Thm.1.42-supp.ae E is a we
group and HEG a closed
Sub
group . Then the
homogene aus
Space % 0dm its a
Unique structure of a Smooth uefd . s.t.IT : G- →
% is
u
Subversion (Smooth t
TGF is
snrj .
Fg EG) .
In particulier , die (4) =
dein (G) die
LH).
More over
, C : Ex G/
# →
% , llg '
,
GH) g)
H ,
is a
smooth left-
adieu
of G an
GIH .
Proof By
Thin .
1.29
,
H, is a we sub
group of G We
wrihe
GEG for the
liesanahgebrawrresp.to H .
Let n) Contract a Smooth atlas for El# .
Cheese a vector
space camp linear t be of G in
J
:
g-
-
ka
G .
Cousine
y :
k ⑦
G → E
(X , Y) n
expk ) exply ) .
By the
proof of Then . 1.29 ,
we know that 7 open
neighbhds W
of O in k with explw) n 17--49
and
aiüäghbhd .
of O in
G s.i.FI U
'
is a
diffeoueorphidm onho an open
neighbled .
U
'
of e in G .
By possinlyshrinkiuy
W
,
we
mag also essen that for Xn , Xz EW
e
xp Kr)
-
'
exp Ke ) e U
'
( by continuity) .
Now
Laender
F : k × H.
→ G
FLX ,
h ): =
exp 4) h
Heim :
F) ja ,
!xH
→ U = : FLWXH ) is a ditfeom .
Outro an
open neighbh.ae . U of e in G .
ty : Assur expkn ) h
,
=
exp Kz ) h
.
Xs , Xz EW
"
^ , hz EH
=) hin ,
= ex pKs )?
explxz) E HAU
'
- =
explv )
=) 7 YEV sie .
exp LY) exrlxn
)
-
?
expl )
4 Knie ) =
411,0 )
=) 4--0 and Xs -
Xz
=) h
hs
i. e F
↳+ µ
iujedive
.
More over
,
FK ,
¥4)) =
y IX. Y )
¢
Loudly oraud Wx 309 we neu wnhe Fas yo Lid , exp
-
1) .
=) Tk ,
is a know is aus
rphin.tt/e-W.--
By
F. Lid ,
f
"
) =
f ! F
,
we warmen that
is an
isomorphem
VX EW ,
KLEH .
=, F is a Local diffeau .
er and
any point in Wx H
Iujediuiey theiugalouddiffeo → U : _ FLWXH)
is open end F : Welt → U di fteau . anno U -
Contract now the athos her G)µ
:
IT :
G →
GIH * LU ) E ↳µ
Is
open (T
-
1
( TW))
P
'
= U
=) ITIU )
Set
y :
W → * LU ) is
open)
4k) - IT
Lexpk))
Haig 4 is a hauen
morphin .
Iujedivihy : X
" , Xz C- W wie
Y, 4) =
4 (Xz)
i. e .
expks ) H =
expkz) H .
=) Xn =
Xz Since Fis in jechwe an
Wx H =) 4 Ninja
surjediv.ly of y
follow> direct
} hautnah ef F .
=) 4 is a
hijedieu and evidentes it is Continuous
as a •
up .
of Continuous
mops .
W
'
CW open sunset
,
then IT
-
'
LYLW
'
) ) = F (It)
is open in G
=) 41W
'
) E GHz vs
open .
-
Fer
ge
G
,
seh
Ug : - IT
Wg LU )) =
{ gexplx) H : XEW }
E GIH und
lel
ug
,
UGTW he
defined hy ug
: -
4-!
log .
,
{ (
Ug ,
ng
) }
,
⇐ a.
is a make ahoi hier EIH .
) k ) =
ug , lgexpk) lt ) =
4
"
Kg '
)
ngexpk)
A)
=p ! ( F
-
1kg
'
)
-
1g explx ) )
-
C-
U
=)
Ug
°
Ug
"
=
pr,
o
F
-
1
.
Ilg, _
Igf
Xp is Smooth .
For this structure
an
% ,
F : G - GI #
n atmenden
Subversion :
Yo Toy : Wxv → W is a Smooth
Hehlereien .=
prn
Sun a sub mer > ran wo,
the Universal pro nerky
that
for any weop f :
GI# → N ho auohew anode mhd .
N
f- is Smooth ← f. IT : G- → GIH → N i ) kuende .
The univ .
in noky imp Lies
wenig
neue
of hie Cb - sv.eu %
Id
hey applyiug it to the
identity LGIH ,
A) → ( GH, Dß )
hier two Smooth runderes A and B .
It remains horhow that l . G x
G) → ↳µ
„ smooth .
leider : G- × G → C- × % Es G/
#
is Smooth
"
-
Ko
III : Ex E → G- →
HH
Since IT Is n
surj .
Sieben .
