Be.
1 .
18 G hie
group
with Lie
algebra g
Then the exponential map of G is
given by
exp =
g
>
G
exp(x):
=
F)(e)
By definition , explo) =
e
Thu.
1 . 19 G hie
group
with lie
alg .
g
and
exp :
g
>
G the exponential map
⑦ The map exp is smooth and Texp :
Tog Te
equals ldg (=
identity
ang).
Heace , exp restricts to a differarpuism from an
open
neighbud of 0
gin g
to on open neighble ofeG
in G.
② For Reg and
geG one has :
FL
*
(g) =
g
.
exp(+X) and FLEX(g) =
exp(tx) +
g
tag) + Leg) is a smooth mop
gxG + TG
(by Prop .
1 .
7)
=> (Kig)+ (0 , hx(g) is a smooth of. a
gx
G
ure
Its integral
curvest+ (X, F((g) and smooth
In particular , (Xit) ++ (X ,
FLE(g) is scooter
and so is exp(x) =
F(4(g) .
Now , FLY (e) =
F(*
*
(e) =
exp(+x)
A (C
: I- G integral curve of Lx ,
thestt clot)
is an integral Curve of aux = x
FaeIR) .
and FL(g) =
gF)(e) =
gexp(+x)
F(a)=
exp(+
x)g
by Prop. 1 . 15.
(exp)(x) =exp)) = G() = X
⑧
-Examples
① Consider the commentative Lie
group (IRo : ).
Itslie
algebra is IR with trivia Lie bracket .
The left-inv . of generated by xeIR is
↳ (a) = ax
11 =>
Integral cure
of Le through
Tax 1 IRyo
hx(C(t)) = d'(t) clo) = 1
11
c(t)X
solution is c(t) = etx usual exponential
map .
Hence , exp :
IR- Ryo is the usual exponential
map
.
② G = GL(u ,
IR) ,
X +
g
= Murn(R) =
gl(u, (R)
(x(1) = AX
↳ (c(+ ) = c'(+
)
G()11
c(t)X clo) = Id
Unique solution #) is the matrix exponential
exp(+x)
=
in
(Voparater norm (111 an MR) 11x1111x174,
while implies that this
power sei es
converges
absolutely and uniformely an
compact sees)
exp(x + y) + exp(x) exp(y) unless [X , 4] = xy -
3x
= 0 .
Hef.
1 . 20 (Exponential coordinates)
G is a lie
group
with lie
alg.
g ,
Veg is a
open neighbled of De Sir .
expl
: - exp(u)= 0
is a differu . anty on
open
neighbud of e- G.
① Ther (U ,
expi) is a local chart for G
with e- Wand (Ng(U) ,
igoexpi) ore
around
ge
G.
Canonical Coordindes
of the first Kind"
② Choose a basis EX ..... Xn] of g ,
ths
v : IR" + G
given by
v(tr , ...
tn) =
exp(tX1) . . . .
exp(tutu)
restrucks to a local differm · from a
neighbud VotOEIRn
onto an
open neighbled W of e in G.
Indeed,
(0) =
Xi
and so
TV ( ..., an) = a
, + +
anX
(U ,
ve) and
(ig() , for) ane
J
wordings around eeG and je G.
Canonical coordinates of the second Kind"
Prop.
1 . 21 Let y : H +G continuous
group homed phille
between lie
groups
I and G . Then
I is smooth .
Prof
We first show this for H = (IR ,
+ ) ,
i . e .
Y is a
continuous 1-parameter subgraup .
im If a : IR-G is a continuous 1-parom .
sunge .,
then h is smoot.
open ball of radius
By Thru .
1. 19
,
Jr > 0 sit. Ir for some inner product
exp
:
Br(o>
exp(Bypa) =:
Ble)
ong
.
g = 1Rh
is a differus · Onto an
open neighbled Barle) of etG.
Since ,
<(0) = e and< is continuous ,
7930
S . r . a (Es ,b) = Byle)
Now let us define
b : [2 .
