C-eodesicsinhyperswtaceslM.gr) EUR
"
; < ,
> ) hype surface Def-6.tt
A (smooth) eure c :
I→ M ,
IEIR internet
,
is Called a
geodesic of IM , g) ,
if Htc I the
be cetera Tien c
"
lt) ( token as a are in IR
"
"
) is
orthogonal to
Teufel E
Tat)R "
? IR
"
1
.
•
It IM - IR"
,
g) = ( IR
"
? < ,
> ) ,
tue
geoebics
one tue affine Lines in IR
"
: at ) =/;) + t (II) c- IR
"
( CH ) =
and c
"
Htt IR
"
⇐
sei
"
Lt) htt .
In vori aus
way
s
geodesics in (
Mg ) one one
Logins of
office Lines in LIR"
.
g) :
• c
"
lt) t Tauft neues that
dloauring acceleration I
only nennen r to Keeps the Curve c in M . Those one
the pathes parties in M to ke when they one in free
toll ( no force is
octiug on it ) .
. c.
"
lt) t Ta) c
"
lt) <
e
"
HI ,
wldt))>wldt)
- = O
w here w is a 6cal mit warnend
neaorfiehd .
Since c is a Curve in M
, < alt) , wldt )) > =D
and different . in t
yields :
<
<
"
lt) ,
wldt) )>
=
< eilt)
.IT#ckt) >
=
< alt )
, ↳ Hh
=
Elekt)
,
<
'
lt)) .
C : I → M
is o
geod .
⇐ C is a Solution
of the secand oder ODE
¥) c.
"
Lt ) t Elekt) ,
eilt))wldt) ) - O .
By Viewing Vos a
fd . de find au au
geusubselöf 11241
(*) is 2nd oder ODE an
gar
Inbset
of IR "-
1 .
Theory of ODE 's that for × c- M
, % C- IM
7
loudly a
Unique
Solution c : I - IR" I
of G) wir
( lo) = x
, c
'
( O ) =
5 ×
( I Novel
conbiniug
O ) .
It is not had to see that dt ) EM htt .
Heule ,
fer
eng
× EM
, ↳ ETXM 7 a
unique maximal
geodesic C : I → M s .
t.ch/--xeudc4d--E .
• Relative ho D :
YET ( TM ) ,
c : IT M a- Curve .
Then
!# fett) ) makes Sense
,
smce
Pszlx) just
-
(
ldepaud> an Sk))
Moreau
,
c
'
(t ) .
q
=
¥2 (dt) )
It follow , that
e) (dt ) ) just demand> •
2£;If if is n wecker Jidd
dang
c ( ie .
R
: I - TM
Camp
Den
BR is
defiuedaudg.in
KNEIF ) .
KK sank weder
find day c .
(Kg) Lt) =L! Lt))
In particulier ,
we
Lenny
form ¥
'
.
Since c ! c
'
= c
"
( by ueusrudia ) ,
tue Gauß eq
.
Impuls LHS of H) =
%
'
G-eodesic
eigneten
↳ I can he wr.hn es
%
'
=D
In partikeln , geodes , es
>
one iutriusic .
S
"
EIR
" "
(5. groß EUR41,9 .
> ) .
dt )
Eeodesic eynatten :
Tuff
"
=
gut)t
(
"
lt) + ICH) ,
<
'
lt ) ) dt ) = (
"
lt ) -
Eckt>µ
=
0
( i ) e
geod .
⇐ <
"
lt ) =
< (
"
( t ) , dt )> dt) .
II: ÷:*
-e. .
c
: tu, {
× it ?
- o
IEEE t.EE/iEitsxF0 .
(
r.DHrxtstf.IT,
-
( lt) = cosllkxllt ) x t Sint " ? " t )
1¥,
←
(
'
It ) =
Hdl
smllkxllt ) x t Null Cos
(Kh t )"¥"
(
"
lt) = lkxlicosllkxllt)
× Nett
sinkst)
, ,
= MIT
dt )-
.
m <
c.
"
Lt ) , dt) ) = -
113×42 (Wilhelm )4×3 tsicilk.at)-
Ein:*
)
=
-
14112 \= 1
=) Great Circles one
gead .
und all gead.
are great Circles,by Uniqueness .
6. 3 Riemann im mfds
(Mi
g) Riemann im mhd m)
7 ! Torsion free
-
an M teed is
uaysotide
wir
g
( Levi - Civita com .
af IM .
g) ) .
n ) lurvahul
of LM .
g) =
Curve
of P ni
T de hermine, o
distihgnidued
dass
afareb)
(
geodesics of Tor (
mg )) ; Def
'
- O
.
-
6.3.lt/tffiwconnectiSupposeMisun-dim.mhdDef6.18-An affine connection on M is a linear connection
on the
long out bundle TM-1M ,
that is n IR- bi linear
cuop
D : TITM) x TITM ) → TLTM )
s.hr .
④
µ
=
f- %
V-s.ge TLTM)
⑥ ! =
C. f)qtf By .
V-fc-caae.IR) .
(cf .
propoties ① and ② of Pop .
6.13 ) .
Rein For
RE
THMI , ⑥ meins ted
Tz is a I! ) - teuer .
Wenn we can also sey
that on office Connection is an IR-
their
map : TLTM) → TLTMQTM) Solis
tyug ⑥ .
1 am (sm
Be ) .
An affine nenne die an M is a device that allows
ho
differentielle neuer fields in direction of never fields
|and Volland ho holte a. at the acceleration
of a done.
