, C), the group of invertible complex n — n-matrices, is a Lie

group; also GL(n, H), the group of all invertible quaternionic n — n-matrices, is

a Lie group, but the quaternionic determinant is a more subtle instrument here.

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4. Lie groups 31

4.5. Example. The orthogonal group O(n, R) is the group of all linear isome-

tries of (Rn , , ), where , is the standard positive de¬nite inner prod-

n

uct on R . The special orthogonal group SO(n, R) := {A ∈ O(n, R) : det A = 1}

is open in O(n, R), since

’1 0

O(n, R) = SO(n, R) SO(n, R),

0 In’1

where Ik is short for the identity matrix IdRk . We claim that O(n, R) and

SO(n, R) are submanifolds of L(Rn , Rn ). For that we consider the mapping

f : L(Rn , Rn ) ’ L(Rn , Rn ), given by f (A) = A.At . Then O(n, R) = f ’1 (In );

so O(n, R) is closed. Since it is also bounded, O(n, R) is compact. We have

df (A).X = X.At + A.X t , so ker df (In ) = {X : X + X t = 0} is the space o(n, R)

of all skew symmetric n — n-matrices. Note that dim o(n, R) = 1 (n ’ 1)n. If

2

A is invertible, we get ker df (A) = {Y : Y.A + A.Y = 0} = {Y : Y.At ∈

t t

o(n, R)} = o(n, R).(A’1 )t . The mapping f takes values in Lsym (Rn , Rn ), the

space of all symmetric n — n-matrices, and dim ker df (A) + dim Lsym (Rn , Rn ) =

1 1 2 n n n n

2 (n ’ 1)n + 2 n(n + 1) = n = dim L(R , R ), so f : GL(n, R) ’ Lsym (R , R )

is a submersion. Since obviously f ’1 (In ) ‚ GL(n, R), we conclude from 1.10

that O(n, R) is a submanifold of GL(n, R). It is also a Lie group, since the group

operations are obviously smooth.

4.6. Example. The special linear group SL(n, R) is the group of all n — n-

matrices of determinant 1. The function det : L(Rn , Rn ) ’ R is smooth and

d det(A)X = trace(C(A).X), where C(A)i , the cofactor of Aj , is the determinant

j i

j

of the matrix, which results from putting 1 instead of Ai into A and 0 in the rest

of the j-th row and the i-th column of A. We recall Cramer™s rule C(A).A =

A.C(A) = det(A).In . So if C(A) = 0 (i.e. rank(A) ≥ n ’ 1) then the linear

functional df (A) is non zero. So det : GL(n, R) ’ R is a submersion and

SL(n, R) = (det)’1 (1) is a manifold and a Lie group of dimension n2 ’ 1. Note

¬nally that TIn SL(n, R) = ker d det(In ) = {X : trace(X) = 0}. This space of

traceless matrices is usually called sl(n, R).

4.7. Example. The symplectic group Sp(n, R) is the group of all 2n — 2n-

matrices A such that ω(Ax, Ay) = ω(x, y) for all x, y ∈ R2n , where ω is the

standard non degenerate skew symmetric bilinear form on R2n .

Such a form exists on a vector space if and only if the dimension is even, and

on Rn —(Rn )— the standard form is given by ω((x, x— ), (y, y — )) = x, y — ’ y, x— ,

n

i.e. in coordinates ω((xi )2n , (y j )2n ) = i=1 (xi y n+i ’ xn+i y i ). Any symplectic

i=1 j=1

form on R2n looks like that after choosing a suitable basis. Let (ei )2n be the

i=1

standard basis in R2n . Then we have

0 In

(ω(ei , ej )i ) = =: J,

j

’In 0

and the matrix J satis¬es J t = ’J, J 2 = ’I2n , J x = ’x in Rn — Rn , and

y

y

ω(x, y) = x, Jy in terms of the standard inner product on R2n .

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32 Chapter I. Manifolds and Lie groups

For A ∈ L(R2n , R2n ) we have ω(Ax, Ay) = Ax, JAy = x, At JAy . Thus

A ∈ Sp(n, R) if and only if At JA = J.

We consider now the mapping f : L(R2n , R2n ) ’ L(R2n , R2n ) given by

f (A) = At JA. Then f (A)t = (At JA)t = ’At JA = ’f (A), so f takes val-

ues in the space o(2n, R) of skew symmetric matrices. We have df (A)X =

X t JA + At JX, and therefore

ker df (I2n ) = {X ∈ L(R2n , R2n ) : X t J + JX = 0}

= {X : JX is symmetric} =: sp(n, R).

We see that dim sp(n, R) = 2n(2n+1) = 2n+1 . Furthermore we have ker df (A) =

2 2

{X : X t JA + At JX = 0} and X ’ At JX is an isomorphism ker df (A) ’

Lsym (R2n , R2n ), if A is invertible. Thus dim ker df (A) = 2n+1 for all A ∈

2

GL(2n, R). If f (A) = J, then At JA = J, so A has rank 2n and is invertible, and

dim ker df (A) + dim o(2n, R) = 2n+1 + 2n(2n’1) = 4n2 = dim L(R2n , R2n ). So

2 2

f : GL(2n, R) ’ o(2n, R) is a submersion and f ’1 (J) = Sp(n, R) is a manifold

and a Lie group. It is the symmetry group of ˜classical mechanics™.

4.8. Example. The complex general linear group GL(n, C) of all invertible

complex n — n-matrices is open in LC (Cn , Cn ), so it is a real Lie group of real

dimension 2n2 ; it is also a complex Lie group of complex dimension n2 . The

complex special linear group SL(n, C) of all matrices of determinant 1 is a sub-

manifold of GL(n, C) of complex codimension 1 (or real codimension 2).

The complex orthogonal group O(n, C) is the set

{A ∈ L(Cn , Cn ) : g(Az, Aw) = g(z, w) for all z, w},

n

where g(z, w) = i=1 z i wi . This is a complex Lie group of complex dimension

(n’1)n

, and it is not compact. Since O(n, C) = {A : At A = In }, we have

2

1 = detC (In ) = detC (At A) = detC (A)2 , so detC (A) = ±1. Thus SO(n, C) :=

{A ∈ O(n, C) : detC (A) = 1} is an open subgroup of index 2 in O(n, C).

The group Sp(n, C) = {A ∈ LC (C2n , C2n ) : At JA = J} is also a complex Lie

group of complex dimension n(2n + 1).