2] >
Br(0)
Eg
r =
expo e &
Bit)
&For
12 we hove
exp(2t) = a (2t) = a(t)a(t)
=
(exp(2b(t)
=>
B(zt)
=
2b(t) = b(z) =
(b(s)EseE9, 2]
By inductions are shows :
B=s
VseEE, E]
EkEIN
=> for kinEIN
a)
= m() =
exp(p())"=
expb(a)
Sinne a(t)
=
= x(t) and explx)
=
exp(x)
-
(x)a(t)
=
exp(+
Eb()) FteSkeN, ne2]=:
S
Since SCIR is dense and both sides
of (*) ane
continuous ,
we deduce that
a(t) = e
/(
+
= (a)) Et
In particular ,
a is smooth ,
because the right-houd
Side is.
Now conside the
general lase
y
: H G
Toke a basis[X .... 1 4n3g . Then
u-1 (ty ,
. .
, tu) =
exphtet) . .
exp(tnXn) defines a
ropen
differu ·
from a
neighbud Of IR" to an open neighbled
ofe elt and its inverse u is a dort
Then
(y0u-1) (te .
.
·
tu) =
4) exp(tem) - -
exp(t=
+a)
=
p(exp(te(n)) .
--
4(exp(tnXn)
Y Y
continuous
1-porouar abgr., lence
smooth
=> you' is smooth and threfere
o s
smooth locally around e
Therefore ,
(in)04
=
Gor is scoot lo
cally around
etH helt .
This shows that
y is smooth locally arand
any
heH
D
-Prop.
1 .
26 For a lie
group
G we da by
G.
I E the connected component of
G containing
etG ,
which is called the connected component
of the identity of G
① Go is a submitd (it is opensubser) of the
same dimension as G.
② Go is a
subgroup .
In fact , it is a
normal subgrees
In particular , GoG is a hie
subgroup of G
and E/Go is also alet called the component
group of E.
Prof
①
② gibt 7 Cocurves
(g)Ch
: [0. 17 >
G
5 . v .
(g(d) =
e =
Clo) and
(g(1)
=
9
and (n(1) = h .
=> ++
(g(t) (y(t) is a Co-curve connecting
e with Ih
=> gue
Go
Since ++ r(cg(t) is a
Co-cure connecting e wie
g-1 ,
also
ge Go for my geto
=> Go IG is a
subgrep .
It is a normal
subgramp ob t : for geGo ,
keG
the KIghtik" is a co-curve connectinge with
KgkeGo
E
Thu
.
1GlG
and H Lie
groups
witdie
e
① If y : G- I is a lie group honomorphisms ,
then
40x4y
=
ex404
where
y' =
Tel :
g -+
G.
② Go coincides with the
subgrap generated
by exp(q)-E.
③ If , p : G + H are lie
group homonarphisms
St .
4 =
y' ,
thae
4)=
In particular , if G is connected , thes 4
=
4
.
Prof ①
Recall that
TH4(g) =
44(g) (in te roo
of Prop- 1.
12)
which impies
yo FL =
Fly ·
Keg .
=>ep(x)) =
y(F(4(e)) = F(y(k)(e) =
my'()
② If E is the
subgr-geerded by exply) ,
the
Go ,
since
tr exp(x) is a Ch Cure
connecting e to exp(x).
To see the converse
, nole ther ,
since
expic
a local differr ·
Ground 0tg ,
exp(g)
and hance E ,
contains an
opens neighbled
UCGof e -G .
=> for ge ,
↓(U) is an
open neighbled of g9
contoinend in E. (we used thatG is a
subgrep).
Heuce, C E is
open.
But &IG is also closed ,
since for ge Gl
Ig(v(U) is an
open neighbled of
a contained in
GIE ·
Hence ,
GIG is
open and therefore
E is closed
= E =
Go .
③ By D ,
4
and
4 voincide on
exply).
Since y and
y are
grouphonou .,
also on
E
.
So the result follows from ②
trample(u , ) - (IRKOS ,.
) is a lie
group
honomorphissen
(det (AB) = der (A) der
(B)
det' =
Tadet =
trace :
gl(u, IR) - IR
By ① of Tum . 1 . 23 :
exp
= e = a(u ,
IR)
+(ex) =
etr(x) >
SL(m,
IR)