Ex.3.tt"
, 9ns ,.
) = (Rum
,
< ,
¥ ja
- en inne
prod .
" " " "
Eeodesics of LH
"
,
gugp
) ?
+
XIXEH"
, ↳ EIH
"
and let c : te ,
c) → M Le the
Unique geod . wir clo) = X
,
c
'
lo) =
5×1=0
Let P he tue 2- dim .
swbspoce spannend by x and
„
{ Ht
#¥ . t.ro IR}
" Ex "
IR
""
=P ⑦ pt ( Pt
ortung.
Laurin .
w - r .
hol, 3)-
S
: IR
" "
-1 Ißut
'
s (x) =
{
× to - e- P
ihr weg µ
-
×
for xzpt
SE 011 , n ) ,
that is
,
it is a linear
isaudry of Minkowski
goce < ,
>
)
Koreaner
,
s Leones H
"
invariant
,
Lance s bestrich
ho an
isaudryof ( Hh ,
gnyp ) i
SoC is also
good .
of Hh hy Pvp .
6.30 .
Since
, c. c) ( o
) - x and (so e)
'
lo) =
Is do) - s
fBy
uurqeeness , Soc =
C .
C
equal> the interaction
of
P with H "
.
=3
( lt ) =
wsh ( 113×11 f) x tsinhllkekt ) .
Is Ny
group of ( Hh .
gnyp ) equals
0+41 , n ) =
{ AE 011in ) : ALH"
) -
Ha}.
( AE 011 , n ) : Ah
'
einer 1- ( Hh ) -
H
"
or A- (HH -
Hu).
Rein Oder models
of LH!
gugp ) geodics
.
Poiuuoe hell model : H
"
-7 B
"
projedier to
" Eis Ireoe < 0=0
project a
pour x e- H "
{+ EIR "
: " ×
!!! .
}ho unters@ die
elf × ! 0 wir the Lene hekweeu
xaall -1
, -0 . o
) .
„ Poincori half
Space model .
Since eoeuotoeu for Kinsey weder fields is an
landete mind
system of PDES ,
on a
general
Riemann.ee
auf ¥ Kung fields (isowekygnaer is
Arvid )
.
Hauce
,
Riem uf
. wie
vergl isautruy ganz
med ho he
very Special .
Thülen Suppen ( M .
g) is
Smplg
ceauecheed
Riem .
ufd.
of dim . n sua that
dimllsaulm.gl/=hlzt-) .
Then (Mig
) is isometrie ( up
to
nescdey of ten were
by a rosine konstant
) to e.her of ten thee :
•
( IR "
,
gene ) Euch der
Spore
•
H .
Grd ) Sphere .
"
UH ,
gun ) ↳ www.cs.ee .
} ¥ )
Remo rk
-
•
lsauevy gar ps of # ) od hansi
hinlegen ( Mg ) :
been any
× ,
y CMig
/ ] felsen (
Mig ) h .
f) y .
→ they are
homogene aus Riem .
uefd :
( R
"
,
gen .
) =
Euch
YOk)
( su .
grd ) =
④ "
%(n)
(Hh ,
grngp) in HH' "
)
↳( n ) .
•
They are the
unique ( up to uaesnar positive aesadhytt g)
the a
unique Complete nugdy
nennen
. Riau hfd
-
of Content sectoaeoharvohre ( O
,
1 ,
-
1) .
Gude. All time war. es are Einstein auch
notwendig
eqmv.to euch eher .
Siege :
Note that Rx - O Fx ELIR
"
, gene ) ConverseCy
,
one
has :
TKm.6.36-kg.mn (M .
g) is a Riem .
ufed . auch + EM .
Then the tolle Wing
Shake meets are
eguivdeul .
.
① The Riemann im a. ru .
R Vereines an an
gen
neighbhd .
efx .
② 7 a Chart (Gu ) oneudxs.lu the Cardinale
neuer Jidds 3¥ ,
_ .
, § } farm an
artenarme
hesis hier TYM , tty E U .
③ 7 an
gen
neigh .