These groups here are the classical complex Lie groups. The groups SL(n, C)

for n ≥ 2, SO(n, C) for n ≥ 3, Sp(n, C) for n ≥ 4, and ¬ve more exceptional

groups exhaust all simple complex Lie groups up to coverings.

4.9. Example. Let Cn be equipped with the standard hermitian inner product

n ii

i=1 z w . The unitary group U (n) consists of all complex n — n-

(z, w) =

matrices A such that (Az, Aw) = (z, w) for all z, w holds, or equivalently U (n) =

t

{A : A— A = In }, where A— = A .

We consider the mapping f : LC (Cn , Cn ) ’ LC (Cn , Cn ), given by f (A) =

A— A. Then f is smooth but not holomorphic. Its derivative is df (A)X =

X — A + A— X, so ker df (In ) = {X : X — + X = 0} =: u(n), the space of all skew

hermitian matrices. We have dimR u(n) = n2 . As above we may check that

f : GL(n, C) ’ Lherm (Cn , Cn ) is a submersion, so U (n) = f ’1 (In ) is a compact

real Lie group of dimension n2 .

The special unitary group is SU (n) = U (n) © SL(n, C). For A ∈ U (n) we

have | detC (A)| = 1, thus dimR SU (n) = n2 ’ 1.

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4. Lie groups 33

4.10. Example. The group Sp(n). Let H be the division algebra of quater-

nions. Then Sp(1) := S 3 ‚ H ∼ R4 is the group of unit quaternions, obviously

=

a Lie group.

Now let V be a right vector space over H. Since H is not commutative, we

have to distinguish between left and right vector spaces and we choose right ones

as basic, so that matrices can multiply from the left. By choosing a basis we get

n

V = Rn —R H = Hn . For u = (ui ), v = (v i ) ∈ Hn we put u, v := i=1 ui v i .

is R-bilinear and ua, vb = a u, v b for a, b ∈ H.

Then ,

An R linear mapping A : V ’ V is called H-linear or quaternionically linear

if A(ua) = A(u)a holds. The space of all such mappings shall be denoted by

LH (V, V ). It is real isomorphic to the space of all quaternionic n — n-matrices

with the usual multiplication, since for the standard basis (ei )n in V = Hn we

i=1

have A(u) = A( i ei ui ) = i A(ei )ui = i,j ej Aj ui . Note that LH (V, V ) is

i

only a real vector space, if V is a right quaternionic vector space - any further

structure must come from a second (left) quaternionic vector space structure on

V.

GL(n, H), the group of invertible H-linear mappings of Hn , is a Lie group,

because it is GL(4n, R) © LH (Hn , Hn ), open in LH (Hn , Hn ).

A quaternionically linear mapping A is called isometric or quaternionically

unitary, if A(u), A(v) = u, v for all u, v ∈ Hn . We denote by Sp(n) the

group of all quaternionic isometries of Hn , the quaternionic unitary group. The

reason for its name is that Sp(n) = Sp(2n, C) © U (2n), since we can decompose

the quaternionic hermitian form , into a complex hermitian one and a

complex symplectic one. Also we have Sp(n) ‚ O(4n, R), since the real part of

is a positive de¬nite real inner product. For A ∈ LH (Hn , Hn ) we put

,

t

A— := A . Then we have u, A(v) = A— (u), v , so A(u), A(v) = A— A(u), v .

Thus A ∈ Sp(n) if and only if A— A = Id.

Again f : LH (Hn , Hn ) ’ LH,herm (Hn , Hn ) = {A : A— = A}, given by f (A) =

A— A, is a smooth mapping with df (A)X = X — A+A— X. So we have ker df (Id) =

{X : X — = ’X} =: sp(n), the space of quaternionic skew hermitian matrices.

The usual proof shows that f has maximal rank on GL(n, H), so Sp(n) = f ’1 (Id)

is a compact real Lie group of dimension 2n(n ’ 1) + 3n.

The groups SO(n, R) for n ≥ 3, SU (n) for n ≥ 2, Sp(n) for n ≥ 2 and

real forms of the exceptional complex Lie groups exhaust all simple compact Lie

groups up to coverings.

4.11. Invariant vector ¬elds and Lie algebras. Let G be a (real) Lie group.

A vector ¬eld ξ on G is called left invariant, if »— ξ = ξ for all a ∈ G, where

a

»— ξ = T (»a’1 )—¦ξ —¦»a as in section 3. Since by 3.11 we have »— [ξ, ·] = [»— ξ, »— ·],

a a a a

the space XL (G) of all left invariant vector ¬elds on G is closed under the Lie

bracket, so it is a sub Lie algebra of X(G). Any left invariant vector ¬eld ξ

is uniquely determined by ξ(e) ∈ Te G, since ξ(a) = Te (»a ).ξ(e). Thus the Lie

algebra XL (G) of left invariant vector ¬elds is linearly isomorphic to Te G, and

on Te G the Lie bracket on XL (G) induces a Lie algebra structure, whose bracket

is again denoted by [ , ]. This Lie algebra will be denoted as usual by g,

sometimes by Lie(G).

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34 Chapter I. Manifolds and Lie groups

We will also give a name to the isomorphism with the space of left invariant

vector ¬elds: L : g ’ XL (G), X ’ LX , where LX (a) = Te »a .X. Thus [X, Y ] =

[LX , LY ](e).

A vector ¬eld · on G is called right invariant, if ρ— · = · for all a ∈ G. If

a

—

ξ is left invariant, then ν ξ is right invariant, since ν —¦ ρa = »a’1 —¦ ν implies

that ρ— ν — ξ = (ν —¦ ρa )— ξ = (»a’1 —¦ ν)— ξ = ν — (»a’1 )— ξ = ν — ξ. The right invariant

a

vector ¬elds form a sub Lie algebra XR (G) of X(G), which is again linearly

isomorphic to Te G and induces also a Lie algebra structure on Te G. Since

ν — : XL (G) ’ XR (G) is an isomorphism of Lie algebras by 3.11, Te ν = ’ Id :

Te G ’ Te G is an isomorphism between the two Lie algebra structures. We will

denote by R : g = Te G ’ XR (G) the isomorphism discussed, which is given by

RX (a) = Te (ρa ).X.

4.12. Lemma. If LX is a left invariant vector ¬eld and RY is a right invariant

one, then [LX , RY ] = 0. Thus the ¬‚ows of LX and RY commute.

