M = G and de¬ne »: G — M ’ M to simply be µ. This action is both free and transitive.

Example 2. Given a homomorphism of Lie groups φ: H ’ G, de¬ne a smooth left

action »: H —G ’ G by the rule »(h, g) = φ(h)g. Then He = ker(φ) and H ·e = φ(H) ‚ G.

In particular, Theorem 1 implies that the kernel of a Lie group homomorphism

is a (closed, normal) Lie subgroup of the domain group and the image of a Lie group

homomorphism is a Lie subgroup of the range group.

Example 3. Any Lie group acts on itself by conjugation: g · g0 = gg0g ’1 . This action

is neither free nor transitive (unless G = {e}). Note that Ge = G and, in general, Gg is

the centralizer of g ∈ G. This action is e¬ective (respectively, almost e¬ective) if and only

if the center of G is trivial (respectively, discrete). The orbits are the conjugacy classes

of G.

Example 4. GL(n, R) acts on Rn as usual by A · v = Av. This action is e¬ective but

is neither free nor transitive since GL(n, R) ¬xes 0 ∈ Rn and acts transitively on Rn \{0}.

Thus, there are exactly two orbits of this action, one closed and the other not.

L.3.2 39

Example 5. SO(n + 1) acts on S n = {x ∈ Rn+1 | x · x = 1} by the usual action

A · x = Ax. This action is transitive and e¬ective, but not free (unless n = 1) since, for

example, the stabilizer of en+1 is clearly isomorphic to SO(n).

Example 6. Let Sn be the n(n + 1)/2-dimensional vector space of n-by-n real sym-

metric matrices. Then GL(n, R) acts on Sn by A · S = A S tA. The orbit of the identity

matrix In is S+ (n), the set of all positive-de¬nite n-by-n real symmetric matrices (Why?).

In fact, it is known that, if we de¬ne Ip,q ∈ Sn to be the matrix

«

Ip 0 0

= 0 0,

’Iq

Ip,q

0 0 0

(where the “0” entries have the appropriate dimensions) then Sn is the (disjoint) union of

the orbits of the matrices Ip,q where 0 ¤ p, q and p + q ¤ n (see the Exercises).

The orbit of Ip,q is open in Sn i¬ p + q = n. The stabilizer of Ip,q in this case is de¬ned

to be O(p, q) ‚ GL(n, R).

Note that the action is merely almost e¬ective since {±In} ‚ GL(n, R) ¬xes every

S ∈ Sn .

Example 7. Let J = J ∈ GL(2n, R) | J 2 = ’I2n . Then GL(2n, R) acts on J on

the left by the formula A · J = AJ A’1 . I leave as exercises for the reader to prove that J

is a smooth manifold and that this action of GL(2n, R) is transitive and almost e¬ective.

The stabilizer of J0 = multiplication by i in Cn ( = R2n ) is simply GL(n, C) ‚ GL(2n, R).

Example 8. Let M = RP1 , denote the projective line, whose elements are the lines

through the origin in R2 . We will use the notation x to denote the line in R2 spanned

y

x

by the non-zero vector .

y

Let G = SL(2, R) act on RP1 on the left by the formula

ab x ax + by

· = .

cd y cx + dy

This action is easily seen to be almost e¬ective, with only ±I2 ∈ SL(2, R) acting trivially.

Actually, it is more common to write this action more informally by using the iden-

ti¬cation RP1 = R∪{∞} which identi¬es x when y = 0 with x/y ∈ R and 1 with ∞.

y 0

With this convention, the action takes on the more familiar “linear fractional” form

ax + b

ab

·x= .

cd cx + d

Note that this form of the action makes it clear that the so-called “linear fractional”

action or “M¨bius” action on the real line is just the projectivization of the usual linear

o

representation of SL(2, R) on R2 .

L.3.3 40

We now turn to the proof of Theorem 1.

Proof of Theorem 1: Fix m ∈ M and de¬ne φ: G ’ M by φ(g) = »(g, m) as in the

theorem. Since Gm = φ’1 (m), it follows that Gm is a closed subset of G. The axioms for

a left action clearly imply that Gm is closed under multiplication and inverse, so it is a

subgroup.