U
of x and on
Denny f. (
Kg )
→ (Vi
gen ) , arlene VER"
open Subset ,
Prof Evi
denny ,
② ⇐ ③ : It ② ,
u : U → ul O)
is on is
andy ,
and if
③ f : LU ,
g) Fluegenc)
Is u wahr wehe
reqwreed
moresby .
Since lerv .
of ( IR "
.
gen ) meines auch ② of Pop -
6.30
↳uds , also
③ ① .
It remains ↳ aww hat ① ② :
Luce this is a Loud question it suftrces to grow it loiedhy
arend 0 EIR"
fuer nen cehirory wehe
9 am IR
"
wie veniaiy anruehre .
We wrie ( x ! . .
, ×
n ) hier her werdendes ie , IR"
.
Cuoose an artenarme has is 9410) ,
_ .
, Salo) } of IR
"
W - r . ho
g
.
We war
exheuoh than ho ho ach weder Trends an IR"
as Lollar:
Fix i
: To
get sik? . .
, xu ) lass podl.ly konstant
3 ilo)
day
the ten t - lt.
o . .
.
o
) toten Paul 416. .
)
then
peedlely kassiert 3 ; 41,0 . .
,
o)
day
the Khe
+ → ( x
? t
,
o . .
. o ) to 4^12,
o .
. o ) and no •
.
Chi Ss , . .
,
su are parallel w . r .
wo Leni Liu
.com .
-
0 .
By umstritten
, Si is parallel
donegal liest-19?
-5¥
Psi =D .
Seine
argument
shows that
d
Psi
d-
Vereines an den subsroce
out
of noirs l
o) .
Now [
%. .
¥ ] -0 and R
! .
¥.
)
uns
¥ !!: = °
-
=)
%);
is parallel day old Lone Ex ly ? _
,
y
"
;)
and vanitas at t - O -
Heule
, % : = 0 .
€-1
Next ,
i vereinend ohh puh , (y
'
,
_
„ y
"
-40,0)e.
Dxu- z
-
and
rainy of Riuplies (as
done ) that it
von: des dr de Ports .
Heaney
, au
gets Psi
9%
= 0
ttiig .
W nehmen ,
%,
= 0 Viii → ES i. Sg] =D
hose-
Greenes,
* '
' I i
go
Hand
,
Laune 3.30 ( hetw Froh . Tun .
) ,
7
a Learn ( U , n
) ward Ost -
Silo
¥
,
whin impuls ② by Pvp .
6.28 .
D .
Def.6.37-LM.mg) Riem . ufal .
,
c : Ea , b) → M G- anne .
Then the here) bugtu of c (w . r .
↳
g) is
Idee hy
Llc) : =
] Hält)
Ilgdt
=
!SÄT dt
( For a
pieaew in G Lire
( i e
. a to
< . .
< tu =
b s .
L .
4
,
is a * i
,
Llc) §
Kei ) .
)
Fw u
geods.cc ,
Halt)
11g
=
lautet in t ( Prop.
b.
2)=) Cteodeiics are
paranormal proportional to acre beugte :
Llc) =
} hält) Hd t = hast (b- a ) :
Recall also that if to : Ee '
ist] → Eau ] weißen .
wih
Iölt) > 0
,
then Llc) = Ll cool ) .
TKm.6.38-LM.gl a come and Riem . utah .
Er ×
,
y EM ht
dg
( x.
y
) : = in f- { Llc) : c : Eo .BA M piece-w.se .
Wie Cooke Smooth .
andy - ch )
} .
Then ( M
,
dg ) is a mehr c
space
and tue weder eeh
topeology Gino des Win the wfed .
kopobogug .
the see Literature .
Minderung
purer
fies of geodsrcs
,
Def.6.38-IM.gr) Riem .
ufd .
We well a
Cure
c : Ea ,
b) → M im
inimiziueg ,
if it is tue hartes
mehr hehueeu its and nönnls
,
i. e .
dglet) ,
clb) ) Llc)
.
Def.6.39-LM.gl Riem . what .
,
× c- M .
① A normal
neigen .
U
of × is
of Le Leon
U =
exp.lv) her an an
neigt .
of O C- IM .
② By Tun .