Proof. We consider 0 — LX ∈ X(G — G), given by (0 — LX )(a, b) = (0a , LX (b)).

Then T(a,b) µ.(0a , LX (b)) = Ta ρb .0a + Tb »a .LX (b) = LX (ab), so 0 — LX is µ-

related to LX . Likewise RY —0 is µ-related to RY . But then 0 = [0—LX , RY —0]

is µ-related to [LX , RY ] by 3.10. Since µ is surjective, [LX , RY ] = 0 follows.

4.13. Let • : G ’ H be a homomorphism of Lie groups, so for the time being

we require • to be smooth.

Lemma. Then • := Te • : g = Te G ’ h = Te H is a Lie algebra homomor-

phism.

Proof. For X ∈ g and x ∈ G we have

Tx •.LX (x) = Tx •.Te »x .X = Te (• —¦ »x ).X =

Te (»•(x) —¦ •).X = Te (»•(x) ).Te •.X = L• (X) (•(x)).

So LX is •-related to L• (X) . By 3.10 the ¬eld [LX , LY ] = L[X,Y ] is •-related

to [L• (X) , L• (Y ) ] = L[• (X),• (Y )] . So we have T • —¦ L[X,Y ] = L[• (X),• (Y )] —¦ •.

If we evaluate this at e the result follows.

Now we will determine the Lie algebras of all the examples given above.

4.14. For the Lie group GL(n, R) we have Te GL(n, R) = L(Rn , Rn ) =: gl(n, R)

and T GL(n, R) = GL(n, R) — L(Rn , Rn ) by the a¬ne structure of the sur-

rounding vector space. For A ∈ GL(n, R) we have »A (B) = A.B, so »A

extends to a linear isomorphism of L(Rn , Rn ), and for (B, X) ∈ T GL(n, R)

we get TB (»A ).(B, X) = (A.B, A.X). So the left invariant vector ¬eld LX ∈

XL (GL(n, R)) is given by LX (A) = Te (»A ).X = (A, A.X).

Let f : GL(n, R) ’ R be the restriction of a linear functional on L(Rn , Rn ).

Then we have LX (f )(A) = df (A)(LX (A)) = df (A)(A.X) = f (A.X), which we

may write as LX (f ) = f ( .X). Therefore

L[X,Y ] (f ) = [LX , LY ](f ) = LX (LY (f )) ’ LY (LX (f )) =

= LX (f ( .Y )) ’ LY (f ( .X)) =

= f ( .X.Y ) ’ f ( .Y.X) = LXY ’Y X (f ).

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4. Lie groups 35

So the Lie bracket on gl(n, R) = L(Rn , Rn ) is given by [X, Y ] = XY ’ Y X, the

usual commutator.

4.15. Example. Let V be a vector space. Then (V, +) is a Lie group, T0 V = V

is its Lie algebra, T V = V —V , left translation is »v (w) = v+w, Tw (»v ).(w, X) =

(v + w, X). So LX (v) = (v, X), a constant vector ¬eld. Thus the Lie bracket is

0.

4.16. Example. The special linear group is SL(n, R) = det’1 (1) and its Lie

algebra is given by Te SL(n, R) = ker d det(I) = {X ∈ L(Rn , Rn ) : trace X =

0} = sl(n, R) by 4.6. The injection i : SL(n, R) ’ GL(n, R) is a smooth

homomorphism of Lie groups, so Te i = i : sl(n, R) ’ gl(n, R) is an injective

homomorphism of Lie algebras. Thus the Lie bracket is given by [X, Y ] =

XY ’ Y X.

The same argument gives the commutator as the Lie bracket in all other

examples we have treated. We have already determined the Lie algebras as Te G.

4.17. One parameter subgroups. Let G be a Lie group with Lie algebra g.

A one parameter subgroup of G is a Lie group homomorphism ± : (R, +) ’ G,

i.e. a smooth curve ± in G with ±(0) = e and ±(s + t) = ±(s).±(t).

Lemma. Let ± : R ’ G be a smooth curve with ±(0) = e. Let X = ±(0) ∈ g.

™

Then the following assertions are equivalent.

(1) ± is a one parameter subgroup.

±(t) = FlLX (t, e) for all t.

(2)

±(t) = FlRX (t, e) for all t.

(3)

x.±(t) = FlLX (t, x), or FlLX = ρ±(t) , for all t.

(4) t

±(t).x = FlRX (t, x), or FlRX = »±(t) , for all t.

(5) t

Proof. (1) =’ (4).

d d d d

ds |0 x.±(t ds |0 x.±(t).±(s) ds |0 »x.±(t) ±(s)

dt x.±(t) = + s) = =

d

= Te (»x.±(t) ). ds |0 ±(s) = LX (x.±(t)).

By uniqueness of solutions we get x.±(t) = FlLX (t, x).

(4) =’ (2). This is clear.

d d d

(2) =’ (1). We have ds ±(t)±(s) = ds (»±(t) ±(s)) = T (»±(t) ) ds ±(s) =

T (»±(t) )LX (±(s)) = LX (±(t)±(s)) and ±(t)±(0) = ±(t). So we get ±(t)±(s) =

FlLX (s, ±(t)) = FlLX FlLX (e) = FlLX (t + s, e) = ±(t + s).

s t —

(4) ⇐’ (5). We have Flν ξ = ν ’1 —¦ Flξ —¦ν by 3.14. Therefore we have by 4.11

t t

—

(FlRX (x’1 ))’1 = (ν —¦ FlRX —¦ν)(x) = Flν RX

(x)

t t t

= FlLX (x) = x.±(’t).

’t

So FlRX (x’1 ) = ±(t).x’1 , and FlRX (y) = ±(t).y.

t t

(5) =’ (3) =’ (1) can be shown in a similar way.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

36 Chapter I. Manifolds and Lie groups

An immediate consequence of the foregoing lemma is that left invariant and

the right invariant vector ¬elds on a Lie group are always complete, so they

have global ¬‚ows, because a locally de¬ned one parameter group can always be

extended to a globally de¬ned one by multiplying it up.

4.18. De¬nition. The exponential mapping exp : g ’ G of a Lie group is

de¬ned by

exp X = FlLX (1, e) = FlRX (1, e) = ±X (1),

where ±X is the one parameter subgroup of G with ±X (0) = X.

™

Theorem.

exp : g ’ G is smooth.

(1)

exp(tX) = FlLX (t, e).

