nseootu
Klein
geometry = M a- mfd . + transitive left -
actien
of a we
group
G- .
M =
% and left oder he neues 6ft
ueultpl .
by G an GI# .
.. Euclid ean
geometry
.
M =
IR
"
equipneed wir shauaherd teuer rrodrd <
, >
Euch ) =
{ f : IR "
→ IR
"
:
f preserve the distance G
T
distky): =
Hx yll•
Affine
geometry
•
$"
,
grad ) , LH
"
,
gngp )
.
IRP
"
⑥ Con) : der IR
" "
'
! IR
" "
aguipredw.tn Lorentzen
immer product < x. y
> : = xt
¥"
a)
y .
[ ⇐ { × ER
"""
: × # ,
< x. × > = 0
}
„
=
well Lean on
light
-
Laue
+
it - -
txn,
r
-
-
-
-
.
-
xp,
sm
:c: :
'
: :c ::c
:#y Smooth
_
.
- -
-
.
.
-(
xry : A by
, sie IR )
Ever . ixn.
) ¥.lu
.
. . .
.mn ) (II!! ) max .
)=
IC =
{ y
ER
""
: <
y ,
× > = 0
}
U
IRX
=
IT : IC -
Fuß
"
iuduces en is and rein
T.%EF.ie
"
¥ )
Since × is well
, < , > Indices
u postiere definite neuer
product
on
TXCIIRX :
< yt IR×
, y
'
+ IRX > =
<
y.gl ) is welldef
.
V-yijc-IC.vngive, rise to inner
prodecl wie E)
of on
TITK}
"
.
Replociug ×
hy Sx Ist Ried ) Changes
the isomorphen (* )
↳ [
↳×
5 Ttt ,)
"
=
T.tk!
"
by a men-
Zero
uueltplie .
-
Heule
,
tue iueduced immer rneeeeed on
Titus
"
Changes by J ?
=) < ,
> indnces on S
"
= IPC
only a Riemann: en
wahre up to heult
pl.ua/ieuhyascuootufd .
→ (S
"
,
[ groß ) Tv wehrics
giadgae
aufercuolly
µ
"
laufend
eguivdeuce class
of grad
9 "iv.
, if
7 Id
. f
s -
t
g
f
ge .
.
( : O Lux 1
, 1) × S
"
→
S
"
transitive Left -
ochieu
S
"
=
Ip
µ
"
Ihohilizer of a null - Line
a a
weiter und
mtd .
Ohh , 1) =
Couflsn .
Egrd] ) = { f : S4 S "
diffeau . :
f-grd C- [
9rad] }
f-
'
[
groß =
Egrd] .
Geometry in the Sense
of Klein ideas ↳ wenn not
can Prise the other
signifikant generalitat of Euclideae
geometry in the 19 tu .
Center
ry,
haue
leg Riemann im
geometry !
Only homogeneren , Riemann im mf.com he descrikeed
° > Klein
geometrie
.
Com man
generalitatien of the beta
of there nations
of geometry wos
giveu by Certain at the
beginning
of the 20in
ceutrury n
) Cartan
geometry-
1. 5 Fuerther existence results and the classification of hie
greups
lemma1.44-supp.ge
y : C- → H a we
group homomorphismen .
and let K : =
Kerl
y ) E G we the normal hie sub
gr .
ginen by
tue Kernel
of y ①
The hie
algebra of k i ,
giueu by the
tdtowiyided.gg :
k -
kerly
'
) EG .
② Multiplikation an G des cand , to a Smooth unehrlichen
WP
%.
× %-) % , i.e .
% is a hie
group .
Pri
① K =
{ GEE :
ylg ) =
e }
Tek = Ker
Hey ) =
kerly
'
) k =
{ ×
zog
:
expltx ) e- K
„ VTEIR
}
µ =
{ XEG :
explty '
# e
+
tfS
+
Eeg :
4
'
k ) -
O
}.
② By Then . 1.42
, % is secundum hat .
Since K is a war und sub
group , GIK is also a
group Ex
G % ×
% Ä % TT : C- →
%
-
¥47
ITXF =
TI i ) s smarten o ) Loup .
ufsueooteiaops.
=) µ
GK
is suwote.ly universal
prophecy of surj .
Sub meistens .
Recall the
followiuy definition freu
topology :
D
Def.1.IN Suppe p : Y → X is Continuous
uop between
homolog Spaces
X and Y . Then p is Called a Cover
ungMap ,
if for each point × e- × 7 an
open heighbhd V.
f ×
in X
, u
deine Space D and a
houreocuerrwzn
Uy :p -40)
s . t .
FYU ) -4, U xD
→
Uxb
p
! { pr,
kommendes .
Equiveleutly ,
for eacu + c- X 7 on
open
neigen .
U
afx s .
I .
p
-
1
LU ) =
! i
for pair Wise dis joint open subsets V ;
•
fy
and
play
:
Vi → U is a home
auorpeiwn .
