DIFFERENTIAL GEOMETRY
STRUCTURE OF THE COURSE
-
I.LIEGROUPS.BAsic
Theory
•
Representation> of hie
group,
•
Classification of hie group ,
•
Homogeneaus
Spaces ,
klein
geometrie>
I.BUNDLES.FIber bundles ,
Vector bundles and Principe bundles
. Associated Vector bundles
•
Homogenean vector beides
HI.CONNECTIONS.linear connection on neckar bundles ( e.
g. affine connection;
•
Principe Connection , an
principe connection s
•
Geometrie structures
determiniere ( closses ) of di >
tingceisbedaffine connect iaes
Leg .
Riemann iaecufds ,
laufend uetds .
projedive structures . . .
)
¢ Gauß Bonnet
Then .
)
•
Holouauey group
• Cartan Comedians ,
Carta
geometrie .
For
refereucos / literature : see IS →
study material .
Evaluation
-
:
•
Home work ,
Lewy oder week)
•
Ehem . lord I written or beth ? ) .
I. LIE GROUPS ( [ op ,
Lie
group, ( leckere notes ) ) .
-
1.1.BasicTheoryto.eegroup
G we write :
•
µ : Ex G- → G for the multiplikation leeop
( µ (g. h) = :
gh Kg ,
he G) .
. v : G- → G for the inversion wlg ) -
g-
1
9-97-5}
.
EEG for the
identity / neutral element .
} -
e
e.
g- g.
e
= e
V
GEG
Det A topologieal
group
is a t.ro logicdspoce
G- eguipped with a
group
structure ( µ ,
W , e ) s .
l .
µ
and v are Continuous .
Rennen Any abstract
group
nahe made into a to
polog .
group
he
equippiugitwih
the Misere
topohgy .
Det A hie
group is a Smooth man : fold G
equipped with a
group Structure ( m , v. e ) s .
I .
µ and
W are Smooth .
Rleuoj.lu Def . 1.2 it is
enough
to
require
that µ is sueade
,
since w is then arkona Lindy smooth
by app
Ying
Kee Im point Fa .
Tun .
to the Toten µ leg,
vlg ) ) e
.
Det
① A homomorphismen between to
pdog .
group ( resp .
Liegraep)
G- and A is a
Continuous ( resp .
smooth ) wo
p
y : G- → H that is also
group homomorphismen
( ylgh) ylg)
ylh) kg , he G) .
② A
homomorphen y : G- → H es in ① is called
9µsan
isomorphen
'
of topdog .
(resp .
Liegreugs) ,
if
if is a
haeeeoueorpuizn ( resp .
aliffeouuopcize) -
Note that in this COM
4-
^
is also a
group homomorphisn .
Notation Two hie
group , G and It are called Isomorphie
it I a We
group isauopti.su
between them .
We
wie G = H in this lose .
Grau ps of greahest inter est in mathe an . and
physics
Lens ist of hijecti aus f. M → M of a seh M to it Self
with
group multiplikation giulu by uaepositieee
pelf,
Ä ) : =
f. Ä
t.IE/3ijlM):-setofhijech-iaesofM-
Innen wlt ) = f- ^
und es =
Idee
.
Examples-ltopolog.geenges ) .
If M hos seine extra Structure
,
one can Con sicher
sub
group of ( Bijlre ) .
.
) lousishug of hijectiaus
presoviug
tue extra structure .
Suppen
M is a
topolog .
Space
Home o ( M ) : =
{ f :
M → M : t is a home
omopv.im}
Seerose
M is a
smooth mfd .
( resp .
asmoohoneutudu.be.
)
Biff ( M) : {
f. M → M : f is a diffeau .
rptüzn} .
Biff+
( M) =
ff :
M - M : fis a
oieuhatieeerresoveegdifkau}
Suppen
M is a smooth mhd .
eguipreed wie a
glaube
Structure like a
Riemann immer :c
9
Or
a
syuepledic form w .
Hau LM
g) ⇐ Sf : M →
M : f- diffeau -
s.t.f-g.mg}
Symp IM .
w) : = { f : MTM : fdikeeu.s.tn .
für =
w}.
With the
eeceptieuoflsaulM.ge) ,
there
group , ooe
all infinite dimensionalen
d cannot he see a a we
group
( at least not finde dienen
sind aus os we can siehe) .
They are how euer all
nehudlyhopobgicdgraeps.dimlrekdiw.net)Isan LM ,
g) is a hie
group of dim .
E -
Z
( see Global
Analysis) .
Now
seine
example , efochud die
graeps .
Elaine ( hie greups )
① IR ,
CI with respect to + one li e
group > and so
is
any finite dimension
d we do
space over IR er G w.
r .
tot
¢ and any cauplex uecker space one even
cauplexlie-grapsli.e.holomorpu.cmfds .
wie hdoueörpeic gray structure
) .
② IRI hob ,
G) 3 of are hie
group ) W - r .
to multipliziere .
( the Loher is
ogoinauaeple.ve group) .
Also
,
UH): S
^
=
{ z EG : tz 1=1} w r
.
to do
heult minder is a hie
group .
f- and H
③ The product Ex H of two we
group, is
ogain a
We
group .
In
poh.cn her ,
the n dim
.
Torus
TT
is a hie group -
T
"
: =
Vln ) x . .