(2)

FlLX (t, x) = x. exp(tX).

(3)

FlRX (t, x) = exp(tX).x.

(4)

exp(0) = e and T0 exp = Id : T0 g = g ’ Te G = g, thus exp is a

(5)

di¬eomorphism from a neighborhood of 0 in g onto a neighborhood of e

in G.

Proof. (1) Let 0 — L ∈ X(g — G) be given by (0 — L)(X, x) = (0X , LX (x)). Then

pr2 Fl0—L (t, (X, e)) = ±X (t) is smooth in (t, X).

(2) exp(tX) = Flt.LX (1, e) = FlLX (t, e) = ±X (t).

(3) and (4) follow from lemma 4.17.

(5) T0 exp .X = dt |0 exp(0 + t.X) = dt |0 FlLX (t, e) = X.

d d

4.19. Remark. If G is connected and U ‚ g is open with 0 ∈ U , then the

group generated by exp(U ) equals G.

For this group is a subgroup of G containing some open neighborhood of e,

so it is open. The complement in G is also open (as union of the other cosets),

so this subgroup is open and closed. Since G is connected, it coincides with G.

If G is not connected, then the subgroup generated by exp(U ) is the connected

component of e in G.

4.20. Remark. Let • : G ’ H be a smooth homomorphism of Lie groups.

Then the diagram

w

•

g h

u u

expG expH

wH

•

G

commutes, since t ’ •(expG (tX)) is a one parameter subgroup of H and

d G G H

dt |0 •(exp tX) = • (X), so •(exp tX) = exp (t• (X)).

If G is connected and •, ψ : G ’ H are homomorphisms of Lie groups with

• = ψ : g ’ h, then • = ψ. For • = ψ on the subgroup generated by expG g

which equals G by 4.19.

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4. Lie groups 37

4.21. Theorem. A continuous homomorphism • : G ’ H between Lie groups

is smooth. In particular a topological group can carry at most one compatible

Lie group structure.

Proof. Let ¬rst • = ± : (R, +) ’ G be a continuous one parameter subgroup.

Then ±(’µ, µ) ‚ exp(U ), where U is an absolutely convex open neighbor-

hood of 0 in g such that exp |2U is a di¬eomorphism, for some µ > 0. Put

β := (exp |2U )’1 —¦ ± : (’µ, µ) ’ g. Then for |t| < 1 we have exp(2β(t)) =

µ

exp(β(t))2 = ±(t)2 = ±(2t) = exp(β(2t)), so 2β(t) = β(2t); thus β( 2 ) = 1 β(s) s

2

for |s| < µ. So we have ±( 2 ) = exp(β( 2 )) = exp( 1 β(s)) for all |s| < µ and by

s s

2

recursion we get ±( 2s ) = exp( 21 β(s)) for n ∈ N and in turn ±( 2k s) = ±( 2s )k =

n n n n

1 k k

k

exp( 2n β(s)) = exp( 2n β(s)) for k ∈ Z. Since the 2n for k ∈ Z and n ∈ N are

dense in R and since ± is continuous we get ±(ts) = exp(tβ(s)) for all t ∈ R. So

± is smooth.

Now let • : G ’ H be a continuous homomorphism. Let X1 , . . . , Xn be a lin-

ear basis of g. We de¬ne ψ : Rn ’ G as ψ(t1 , . . . , tn ) = exp(t1 X1 ) · · · exp(tn Xn ).

Then T0 ψ is invertible, so ψ is a di¬eomorphism near 0. Sometimes ψ ’1 is called

a coordinate system of the second kind. t ’ •(expG tXi ) is a continuous one

parameter subgroup of H, so it is smooth by the ¬rst part of the proof. We have

(• —¦ ψ)(t1 , . . . , tn ) = (• exp(t1 X1 )) · · · (• exp(tn Xn )), so • —¦ ψ is smooth. Thus

• is smooth near e ∈ G and consequently everywhere on G.

4.22. Theorem. Let G and H be Lie groups (G separable is essential here),

and let • : G ’ H be a continuous bijective homomorphism. Then • is a

di¬eomorphism.

Proof. Our ¬rst aim is to show that • is a homeomorphism. Let V be an

open e-neighborhood in G, and let K be a compact e-neighborhood in G such

that K.K ’1 ‚ V . Since G is separable there is a sequence (ai )i∈N in G such

∞

that G = i=1 ai .K. Since H is locally compact, it is a Baire space (Vi open

and dense implies Vi dense). The set •(ai )•(K) is compact, thus closed.

Since H = i •(ai ).•(K), there is some i such that •(ai )•(K) has non empty

interior, so •(K) has non empty interior. Choose b ∈ G such that •(b) is an

interior point of •(K) in H. Then eH = •(b)•(b’1 ) is an interior point of

•(K)•(K ’1 ) ‚ •(V ). So if U is open in G and a ∈ U , then eH is an interior

point of •(a’1 U ), so •(a) is in the interior of •(U ). Thus •(U ) is open in H,

and • is a homeomorphism.

Now by 4.21 • and •’1 are smooth.

4.23. Examples. The exponential mapping on GL(n, R). Let X ∈ gl(n, R) =

L(Rn , Rn ), then the left invariant vector ¬eld is given by LX (A) = (A, A.X) ∈

GL(n, R) — gl(n, R) and the one parameter group ±X (t) = FlLX (t, I) is given

d

by the di¬erential equation dt ±X (t) = LX (±X (t)) = ±X (t).X, with initial con-

dition ±X (0) = I. But the unique solution of this equation is ±X (t) = etX =

∞ tk k

k=0 k! X . So

∞

expGL(n,R) (X) = eX = 1

Xk.

k=0 k!

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38 Chapter I. Manifolds and Lie groups

If n = 1 we get the usual exponential mapping of one real variable. For all Lie

subgroups of GL(n, R), the exponential mapping is given by the same formula

exp(X) = eX ; this follows from 4.20.

4.24. The adjoint representation. A representation of a Lie group G on a

¬nite dimensional vector space V (real or complex) is a homomorphism ρ : G ’

GL(V ) of Lie groups. Then by 4.13 ρ : g ’ gl(V ) = L(V, V ) is a Lie algebra

homomorphism.

For a ∈ G we de¬ne conja : G ’ G by conja (x) = axa’1 . It is called

the conjugation or the inner automorphism by a ∈ G. We have conja (xy) =

conja (x) conja (y), conjab = conja —¦ conjb , and conj is smooth in all variables.