ReeuorK_r.Lz@iuudauobgueefdirediadevN.sSaxe und teuerich in S .
Er M =
IR
"
ht . .
.
xu ) s
=
Es
:
¥.
iz
-
Eni!
uf .
• IR "
.
By K ) : =
Eirik) #k) f- s -
ZH)
Delius affine connection , ideal the standard fld kenne die
an IR "
.
In
particulier
,
0%4 = 0 .
eguds also tue Levi Civita
nannte
ef URI gen )
C-
Ynys,
>) .
En Levi - Civita Connection of hynerswt.ae we dehmel
µ preview Sectieh ( see
Prep .
6. B) .
Any men : total oder .to an affine connection (we Sow
that
any mhd eduis wa n Rein . mehr:(
good
will see
9
deherwiuesbdish.ngu.net oftime Connection) , which
the
p
Lies it oder its
many ( since
oddeny any K)-
heuser
to a Connection o)
egoiu an
efhiue kenne den) .
Properties o
) D im
ply
: U EM
open Sunset .
Then
9) u
just derart an
Stu end
Klo .
In
lo ad mordendes :L U, n ) : 4h ETC TM ) ,
SIE ES
'
¥;
Ei :-c:*. .
" ÄE
„
lonnectien coefficieuls of O w.r.to ( Ku)
"
( or Christoffel syubds of →
)
-
Re) !
=
!!
=
Eiji =
=
!
=
.EE?oEt--:iioEE-.:.iiiriiEr.
.
-
P
dRenner
For a wort ( au ) u kenne Äu an
U i
T! T ( Tv) x TLTU) → TLTU) is
gun by
rund :<
÷.ie ¥.
.
!ü¥(Be) lo {sind
! t
Tulsa) .
an M
! .
Suppose c : I → M Curve
and
RE
ACM ) ,
than
the the Loud for nenne Shows that
(%) (dt) C- Tat)M
just demand, on the aeshictieu of y
to lange(c)
( c
'
lt) zb
(dt) ) =
¥25 (dt)) .
Deuce
, if z is anecken b.dat
day c ( re .
q
→ TM
Egg
we
way dehne
(G)H) : =
!5) ( cm)
" "
4k¥37 ) .
Whee Ü is a vector on a
neigh.ge
LI) C- Ms.
t . ÜKLIFRH).
Then % ,
4
is a well def
.
weder tided day c
(iudep .
of the
extensiver) .
Then the del
of never Lied ,
doug
c : I - he is
a we our
space
and a module der C- tot .
fit.
lew.mn?19- ( M , D) mtd .
wite affine connection ,
c : It M
is o O Curve
.
① Then the induced heop ! freu vl .
day a
to
vf .
day
a is IR linear
adf.fr -
f
'
ztf ! 2
-
② If c : IT M ↳s wehe es in a not (U ,
n )
and
z
is n neuer held
day c
,
team
(4) lt ) =
ÄH)
'
lt) #(dt ) ) t
TYEYH.gl#ldtD
=
ÄH)
'
lt ) # ( dt) ) t
"
¥4 )
'
lt ) 254) Fühlt))
GeldD)
when
ylt) =
.IE?iH)f-u..lcltI) ( yi :
IHR G- Kis )
und ci = ni .
c : In IR ¢'
Id
) .
Prof. IR limeuihg
hellen ,
free IR hin
?
of D .
Local
werdende formula fellows been Local Word .
fanden oft
and who product rule fellows eos.by Ken deere .
Def.6.20-LM.ir) wird .
weh altiue comedia .
① A weder bietet ye
TLTM ) is called parallel
W. r .
ho T
, i l
µ = 0 ES E
T (TM) .
② If z is a weder fields
day a Curve c : IT M
,
then y
is Called parallel alaugclw.r.no) ,
ill! 2)Ato
HTEE
① delius an der detern .
System of
PDES , Lance in
general
there are no paddel weder fields .
② of Leuna 6.19 show , that parallel vf .
day
a
Jixed were
always ex ist , since they are seentiees
of a first oder ODE .
Prop.6.21-IM.hn/d.wihceftiiueconnneclieu
① Suppen c : In M is a- are and Rar! Ta! ataget
Wecker d dt. ) ,
when to EI .
Then 7 ! parallel neuer
hehdzaeaegcs.to
quo) =
Rat. )
°
② In the
setting of ① , super Ito ,
ti EI -
Thees
PEEK) :
Tat! →
Tat!
Ratet '→
24 ) ( www y
) .
is the never hadis a linear isemorai.su .
es .
It is called the parallel treu)
Part day
c
ddom.by!
Proof .
①
Suppen ① were
elreodylroued Leer leeres wih CLI )
- louhoihad in tue down
of
a
Single hart.
By langsames : her
ey
t
,
EI ,
ctto ,
t ) ) Can
be Loved
by Jinitaly mag warts in eou
of
whichis deficient by
usswnpniaaudbywn.quonessteedlf.in.
coiuci des on the iutosectens
ef tue ceedbs .
~) we
get well del
. sd .
2 doug
Ito, ti .
If CLI ) is ueuhoiued in a end ( Un ) ,
denen ②
of Leena 6.19. Shows
that Puk ) htt =D is
eepiiv .
to the system of first oder Liner ODES :
µ )
'
(t) t
¥4 )
'
lt) gilt)
"
KH) - O
- Hk
=
1, .
.
) h .
White indies ① und ebo ② .