• =
¥ IR → 51=41)
+ -seit- I
e| p
u
0
If Y and X are Smooth mtds
,
Keen one can Talk about
Smooth love
Sings I Lovers
,
p : Y → × regnier img
everything in Def . 1.45 hohe smooth .
Themis het y : G → H be a hie group homomorph isn .
between von nected
liegen ps
G and H .
① Ihr 4
'
:
G G
is
injectiue ,
then kerly) is a discnet
normal sub
group of G Conte in ed in Z (E) .
② Hey
'
:
g →
G is
surjective ,
then y surjective # .
and des ceeuds to a die
group isomorphisn Eher(y)
③ If y
'
is
hijective ,
then
y : C- → It is a smooth
↳
wring cuop and a Local diffeomorph.im .
Proof
① By Lemma 1.44
,
the hie
algebra of kerly) is Zero .
Kerl y ) E G is a
submtd .
a) dimension 0 .
Submfd Chong hier
kerly) give rise ↳ Üönen rund
U E G- , e EU with Un Kerle) =
les
Forge kehy) ,
Ug : =
bg LU) E G is
open
neighbh.ggwir
Ugnkuly) Sgs
=)suhspace
torology eukerly)-
Ais
diskret .
Since Kerl
y ) is normal
,
the Curve c :
t ↳ expltx ) g expltx ) -1 Htt IR
,
ttxcgHg
cKerle)is
Continuous Win value , in Kerl
y
) -
By
aliscnetness, clt ) =
g
t
TEIR
=)
exr.lt/)g--gexpltt)V-tElR,V-xEg .
= ,
g Everly ) louumhes with all elements of the
kisgnaep
gemieden hy
¥7) , www.coiucides
With G
, since G is com ocheed .
=
kerly ) EZLG ) .
② y
'
:
g.
→
G surjeaie .
y ( ei) =
exply ) =
expleg ) Thai
⇐
yl G) a- H YIG ) -
H
P P
suhgraep
=)
y in duces a
group iseeuerpwzn.lt :
%,
H
.
It is en
isomorph in
of hie
group ' :
p : G- →
%@ (
y)4=4 op
is Smooth t
p Snrj .
SubversionFa
→
4- Is sina.tk -
y
'
:
g P
'
:
G- Hedy )
)
kerly
'
) Ker
(pl )
=) 4! is an
isauornu.sn ( 4=4 -
°
P ) .
= )
Y is
a Loud di Mean .
,
whiuhog.tw
weh
hijediity,
iv.
phil )
Y i
) o
diffeocuorpkisn .
③ By ② we
way assam H Kerl
y ) and y =p . G
-1%4).
is the natural projectien .
By Proof of ① ,
7 an
open neighbkd .
U of e in G sit .
Kerl
y ) n U =3 es .
By Continuity of Mond U
,
7 am
open neighbh .
Vote
s - t .
g.hevzh-tgef-C-luparhw.la,
V EU .
Fug e- G
,
sehr
Vg -
19 (V ) .
• Er
gtg
'
Ekely) one los
vgnvg ,
=
¢ .
h E
Vgnvg,
=) h - V -
g
=
vlg
'
v.v
'
E ✓
=)
k¥1 = v =
vlg
'
g-
1 eV
⇐
w
=) 9
'
g-
'
E Onkels) = les kely) I
=/ 91=9 .
°
P
Iv
: V →
plv) is bijeceiive ( v - v
'
g g ekely)
c- v
=)
JE
! 4)
FUNF Geiß,
Moreno
,
Plug
:
Vg
=
, p (
Vg ) =p Iv ) is a
hijeclie
↳
ud
elifteaen .
und Lance u
difteeeudzuihr .
¥ g
e- G-
,
ie on
open heights.
efgkd4) e-
%),
-
Wik
p
"
( plvg)) ogahueunieu of poirwrse dis joint open
Set -
O
QUESTION
=
:
Suppen C- and It are hie
group
and
4 :
J G
homomorphin
hehween their lie
algebra s .
Does there 7 a hie
group homomorphen 4 : GTH
sie .
y
'
=
y ?
Def.1-47-G.tt
hie
greups .
A local homomorphe ihn
freu E to It is
give by on
open neighbhelrfe in G
and a O -
map
y : U - H
EG
s.t.yle-e.iq/gh)=ylg)y/h)wheuewg,hTdgVhliein0.
Note that Tey = :
y
'
:
g-)
G is
vegan
hie
dgenne
haus war
phizn Tumult
Lel E and It we hie
group,
wih die
algebra ,
J und
fand lel 4 :
JTG he a homomorph in
of
hie
algehnas .
Then I a Local homomorphim
y
:
%-)
H Srt -
41=4 .
If E is
simply nenne chad ,
then
'
there a ists a
homomorphen of hie
graues y : E - H s. t .
4
'
-
4 .
D