Next we de¬ne for a ∈ G the mapping Ad(a) = (conja ) = Te (conja ) : g ’ g.

By 4.13 Ad(a) is a Lie algebra homomorphism, so we have Ad(a)[X, Y ] =

[Ad(a)X, Ad(a)Y ]. Furthermore Ad : G ’ GL(g) is a representation, called

the adjoint representation of G, since Ad(ab) = Te (conjab ) = Te (conja —¦ conjb ) =

Te (conja ) —¦ Te (conjb ) = Ad(a) —¦ Ad(b). We will use the relations Ad(a) =

Te (conja ) = Ta (ρa’1 ).Te (»a ) = Ta’1 (»a ).Te (ρa’1 ).

Finally we de¬ne the (lower case) adjoint representation of the Lie algebra g,

ad : g ’ gl(g) = L(g, g), by ad := Ad = Te Ad.

Lemma. (1) LX (a) = RAd(a)X (a) for X ∈ g and a ∈ G.

(2) ad(X)Y = [X, Y ] for X, Y ∈ g.

Proof. (1) LX (a) = Te (»a ).X = Te (ρa ).Te (ρa’1 —¦ »a ).X = RAd(a)X (a).

(2) Let X1 , . . . , Xn be a linear basis of g and ¬x X ∈ g. Then Ad(x)X =

n ∞

i=1 fi (x).Xi for fi ∈ C (G, R) and we have in turn

Ad (Y )X = Te (Ad( )X)Y = d(Ad( )X)|e Y = d( fi Xi )|e Y =

dfi |e (Y )Xi =

= LY (fi )(e).Xi .

LX (x) = RAd(x)X (x) = R( fi (x)Xi )(x) = fi (x).RXi (x) by (1).

[LY , LX ] = [LY , fi .RXi ] = 0 + LY (fi ).RXi by 3.4 and 4.12.

[Y, X] = [LY , LX ](e) = LY (fi )(e).RXi (e) = Ad (Y )X = ad(Y )X.

4.25. Corollary. From 4.20 and 4.23 we have

Ad —¦expG = expGL(g) —¦ ad

∞

G

(ad X)k Y = ead X Y

1

Ad(exp X)Y = k!

k=0

1 1

]]] + · · ·

= Y + [X, Y ] + 2! [X, [X, Y ]] + 3! [X, [X, [X, Y

4.26. The right logarithmic derivative. 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 δf : T M ’ g by the formula δf (ξx ) := Tf (x) (ρf (x)’1 ).Tx f.ξx .

Then δf is a g-valued 1-form on M , δf ∈ „¦1 (M, g), as we will write later. We

call δf the right logarithmic derivative of f , since for f : R ’ (R+ , ·) we have

δf (x).1 = f (x) = (log —¦f ) (x).

(x)

f

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4. Lie groups 39

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

δ(f.g)(x) = δf (x) + Ad(f (x)).δg(x).

Proof. We just compute:

δ(f.g)(x) = T (ρg(x)’1 .f (x)’1 ).Tx (f.g) =

= T (ρf (x)’1 ).T (ρg(x)’1 ).T(f (x),g(x)) µ.(Tx f, Tx g) =

= T (ρf (x)’1 ).T (ρg(x)’1 ). T (ρg(x) ).Tx f + T (»f (x) ).Tx g =

= δf (x) + Ad(f (x)).δg(x).

Remark. The left logarithmic derivative δ left f ∈ „¦1 (M, g) of a smooth mapping

f : M ’ G is given by δ left f.ξx = Tf (x) (»f (x)’1 ).Tx f.ξx . The corresponding

Leibnitz rule for it is uglier than that for the right logarithmic derivative:

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

The form δ left (IdG ) ∈ „¦1 (G; g) is also called the Maurer-Cartan form of the Lie

group G.

ez ’ 1

4.27. Lemma. For exp : g ’ G and for g(z) := we have

z

∞

(ad X)p = g(ad X).

1

δ(exp)(X) = T (ρexp(’X) ).TX exp = (p+1)!

p=0

Proof. We put M (X) = δ(exp)(X) : g ’ g. Then

(s + t)M ((s + t)X) = (s + t)δ(exp)((s + t)X)

= δ(exp((s + t) ))X by the chain rule,

= δ(exp(s ). exp(t )).X

= δ(exp(s )).X + Ad(exp(sX)).δ(exp(t )).X by 4.26,

= s.δ(exp)(sX) + Ad(exp(sX)).t.δ(exp)(tX)

= s.M (sX) + Ad(exp(sX)).t.M (tX).

Now we put N (t) := t.M (tX) ∈ L(g, g), then the above equation gives N (s+t) =

d

N (s) + Ad(exp(sX)).N (t). We ¬x t, apply ds |0 , and get N (t) = N (0) +

ad(X).N (t), where N (0) = M (0) + 0 = δ(exp)(0) = Idg . So we have the

di¬erential equation N (t) = Idg + ad(X).N (t) in L(g, g) with initial condition

N (0) = 0. The unique solution is

∞

ad(X)p .sp+1 ,

1

N (s) = and so

(p+1)!

p=0

∞

ad(X)p .

1

δ(exp)(X) = M (X) = N (1) = (p+1)!

p=0

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40 Chapter I. Manifolds and Lie groups

4.28. Corollary. TX exp is bijective if and only if no eigenvalue of ad(X) :

√

g ’ g is of the form ’1 2kπ for k ∈ Z \ {0}.

√

z

’1

Proof. The zeros of g(z) = e z are exactly z = ’1 2kπ for k ∈ Z \ {0}. The

linear mapping TX exp is bijective if and only if no eigenvalue of g(ad(X)) =

T (ρexp(’X) ).TX exp is 0. But the eigenvalues of g(ad(X)) are the images under

g of the eigenvalues of ad(X).

4.29. Theorem. The Baker-Campbell-Hausdor¬ formula.

Let G be a Lie group with Lie algebra g. For complex z near 1 we consider the

n

function f (z) := log(z) = n≥0 (’1) (z ’ 1)n .

z’1 n+1

Then for X, Y near 0 in g we have exp X. exp Y = exp C(X, Y ), where

1

f (et. ad X .ead Y ).X dt

C(X, Y ) = Y +

0

n

1

(’1)n tk

(ad X)k (ad Y )

=X +Y + X dt

n+1 k! !