,
so is id × IT and Jo the
Univ pro
nerhy of Summer sei aus ruplies
l i ) Smooth O
.
It is not difticulrko see that :
Then . 1.43
=
Suppen M is a
smooth uufd .
eguirneed with
l : Ex MTM
•
Smooth Marine Left udrieurof
a hie
group G- .
Then to
any × e- M ,
¥ is aosed a schied
suehgreup of G
-
:
and the naturel hr jeder
-0%6 -7 C-* =
M is a di
ffeauopu.sn .
×
-
9 Ex -
g-×
Until the 19h
ceuhury (heitere Riemann im
geometry) I
people under >hood
hy „ geometry
"
denn
exckesiudy
Euch denn
geometry .
In oder to in
Corporate non -
Enclidhae
gedehntes
(parallel postenhabe das not hold) F. Klein Propanol
in its
Erlangen programm bruder mehren
of the
geometry .
Geometry in the Sense
of Klein ( Klein
geometry) EE
Gmtd
.
M equipment when a
Smooth transitive Left -
adieu
of a we
group .
If we fix a hose mit × c- M and sel
§ = : H ,
then M =
% is n
homogenen hspoe wir G
adiug
an GIH by Left multiplikatoren .
The
geometry srecikedkysua.eu adieu is the
Steady of figures Ipropertres of tigures that are Left inuoieub
under the
group . So the
geometrie studiere an
M
"
Speak ed
iudinechly hy Soyrug
what its on home
air ms
(
syuuehes) one .
Exocet
① High- school
geometry / Euch denn
geometry .
M -
IR
"
-
G-
EY
) {
¥ Axtb :
IIII! } .
equippeed wir shaudud
lsom LIR"
,
gen) .
immer
product < , > .
G ×
M → M transitive left odieuef G an M .
( ( A
, b) ,
×
) in Axtb
× - OEIR
"
G. = Ola) IR
"
=
Euch
)P
0h ) .
② Affine
geometry
M - IR
"
G = Afflu) = { xt)
Axt
: At GUu.IR ),
4- IR}
Ä {
↳ . . .
, a) e- IR
" "
: x.
=
1} ± IR
" "
ohne
-
uywpwe.ee.
Elements
of GLIut1.IR ) that Preserve A "
are
of hee
£""
{ %-) : be IR
"
, A- e- Gun .IR/3EGLheHiR)
%
⇐ Afk)
Aftlu ) 4) EA "
Atf"
)) /
- •
GLK.gg
Afflu ) x A-
"
→ A
"
transitive bfsodieu .
= IR "
.
No measure of distance and
Kugler ,
but the
concept
of parallel Wies ,
Co linear
ihy .
. . still leute .
③ M = S
"
E IR " "
eguipreed with wund mehre
Ira
-
-
Olnt
1) = Isaak"
.
grd ) i
0 Int n ) x S "
→ S"
ads
transitweg an
S "
.
S
"
=
Okt
)au )
-
GF
Ola )
.
es =/? Orr , o)
C- IR
" t
,
Analogwe of the parallel rushulölehe for the
endogene
of "
line ,
"
en
S"
, handy
tue
geodlsics of g.a
I
does not hold !
Any two
great Circles me et et kenn raus .
④ IR
" "
= IM
" "
eguipradw.tn standard ↳entziehe
× Ko
. . .
, au ) inner product
Cry> = -
x.
g.
t
nxiyi
= xt [
'
n
.
.
) y
HI Ex e- IR" "
: < ×
,
× ) = -
1
,
× .
> 0
}
↳ n -
dein
ky neben space equipment with its standard
mehr C
9 nyp
*
.
„
Parallel postulate
"
deoe ) hat bald hier
geodes.cc
does und bald : 7
infiurhelymoygeodesi.es through
•
Minh not
iuwsed.my a
9in an .
⑤ Classical projectine geometry .
IRP
"
=
n dein
.
proj .
Space = 1- dich .
kebsnoce
of IR htt
.
IT : IR
" "
hol → IRP"
-
Project
we Lines
of IRP
"
= im
Gls of 2- dem .
subgpoceg
of 11241 under IT
C- LLut1.IR) × IRP "
→ IR Ph- transitive Left edien
.
( A
,
[ ×
] ) -7 [Ax] Ex ]
2- ( GLIutn.IR ) ) =
( IRIS d) I.„ + ,
/ "
:
g) × - Ix
PGL Luxe ,
IR ) : =
G"" "
'
"
%( Eda +1
,
IR )
µ
"
mond
subgneup .
pnejechve linear
group
.
=p left adieu
¥42) × IRP "
IRP "
PGLlutn.IR) ⇐ { f- : IRP) IRPN : f- mops
rrejea.ve Lines= IRP"
.
ho
psosechwelimsßf.
PGLlutt.IR)# ¥) ]
"
stabil zu
of
line Een]