I claim that Gm is a submanifold of G. To see this, let gm ‚ g = Te G be the kernel

of the mapping φ (e): Te G ’ Tm M. Since φ —¦ Lg = »g —¦ φ for all g ∈ G, the Chain Rule

yields a commutative diagram:

Lg (e)

’’ Tg G

g

¦ ¦

¦ ¦

φ (e) φ (g)

»g (m)

’’

Tm M Tg·m M

Since both Lg (e) and »g (m) are isomorphisms, it follows that ker(φ (g)) = Lg (e)(g m ) for

all g ∈ G. In particular, the rank of φ (g) is independent of g ∈ G. By the Implicit

Function Theorem (see Exercise 2), it follows that φ’1 (m) = Gm is a smooth submanifold

of G.

It remains to show that the orbit G · m can be given the structure of a smooth

submanifold of M with the stated properties. That is, that G · m can be given a second

countable, Hausdor¬, locally Euclidean topology and a smooth structure for which the

inclusion map G · m ’ M is a smooth immersion and for which the map φ: G ’ G · m is

a submersion.

Before embarking on this task, it is useful to remark on the nature of the ¬bers of

the map φ. By the axioms for left actions, φ(h) = h · m = g · m = φ(g) if and only if

g ’1 h · m = m, i.e., if and only if g ’1 h lies in Gm . This is equivalent to the condition that

h lie in the left Gm -coset gGm . Thus, the ¬bers of the map φ are the left Gm -cosets in G.

¯

In particular, the map φ establishes a bijection φ: G/Gm ’ G · m.

First, I specify the topology on G· m to be quotient topology induced by the surjective

map φ: G ’ G · m. Thus, a set U in G · m is open if and only if φ’1 (U) is open in G. Since

φ: G ’ M is continuous, the quotient topology on the image G · m is at least as ¬ne as

the subspace topology G · m inherits via inclusion into M. Since the subspace topology is

Hausdor¬, the quotient topology must be also. Moreover, the quotient topology on G · m

is also second countable since the topology of G is. For the rest of the proof, “the topology

on G · m” means the quotient topology.

I will both establish the locally Euclidean nature of this topology and construct a

smooth structure on G · m at the same time by ¬nding the required neighborhood charts

and proving that they are smooth on overlaps. First, however, I need a lemma establishing

the existence of a “tubular neighborhood” of the submanifold Gm ‚ G. Let d = dim(G) ’

dim(Gm ). Then there exists a smooth mapping ψ: B d ’ G (where B d is an open ball

about 0 in Rd ) so that ψ(0) = e and so that g is the direct sum of the subspaces gm and

V = ψ (0)(Rd ). By the Chain Rule and the de¬nition of gm , it follows that (φ—¦ψ) (0): Rd ’

L.3.4 41

Tm M is injective. Thus, by restricting to a smaller ball in Rd if necessary, I may assume

henceforth that φ —¦ ψ: B d ’ M is a smooth embedding.

Consider the mapping Ψ: B d — Gm ’ G de¬ned by Ψ(x, g) = ψ(x)g. I claim that Ψ

is a di¬eomorphism onto its image (which is an open set), say U = Ψ(B d — Gm ) ‚ G.

(Thus, U forms a sort of “tubular neighborhood” of the submanifold Gm in G.)

To see this, ¬rst I show that Ψ is one-to-one: If Ψ(x1 , g1 ) = Ψ(x2 , g2 ), then

(φ —¦ ψ)(x1 ) = ψ(x1 ) · m = (ψ(x1 )g1 ) · m = (ψ(x2 )g2 ) · m = ψ(x2 ) · m = (φ —¦ ψ)(x2 ),

so the injectivity of φ —¦ ψ implies x1 = x2 . Since ψ(x1 )g1 = ψ(x2 )g2 , this in turn implies

that g1 = g2 .