0 k, ≥0

n≥1

k+ ≥1

(’1)n (ad X)k1 (ad Y ) 1 . . . (ad X)kn (ad Y ) n

=X +Y + X

(k1 + · · · + kn + 1)k1 ! . . . kn ! 1 ! . . . n !

n+1

k1 ,...,kn ≥0

n≥1

1 ,... n ≥0

ki + i ≥1

= X + Y + 1 [X, Y ] + 1

]] ’ [Y, [Y, X]]) + · · ·

12 ([X, [X, Y

2

Proof. Let C(X, Y ) := exp’1 (exp X. exp Y ) for X, Y near 0 in g, and let C(t) :=

C(tX, Y ). Then

™

d

T (ρexp(’C(t)) ) dt (exp C(t)) = δ exp(C(t)).C(t)

™ ™

1

(ad C(t))k C(t) = g(ad C(t)).C(t),

= k≥0 (k+1)!

z k

’1

where g(z) := e z = z

k≥0 (k+1)! . We have exp C(t) = exp(tX) exp Y and

exp(’C(t)) = exp(C(t))’1 = exp(’Y ) exp(’tX), therefore

d d

T (ρexp(’C(t)) ) dt (exp C(t)) = T (ρexp(’Y ) exp(’tX) ) dt (exp(tX) exp Y )

d

= T (ρexp(’tX) )T (ρexp(’Y ) )T (ρexp Y ) dt exp(tX)

= T (ρexp(’tX) ).RX (exp(tX)) = X, by 4.18.4 and 4.11.

™

X = g(ad C(t)).C(t).

ead C(t) = Ad(exp C(t)) by 4.25

= Ad(exp(tX) exp Y ) = Ad(exp(tX)). Ad(exp Y )

= ead(tX) .ead Y = et. ad X .ead Y .

If X, Y , and t are small enough we get ad C(t) = log(et. ad X .ead Y ), where

n+1

log(z) = n≥1 (’1) (z ’ 1)n , thus we have

n

™ ™

X = g(ad C(t)).C(t) = g(log(et. ad X .ead Y )).C(t).

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5. Lie subgroups and homogeneous Spaces 41

(’1)n

log(z)

’ 1)n , which satis¬es

For z near 1 we put f (z) := = n≥0 n+1 (z

z’1

g(log(z)).f (z) = 1. So we have

™ ™

X = g(log(et. ad X .ead Y )).C(t) = f (et. ad X .ead Y )’1 .C(t),

™

C(t) = f (et. ad X .ead Y ).X,

C(0) = Y.

Passing to the de¬nite integral we get the desired formula

1

™

C(X, Y ) = C(1) = C(0) + C(t) dt

0

1

f (et. ad X .ead Y ).X dt

=Y +

0

n

1

(’1)n tk

(ad X)k (ad Y )

=X +Y + X dt.

n+1 k! !

0 k, ≥0

n≥1

k+ ≥1

Remark. If G is a Lie group of di¬erentiability class C 2 , then we may de¬ne

T G and the Lie bracket of vector ¬elds. The proof above then makes sense

and the theorem shows, that in the chart given by exp’1 the multiplication

µ : G — G ’ G is C ω near e, hence everywhere. So in this case G is a real

analytic Lie group. See also remark 5.6 below.

4.30. Convention. We will use the following convention for the rest of the

book. If we write a symbol of a classical group from this section without indi-

cating the ground ¬eld, then we always mean the ¬eld R ” except Sp(n). In

particular GL(n) = GL(n, R), and O(n) = O(n, R) from now on.

5. Lie subgroups and homogeneous spaces

5.1. De¬nition. Let G be a Lie group. A subgroup H of G is called a Lie

subgroup, if H is itself a Lie group (so it is separable) and the inclusion i : H ’ G

is smooth.

In this case the inclusion is even an immersion. For that it su¬ces to check

that Te i is injective: If X ∈ h is in the kernel of Te i, then i —¦ expH (tX) =

expG (t.Te i.X) = e. Since i is injective, X = 0.

From the next result it follows that H ‚ G is then an initial submanifold in

the sense of 2.14: If H0 is the connected component of H, then i(H0 ) is the Lie

subgroup of G generated by i (h) ‚ g, which is an initial submanifold, and this

is true for all components of H.

5.2. Theorem. Let G be a Lie group with Lie algebra g. If h ‚ g is a Lie

subalgebra, then there is a unique connected Lie subgroup H of G with Lie

algebra h. H is an initial submanifold.

Proof. Put Ex := {Te (»x ).X : X ∈ h} ‚ Tx G. Then E := x∈G Ex is a

distribution of constant rank on G, in the sense of 3.18. The set {LX : X ∈ h}

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42 Chapter I. Manifolds and Lie groups

is an involutive set in the sense of 3.23 which spans E. So by theorem 3.25 the

distribution E is integrable and by theorem 3.22 the leaf H through e is an initial

submanifold. It is even a subgroup, since for x ∈ H the initial submanifold »x H

is again a leaf (since E is left invariant) and intersects H (in x), so »x (H) = H.

Thus H.H = H and consequently H ’1 = H. The multiplication µ : H — H ’ G

is smooth by restriction, and smooth as a mapping H — H ’ H, since H is an

initial submanifold, by lemma 2.17.

5.3. Theorem. Let g be a ¬nite dimensional real Lie algebra. Then there

exists a connected Lie group G whose Lie algebra is g.

Sketch of Proof. By the theorem of Ado (see [Jacobson, 62] or [Varadarajan, 74,

p. 237]) g has a faithful (i.e. injective) representation on a ¬nite dimensional

vector space V , i.e. g can be viewed as a Lie subalgebra of gl(V ) = L(V, V ).

By theorem 5.2 above there is a Lie subgroup G of GL(V ) with g as its Lie

algebra.

This is a rather involved proof, since the theorem of Ado needs the struc-

ture theory of Lie algebras for its proof. There are simpler proofs available,

starting from a neighborhood of e in G (a neighborhood of 0 in g with the

Baker-Campbell-Hausdor¬ formula 4.29 as multiplication) and extending it.

5.4. Theorem. Let G and H be Lie groups with Lie algebras g and h, re-

spectively. Let f : g ’ h be a homomorphism of Lie algebras. Then there

is a Lie group homomorphism •, locally de¬ned near e, from G to H, such

that • = Te • = f . If G is simply connected, then there is a globally de¬ned

homomorphism of Lie groups • : G ’ H with this property.