× U L1 )
For men E Z
> •
also IR
"
×
Tn is a
we
group The
↳ the ex haust all come cted kommenden hie
groupS ④
If G is a hie
group ,
then a
hie Sub
group Hoff
is a sub
group of G that is also asubmfd .
Since the multpl . on It is just the restrictien
of tue
die an E-
,
it is Smooth and to It is a hie
group .
⑤ Seppose V is a red or
Loup lex vector
Space (dimv
)
C-LU) : = { linear isomorphes of V } E End W) =
/ { linea uops
is a hie
group w . r .
to to VTVJ .
open
subsetdf neuer
waupositieu of know
maps . Mae End (v ) .
( If V is a
couplet war Ignace ,
teuer GLLV) is a
wauplex hie
group) .
Via a
more of has is
of V
,
we can
identity Volk
"
for IK =
IR or E und ELN) with .
G- Lln ,
IK ) : =
{ AE Mnllk) : A innehabe
}
µ
hin wahllos euer IK
and can
positiv of Ii war
ueop, be comes matrix
heult pliuetieu .
It is called the
general Linear
group .
⑥ Matrix: (also called line ) one
hie sub
group of GLLV) resp .
G- Lln ,
IK )
( IK =
Q.IR) .
Recall freue Global
Analysis ,
• Special linear group
SLln.lk) - { AEGLLn.lk ) :
det.LA)-
1 }
( Sun
, d) is even uauplex die
group ) .
•
Orthogonal group> :
Ing : =
(IP %)
c- Mulk)
n =
pt 9 .
defilees standard inner product on IR
"
of
Signature (
ng ) :
< x.
y
) : = xt Ip , qy V-x.ge IR "
.
01mg) =
{ AEELK.IR ) : At
Ing A =
Ip,
}
=
9 AEGLln.IR) : < Ax ,
Ay > =
< x. y >
V-x.DE IR "
} .
linear
arthog .
grey of Sign .
( ng ) .
q
- O : Olp ,
O) = Oln ) =3 AEGLlu.IR) :
AEA "
}.
q
- 1 : Oh ,
1) Lorentzen
gnap ,
( =
linear isometrie) e
)
Minkowski
lpoce) .
>
linear
sywplectic grap
:
In =
du
'
! ) EM.dk)
defilees o
sywpcedic Structure an IR "
:
WK ,
y) =
xt
Ing
V-x.ge/R4.Spl2n,IR)--fAEGLI2n,iR) : At In A In
}
=
2 A E /
,
- :
wkiy) - wltx,
1-g)Feiger}
Suppen l E , m .
W
,
e) is a
hie
group .
Then we elende by z
•
Ig : G → G left ueultpciuetien by ge
G .
Igh) =
µ (
g , h) =
gh the G .
•
f
9
:
E → E
right ueultplicatioa by GE
G .
19h ) =
µ ( h .
g) =
hg the G .
Leutnant
For
any GE G ,
Jg ( resp. 19 ) is a
diffeauorph.su
with inverse
bge (
resp .
191 ) .
In
particulier ,
if
U E G- is on
open neighborhood of GEG ,
then
juhu) ( resp .
f
"
IU) ) i ) an open
neighbhd .
of kg
( resp .
gh) .
More over
, for g. ne E we home
bye! -
Ignund
f ! p
"
-
f
"
9
.
Profen Jg resp .
f
9 can he written es coeepositieees
of Smooth
Koop) i k G - Gx G -
G
u - Lg , h ) KG
-
Igand
siwühdy bei
f
9
:
hm, In .
g) Ä
hg .
Cleary ,
they are difteeau .
wir
neueres
by,
und
f 5 ?
Rest is also der freie definitions .
/
Lemma1.5_SupposeGisoliegreupr@Ferg.hEG- ,
SETGG , ZEIG one has
(g)µ Isin ) Tigre
tgehs .
-
-
(
m : Ex E → G Tm :
T ( G- + G) → TG
)②
For any g c- E
,
=
TEXTE
Tgv = Telmo
Tgbg,
= Teigig
19
"
.
In particulier , Teu :
TEE → Te G land ) -
IDTEE
'
Proof .
g.nl Ex G)
-
①
Tun, µ :
Tg G- ×
TNG →
Tgn
G is linear
weg .
Iggy kn) Ignite
( s
,
o ) +
Taille) .
Lev
c :LE.a)
→
E ke a Curve
regeres . 9
( ie .
do ) =
g , c
'
lo) =
s ) .
-
Then
nesperat tu
kaugal neuer 14,0)
EINE;D
Toni " " ) -
G)*
µ# Hin ) =D! !!!lett
) =
=
Tgl)
Similarly ,
T.g.ng.MLO.ie/=dd-I+M=o9idt))=ddFfIgldaD---TndghWherec : l E.
a) → G is a Curve terres .
2
( do) - h
,
c
'
lo) =L )
j - ↳
② e =
µ ( g , wlg)) g(
g. vlg ) ) -7 GDifferenzierung
=, O Tag!
)
(Tglds ,
Tgvs)
Tg 99) +
Tgigtgvs the
Tgf .
- -
=)
Tgeig
Tgus = Tg
154 ⇐ Tgvs = -
Teigige;
Seward konnte tollaus
siuileeiy been di ft .
e- µ
Ivlglg).
D .