Second, I must show that the derivative

Ψ (x, g): Tx Rd • Tg Gm ’ Tψ(x)g G

is an isomorphism for all (x, g) ∈ B d — Gm . However, from the beginning of the proof,

ker(φ (ψ(x)g)) = Lψ(x)g (e)(gm ) and this latter space is clearly Ψ (x, g)(0 • Tg Gm ). On

the other hand, since φ(Ψ(x, g)) = φ —¦ ψ(x), it follows that

φ (Ψ(x, g)) Ψ (x, g)(Tx Rd • 0) = (φ —¦ ψ) (x)(Tx Rd )

and this latter space has dimension d by construction. Hence, Ψ (x, g)(Tx Rd • 0) is

a d-dimensional subspace of Tψ(x)g G which is transverse to Ψ (x, g)(0 • Tg Gm ). Thus,

Ψ (x, g): Tx Rd • Tg Gm ’ Tψ(x)g G is surjective and hence an isomorphism, as desired.

This completes the proof that Ψ is a di¬eomorphism onto U. It follows that the inverse

of Ψ is smooth and can be written in the form Ψ’1 = π1 — π2 where π1 : U ’ B d and

π2 : U ’ Gm are smooth submersions.

Now, for each g ∈ G, de¬ne ρg : B d ’ M by the formula ρg (x) = φ gψ(x) . Then

ρg = »g —¦ φ —¦ ψ, so ρg is a smooth embedding of B d into M. By construction, U =

φ’1 (φ —¦ ψ(B d )) = φ’1 (ρe (B d )) is an open set in G, so it follows that ρe (B d ) is an open

neighborhood of e · m = m in G · m. By the axioms for left actions, it follows that

φ’1 ρg (B d ) = Lg (U) (which is open in G) for all g ∈ G. Thus, ρg (B d ) is an open

neighborhood of g · m in G · m (in the quotient topology). Moreover, contemplating the

commutative square Lg

U ’’ Lg (U)

¦ ¦

¦ ¦φ

π

1

ρg

’’

Bd ρg (B d )

whose upper horizontal arrow is a di¬eomorphism which identi¬es the ¬bers of the vertical

arrows (each of which is a topological identi¬cation map) implies that ρg is, in fact, a

homeomorphism onto its image. Thus, the quotient topology is locally Euclidean.

Finally, I show that the “patches” ρg overlap smoothly. Suppose that

ρg (B d ) © ρh (B d ) = ….

L.3.5 42

Then, because the maps ρg and ρh are homeomorphisms,

ρg (B d ) © ρh (B d ) = ρg (W1 ) = ρh (W2 )

where Wi = … are open subsets of B d . It follows that

Lg Ψ(W1 — Gm ) = Lh Ψ(W2 — Gm ) .

Thus, if „ : W1 ’ W2 is de¬ned by the rule „ = π1 —¦ Lh’1 —¦ Lg —¦ ψ, then „ is a smooth map

with smooth inverse „ ’1 = π1 —¦ Lg’1 —¦ Lh —¦ ψ and hence is a di¬eomorphism. Moreover, we

have ρg = ρh —¦ „ , thus establishing that the patches ρg overlap smoothly and hence that

the patches de¬ne the structure of a smooth manifold on G · m.

That the map φ: G ’ G· m is a smooth submersion and that the inclusion G· m ’ M

is a smooth one-to-one immersion are now clear.

It is worth remarking that the proof of Theorem 1 shows that the Lie algebra of Gm

is the subspace gm . In particular, if Gm = {e}, then the map φ: G ’ M is a one-to-one

immersion.

The proof also brings out the fact that the orbit G · m can be identi¬ed with the left

coset space G/Gm , which thereby inherits the structure of a smooth manifold. It is natural

to wonder which subgroups H of G have the property that the coset space G/H can be

given the structure of a smooth manifold for which the coset projection π: G ’ G/H is a

smooth map. This question is answered by the following result. The proof is quite similar

to that of Theorem 1, so I will only provide an outline, leaving the details as exercises for

the reader.