Proof. Let k := graph(f ) ‚ g — h. Then k is a Lie subalgebra of g — h, since f is a

homomorphism of Lie algebras. g — h is the Lie algebra of G — H, so by theorem

5.2 there is a connected Lie subgroup K ‚ G — H with algebra k. We consider

the homomorphism g := pr1 —¦ incl : K ’ G — H ’ G, whose tangent mapping

satis¬es Te g(X, f (X)) = T(e,e) pr1 .Te incl.(X, f (X)) = X, so is invertible. Thus

g is a local di¬eomorphism, so g : K ’ G0 is a covering of the connected

component G0 of e in G. If G is simply connected, g is an isomorphism. Now we

consider the homomorphism ψ := pr2 —¦ incl : K ’ G — H ’ H, whose tangent

mapping satis¬es Te ψ.(X, f (X)) = f (X). We see that • := ψ —¦ (g|U )’1 : G ⊃

U ’ H solves the problem, where U is an e-neighborhood in K such that g|U is a

di¬eomorphism. If G is simply connected, • = ψ —¦ g ’1 is the global solution.

5.5. Theorem. Let H be a closed subgroup of a Lie group G. Then H is a Lie

subgroup and a submanifold of G.

Proof. Let g be the Lie algebra of G. We consider the subset h := {c (0) : c ∈

C ∞ (R, G), c(R) ‚ H, c(0) = e}.

Claim 1. h is a linear subspace.

If ci (0) ∈ h and ti ∈ R, we de¬ne c(t) := c1 (t1 .t).c2 (t2 .t). Then c (0) =

T(e,e) µ.(t1 .c1 (0), t2 .c2 (0)) = t1 .c1 (0) + t2 .c2 (0) ∈ h.

Claim 2. h = {X ∈ g : exp(tX) ∈ H for all t ∈ R}.

Clearly we have ˜⊇™. To check the other inclusion, let X = c (0) ∈ h and consider

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5. Lie subgroups and homogeneous Spaces 43

v(t) := (expG )’1 c(t) for small t. Then we get X = c (0) = dt |0 exp(v(t)) =

d

1 1 1

v (0) = limn’∞ n.v( n ). We put tn = n and Xn = n.v( n ), so that exp(tn .Xn ) =

1 1

exp(v( n )) = c( n ) ∈ H. By claim 3 below we then get exp(tX) ∈ H for all t.

Claim 3. Let Xn ’ X in g, 0 < tn ’ 0 in R with exp(tn Xn ) ∈ H. Then

exp(tX) ∈ H for all t ∈ R.

Let t ∈ R and take mn ∈ ( tt ’ 1, tt ] © Z. Then tn .mn ’ t and mn .tn .Xn ’ tX,

n n

and since H is closed we may conclude that exp(tX) = limn exp(mn .tn .Xn ) =

limn exp(tn .Xn )mn ∈ H.

Claim 4. Let k be a complementary linear subspace for h in g. Then there is

an open 0-neighborhood W in k such that exp(W ) © H = {e}.

If not there are 0 = Yk ∈ k with Yk ’ 0 such that exp(Yk ) ∈ H. Choose a

norm | | on g and let Xn = Yn /|Yn |. Passing to a subsequence we may assume

that Xn ’ X in k, then |X| = 1. But exp(|Yn |.Xn ) = exp(Yn ) ∈ H and

0 < |Yn | ’ 0, so by claim 3 we have exp(tX) ∈ H for all t ∈ R. So by claim 2

X ∈ h, a contradiction.

Claim 5. Put • : h — k ’ G, •(X, Y ) = exp X. exp Y . Then there are 0-

neighborhoods V in h, W in k, and an e-neighborhood U in G such that • :

V — W ’ U is a di¬eomorphism and U © H = exp(V ).

Choose V , W , and U so small that • becomes a di¬eomorphism. By claim

4 W may be chosen so small that exp(W ) © H = {e}. By claim 2 we have

exp(V ) ⊆ H © U . Let x ∈ H © U . Since x ∈ U we have x = exp X. exp Y for

unique (X, Y ) ∈ V — W . Then x and exp X ∈ H, so exp Y ∈ H © exp(W ), thus

Y = 0. So x = exp X ∈ exp(V ).

Claim 6. H is a submanifold and a Lie subgroup.

(U, (•|V — W )’1 =: u) is a submanifold chart for H centered at e by claim 5.

For x ∈ H the pair (»x (U ), u —¦ »x’1 ) is a submanifold chart for H centered at

x. So H is a closed submanifold of G, and the multiplication is smooth since it

is a restriction.

5.6. Remark. The following stronger results on subgroups and the relation

between topological groups and Lie groups in general are available.

Any arc wise connected subgroup of a Lie group is a connected Lie subgroup,

[Yamabe, 50].

Let G be a separable locally compact topological group. If it has an e-

neighborhood which does not contain a proper subgroup, then G is a Lie group.

This is the solution of the 5-th problem of Hilbert, see the book [Montgomery-

Zippin, 55, p. 107].

Any subgroup H of a Lie group G has a coarsest Lie group structure, but

it might be non separable. To indicate a proof of this statement, consider all

continuous curves c : R ’ G with c(R) ‚ H, and equip H with the ¬nal topology

with respect to them. Then the component of the identity satis¬es the conditions

of the Gleason-Yamabe theorem cited above.

5.7. Let g be a Lie algebra. An ideal k in g is a linear subspace k such that

[k, g] ‚ k. Then the quotient space g/k carries a unique Lie algebra structure

such that g ’ g/k is a Lie algebra homomorphism.

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44 Chapter I. Manifolds and Lie groups

Lemma. A connected Lie subgroup H of a connected Lie group G is a normal

subgroup if and only if its Lie algebra h is an ideal in g.

Proof. H normal in G means xHx’1 = conjx (H) ‚ H for all x ∈ G. By remark

4.20 this is equivalent to Te (conjx )(h) ‚ h, i.e. Ad(x)h ‚ h, for all x ∈ G. But

this in turn is equivalent to ad(X)h ‚ h for all X ∈ g, so to the fact that h is an

ideal in g.

5.8. Let G be a connected Lie group. If A ‚ G is an arbitrary subset, the

centralizer of A in G is the closed subgroup ZA := {x ∈ G : xa = ax for all a ∈

A}.