6.23
,
JE > o Sat .
exp×
:
Bdo) → U
i ) o
elften .
Lon Belo) EE
+ EIN : 1)
Sally
< E }
ho an gen
neigt .
U
of × .
Ära (Bdo) ) = :
Bek)
is Called
gejuckt wie Gate
X and ni alias E DE
BEK) war he joiaed ↳ × by geod .
c : Eon] → M
"
Cxpxkx ) c
4) =
exprltsx )
G-eoebics in Bek )
auaeeheg
Leer x are called
radial
geoeltes
.
③ Fw u
0<5 < E wir E es in ② .
So 4) =
expxlso.to) ) wir Solo){
KEIM :
" KH =
)
is called tue
geoderc (er normal ) steuere
d ×
ef radius 5 .
Lennie (Gauß Leuna ) Letx EM and VEIM
sie.
expxlv) is defiual .
Lel WE IM .
Then
I (Tex ! v. texpw ) =
glyw) *)
Wwe
Iexpen : I IM → IM is wewedo '
•
uuop IM > IM
vie Tu IM - IM .
Prof VEIM IM = Rv ⑦ (w.r.bg .
)
w =
WE ⑦
www.r-hokusdecoup .
Since
Trek is line and
by def .
efexpx ,
gLTexp.v.Iexp.is ) =
glv, wt )
- x
( GK c :
t-eap.fr ) ch ) -
Iexgzv
- glcko) .
( Yo) )
( 40 ) = v
=
gldlol.cl/o))--glyu ) .
→
It subtiles kenne ¥ ) her w we
# 0 .
Since expelv ) is auf . 7 E > O s.
h .
% Lu ) is
def . her u = tvls ) OETE 1
,
E
< 5 ,
< E .
wer vls) is a
lerne in
IM Wd vlo ) v
,
Who ) -
my
and llvls)
kg
Casual
.
Heule, we can
Cancer the Haram -
Inter
f : A → M A -
50,1] +
C- E. c)
↳ flts ) =
exgzltuls)) .
+ → flt, so ) one
geodrcs for fixeed So .
t.EE#T
|
ein
LEIF
g (¥4.0) ,
0¥40)) =
glIexr.nu , Texas.
)
E.
gEF.EE?;...gl!Ei! )
→
+
gl! !! )
=
917¥ .
¥)
-
o
o
0¥ is
toisantee
haengtne eher
of
÷.
EE.gl#iE)--:-:::e.
Ge oder one to
aelly mimimi
zug
:
Prop.6.41-LM.gr ) Riem . mhd .
, x EM
,
U normal
heiß.
g × and B c U
geod .
hell wie Cee her × .
Lee t : [0,17 → B '
he a
geod .
Wah Ho) ×
and
c : [ 0,13 → M a nie <e-Wise H we
do# auch
< IN =
ru ) ,
then Ut) E Llc) .
11 adelig
hdds ,
tun
g. LEONI ) =
( EKD) .
Ptb Use heure
of Gauß .
Prop . es hat
gdeäsellg
true ( cl Sphere
) .
On the other hand :
Paid If a
pieee-w.se sina.hu c : Ea .
→ ae
Weh paar .
proport .
ho are
heugth ( Llc ) = hast.
( b -
e
))
is
'
mimimi
zeug ,
teuer c is n
geodcsic.lu parkour
,
( i s Smooth .
PI.
see loherakee .
Munich ( Hopf - Ricow Tun .
) LM .
g) is
a come cheat Riem .
leefd .
Then den
Talhang
are
egmr
-
l
① (M
.
g) is
loupiehe
② ] + EM s.ir .
expx is
abfinden ellaf IM .
③ ( M
,
dg) is
wartete es a were space .
④ Closed und badend Schieds in M are
Loup od .
In additiv , it
any of Lese
Statements oesdeistied
+ war ein Los
⑤ For
any
x
, y
EM 7 o
geodes.cc jene ]
X and
y
s . L . Llc) =
dlxy) .
Prof. See Uhren .
=
) g (0¥ ,
0¥ ) is unstet ict .
-
Snce
¥! ( to) =
0
g ( ¥ (as ), 8¥ ( as ) )- O
D .