Theorem 2: If H is a closed subgroup of a Lie group G, then the left coset space G/H

can be given the structure of a smooth manifold in a unique way so that the coset mapping

π: G ’ G/H is a smooth submersion. Moreover, with this smooth structure, the left action

»: G — G/H ’ G/H de¬ned by »(g, hH) = ghH is a transitive smooth left action.

Proof: (Outline.) If the coset mapping π: G ’ G/H is to be a smooth submersion,

elementary linear algebra tells us that the dimension of G/H will have to be d = dim(G) ’

dim(H). Moreover, for every g ∈ G, there will have to exist a smooth mapping ψg : B d ’ G

with ψg (0) = g which is transverse to the submanifold gH at g and so that the composition

π —¦ ψ: B d ’ G/H is a di¬eomorphism onto a neighborhood of gH ∈ G/H. It is not

di¬cult to see that this is only possible if G/H is endowed with the quotient topology. The

hypothesis that H be closed implies that the quotient topology is Hausdor¬. It is automatic

that the quotient topology is second countable. The proof that the quotient topology is

locally Euclidean depends on being able to construct the “tubular neighborhood” U of H

as constructed for the case of a stabilizer subgroup in the proof of Theorem 1. Once this

is done, the rest of the construction of charts with smooth overlaps follows the end of the

proof of Theorem 1 almost verbatim.

L.3.6 43

Group Actions and Vector Fields. A left action »: R — M ’ M (where R has

its usual additive Lie group structure) is, of course, the same thing as a ¬‚ow. Associated

to each ¬‚ow on M is a vector ¬eld which generates this ¬‚ow. The generalization of this

association to more general Lie group actions is the subject of this section.

Let »: G — M ’ M be a left action. Then, for each v ∈ g, there is a ¬‚ow Ψ» on M

v

de¬ned by the formula

Ψ» (t, m) = etv · m.

v

This ¬‚ow is associated to a vector ¬eld on M which we shall denote by Yv» , or simply

Yv if the action » is clear from context. This de¬nes a mapping »— : g ’ X(M), where

»— (v) = Yv» .

Proposition 1: For each left action »: G — M ’ M, the mapping »— is a linear anti-

homomorphism from g to X(M). In other words, »— is linear and

»— ([x, y]) = ’[»— (x), »— (y)].

Proof: For each v ∈ g, let Yv denote the right invariant vector ¬eld on G whose value

at e is v. Then, according to Lecture 2, the ¬‚ow of Yv on G is given by the formula

Ψv (t, g) = exp(tv)g. As usual, let ¦v denote the ¬‚ow of the left invariant vector ¬eld Xv .

Then the formula

’1

Ψv (t, g) = ¦’v (t, g ’1 )

is immediate. If ι— : X(G) ’ X(G) is the map induced by the di¬eomorphism ι(g) = g ’1 ,

then the above formula implies

ι— (X’v ) = Yv .

In particular, since ι— commutes with Lie bracket, it follows that

[Yx , Yy ] = ’Y[x,y]

for all x, y ∈ g.

Now, regard Yv and Ψv as being de¬ned on G— M in the obvious way, i.e., Ψv (g, m) =

(etv g, m). Then » intertwines this ¬‚ow with that of Ψ» :

v

» —¦ Ψv = Ψ» —¦ ».

v

It follows that the vector ¬elds Yv and Yv» are »-related. Thus, [Yx , Yy» ] is »-related to

»

[Yx , Yy ] = ’Y[x,y] and hence must be equal to ’Y[x,y] . Finally, since the map v ’ Yv is

»

clearly linear, it follows that »— is also linear.

The appearance of the minus sign in the above formula is something of an annoyance

and has led some authors (cf. [A]) to introduce a non-classical minus sign into either the

de¬nition of the Lie bracket of vector ¬elds or the de¬nition of the Lie bracket on g in order

to get rid of the minus sign in this theorem. Unfortunately, as logical as this revisionism

is, it has not been particularly popular. However, let the reader of other sources beware

when comparing formulas.