The Lie algebra zA of ZA then consists of all X ∈ g such that a. exp(tX).a’1 =

exp(tX) for all a ∈ A, i.e. zA = {X ∈ g : Ad(a)X = X for all a ∈ A}.

If A is itself a connected Lie subgroup of G, then zA = {X ∈ g : ad(Y )X =

0 for all Y ∈ a}. This set is also called the centralizer of a in g. If A = G then

ZG is called the center of G and zG = {X ∈ g : [X, Y ] = 0 for all Y ∈ g} is then

the center of the Lie algebra g.

5.9. The normalizer of a subset A of a connected Lie group G is the subgroup

NA = {x ∈ G : »x (A) = ρx (A)} = {x ∈ G : conjx (A) = A}. If A is closed then

NA is also closed.

If A is a connected Lie subgroup of G then NA = {x ∈ G : Ad(x)a ‚ a} and

its Lie algebra is nA = {X ∈ g : ad(X)a ‚ a} is then the idealizer of a in g.

5.10. Group actions. A left action of a Lie group G on a manifold M is a

smooth mapping : G — M ’ M such that x —¦ y = xy and e = IdM , where

x (z) = (x, z).

A right action of a Lie group G on a manifold M is a smooth mapping

r : M — G ’ M such that rx —¦ ry = ryx and re = IdM , where rx (z) = r(z, x).

A G-space is a manifold M together with a right or left action of G on M .

We will describe the following notions only for a left action of G on M . They

make sense also for right actions.

The orbit through z ∈ M is the set G.z = (G, z) ‚ M . The action is called

transitive, if M is one orbit, i.e. for all z, w ∈ M there is some g ∈ G with

g.z = w. The action is called free, if g1 .z = g2 .z for some z ∈ M implies already

g1 = g2 . The action is called e¬ective, if x = y implies x = y, i.e. if : G ’

Di¬(M ) is injective, where Di¬(M ) denotes the group of all di¬eomorphisms of

M.

More generally, a continuous transformation group of a topological space M

is a pair (G, M ) where G is a topological group and where to each element x ∈ G

there is given a homeomorphism x of M such that : G—M ’ M is continuous,

and x —¦ y = xy . The continuity is an obvious geometrical requirement, but

in accordance with the general observation that group properties often force

more regularity than explicitly postulated (cf. 5.6), di¬erentiability follows in

many situations. So, if G is locally compact, M is a smooth or real analytic

manifold, all x are smooth or real analytic homeomorphisms and the action is

e¬ective, then G is a Lie group and is smooth or real analytic, respectively,

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5. Lie subgroups and homogeneous Spaces 45

see [Montgomery, Zippin, 55, p. 212]. The latter result is deeply re¬‚ected in the

theory of bundle functors and will be heavily used in chapter V.

5.11. Homogeneous spaces. Let G be a Lie group and let H ‚ G be a closed

subgroup. By theorem 5.5 H is a Lie subgroup of G. We denote by G/H the

space of all right cosets of G, i.e. G/H = {xH : x ∈ G}. Let p : G ’ G/H

be the projection. We equip G/H with the quotient topology, i.e. U ‚ G/H is

open if and only if p’1 (U ) is open in G. Since H is closed, G/H is a Hausdor¬

space.

G/H is called a homogeneous space of G. We have a left action of G on G/H,

¯

which is induced by the left translation and is given by »x (zH) = xzH.

Theorem. If H is a closed subgroup of G, then there exists a unique structure

of a smooth manifold on G/H such that p : G ’ G/H is a submersion. So

dim G/H = dim G ’ dim H.

Proof. Surjective submersions have the universal property 2.4, thus the manifold

structure on G/H is unique, if it exists. Let h be the Lie algebra of the Lie

subgroup H. We choose a complementary linear subspace k such that g = h • k.

Claim 1. We consider the mapping f : k — H ’ G, given by f (X, h) := exp X.h.

Then there is an open 0-neighborhood W in k and an open e-neighborhood U in

G such that f : W — H ’ U is a di¬eomorphism.

By claim 5 in the proof of theorem 5.5 there are open 0-neighborhoods V in

h, W in k, and an open e-neighborhood U in G such that • : W — V ’ U is a

di¬eomorphism, where •(X, Y ) = exp X. exp Y , and such that U © H = exp V .

Now we choose W in k so small that exp(W )’1 . exp(W ) ‚ U . We will check

that this W satis¬es claim 1.

Claim 2. f |W — H is injective.

f (X1 , h1 ) = f (X2 , h2 ) means exp X1 .h1 = exp X2 .h2 , consequently we have

h2 h’1 = (exp X2 )’1 exp X1 ∈ exp(W )’1 exp(W ) © H ‚ U © H = exp V . So

1

there is a unique Y ∈ V with h2 h’1 = exp Y . But then •(X1 , 0) = exp X1 =

1

’1

exp X2 .h2 .h1 = exp X2 . exp Y = •(X2 , Y ). Since • is injective, X1 = X2 and

Y = 0, so h1 = h2 .

Claim 3. f |W — H is a local di¬eomorphism.

The diagram

w

Id — exp

W —V W — (U © H)

•

u u

f

wU

incl

•(W — V )

commutes, and IdW — exp and • are di¬eomorphisms. So f |W — (U © H) is

a local di¬eomorphism. Since f (X, h) = f (X, e).h we conclude that f |W — H

is everywhere a local di¬eomorphism. So ¬nally claim 1 follows, where U =

f (W — H).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

46 Chapter I. Manifolds and Lie groups

Now we put g := p —¦ (exp |W ) : k ⊃ W ’ G/H. Then the following diagram

w

commutes: f

W —H U

p

u u

pr1

w G/H.

g

W

¯

Claim 4. g is a homeomorphism onto p(U ) =: U ‚ G/H.

Clearly g is continuous, and g is open, since p is open. If g(X1 ) = g(X2 ) then

exp X1 = exp X2 .h for some h ∈ H, so f (X1 , e) = f (X2 , h). By claim 1 we get

¯

X1 = X2 , so g is injective. Finally g(W ) = U , so claim 4 follows.

¯¯ ¯

¯ ¯

For a ∈ G we consider Ua = »a (U ) = a.U and the mapping ua := g ’1 —¦ »a’1 :

¯

Ua ’ W ‚ k.

¯

¯ ¯

Claim 5. (Ua , ua = g ’1 —¦ »a’1 : Ua ’ W )a∈G is a smooth atlas for G/H.