M mfd .
equipment with on affine connection D
m
) notiere of parallel
trans Part of taugen weder s
dany Curves C
: Ito ,
ti - M
PEI(e ) :
Tea! →
Ta) linear rin Dauernsu .
Reine
T µ
① There ex ist unstrut nation of parallelTransport
day eures
of M and it con Le shawn that it is
eguiudah.toan affine an M .
② Einen on affine connection D ,
then it inaeuces
a Linear connection on
any
Tensor Knalle :
T : Alm) ×
FILM ) →
FILM)
P
Bt
( hihi now over
IR
,
teurer ich ins ( recall ④ of Def . 6.18 )
+ ⑥ of Det . 6.18 .
htt =
f- Yt +
G. f)t
)
an trivial kuehle Mx IR → M is just d
and
you regierte Tt ① s =
PE ) ② s tt ⑦ Ds .
③ In the be
of M = IR
"
wie tue shandwd connection
,
the parallel Heespen is independent of a and
it is
gruen by
Mens hatten .
LIIR
"
= R
"
Ty
IR
"
x.ge/Rn).
On a mhd M wie affine we can Talk about the
acceleration of a smooth Cure in :
Note Und on a mfd M
←
(
'
ltxh ) -
c
'
lt) ¥ )
c
"
Lt) = line h→
o
k
wo kes ho senge , since c
'
ltth ) e III,
and c
'
G) C-
Tante
wie indifferent . Woher spaces ,
so tue minus in G)
hinke , no Sense .
How ever , if k is equiptadwih an ohne Connect!er
°
( equir
- a nation
of parallel tzesp . ) ,
then we
Can defwe the acceleration of c w - r -
too
↳
E.c
'
=
liy.PK/kI)-.hDef.6.Z2-(Mi) mhd .
wih affine connection . A E-arne
c : I → M LIEIR open iuwvd ) is called a
geodes.CI B if
(B. c
'
) lt ) = 0 kt E I .
Reinen
• M = IR
"
C
"
= c) .
c
'
=
Pc ,
C
'
fer D stuende rd
Launcher an IR "
.
~)
geodes.es are affine Lies .
. mhd .
wir alten comedians
generativ office geometry
home
setting of mtds .
In Loud Coordinator the geodesic egnatia kosten
follow iug
fern : c : IT M wihvelues in a aeart
(U, n
) und ci =
ni . C then Laune 6.19
iueplieg
✓(
k
that
+
+
⇐TÄGIGE = 0 tun .
-
in
-
"
geodesic eynatten .
Theory of ODE s
impuls :
Thm.6.23_ (Mi) het d .
wih affine connection ①
G-Neu x EM ,
KEIM ,
then 7 ! maximal iuherual
IEIR wir OEI and a
unique ( maximal) geoderc
E : I -1M sie . do ) = × (
'
( o) =L
② G- Neu x EM ,
7 an open
heighbh .
U of 0inIM s - t .
fuer euch ↳ EU the internet I in
① cantons [0,1]
und
e xp.
: U → M is Smooth .
expx 14 ) = c- (1) It is called the exponate
wo p atx of T .
④ exnxsetistiesexp.to) =
x and Toexp : IM →
IM
y
is the Identity an Tx M .
E.IM = IM ) .
vonHaie , en is a Local elifteeau .
freu a
neigen .
U of 0 in IM ho an gen neigte b. Vote in M .
④ 7 an open
neigen .
Ü of the zero >ecken
of p : TM-1M
sie .
for any 3 E ÜETM ,
expp ,
!}
) E M is alefiweah .
Moreau ,
Clooney
Ü Sunde
eaogh , Ip , exp ) : Ü → MXM
is a eliffean .
euho an open
neighbhd .
of tue diagonal
in Meth .
Pri
① Theory of ODES ( af .
also Prep .
6.20 ) .
② Follow, f. neunten trachtend sehnten, ↳ ODES elegant
Suuoothly on the initial data and the too that
for a
geodesrc
c : I → M ,
OEI
,
ÖH ) : = elst)
hier
eng
SEIR is
ogein a
geoder c weh Elo) c
(a)
und E
'
Lb) =
se
'
lo ) .
③ Since leastout Curve CH) - x Et is the
unique
geodesicw.tn do) - x and c
'
lo ) = 0
, etp.to) - x .
✓
For ge U und sucht ,
texpltsx ) is delikat
"
und equds c
"
.
Eku ) Ein ) =
EH)
EE
'
¥!
-
±
→
T.edu =
\
! µ
④
Existente
of ¥ fellows from Smooth demand .
of
solutions of ODES ne initial uauditieees und③ . D
Reino ③ Shows that expx :
¥
V is a diftp.
auto
neigt .
V
of × ru M =
, e
xp;
^
: V → U EIN
= IR
"
defilees distiuguishud Coordinator on
M wir neues in t
, Marple
,
lallend normal • ordinäres .
inGeodsicseeuaeatiug.fr
aux Carne speed in diese cooediedes
ho
strengt Lines
through 0 in IM - IR
"
:
"
Ä
Det.cn ( Mio) wild .
wie Vänäeetia .
( M .
O) is Called Complete .
if F- EM ,
exp,
is
detiued Thx E IM .
Prop.6.cn (M ,
O) wmfd .
Wide affine com .
,
3,4 ,
CETLTM ) .
① Rk ) le) : KEE
-
Ecke -
Es.pe
detines a
(3) Tensor
, called tue Curvatureo
JO .
( R E T ( AFM ⑦ TM ④ TM ) =
TLIPTM ① End HM ))
is ne Heel fleet ,
.it Rx - O Ex e- M .
|
② Tk ) =
Bz Gs
-
Es ,
q
] is a (!)-
teuer
Called the
tersieuafv.LT E T ( MT MQTM
) .
is called to> ien free
,
if I =D K - EM
( is
eyuiv .
ho
Ff =
Tjf Hi , j ) .
-
Proof.
① Exercise .
.
② Tl -
,
_
) IR hihi
nein
,
since Eand
E- , -3
we ft
cnn.IR) Ä
TH ) =
II
GE
-
Ef :P =
the-
f)
=
f- Tls , n) -4¥ tun
and
by skew
syuuety
also Tls ,
tz) =
i T (fq,
g)
=
filz
, s)
f-
Tls
,
5) .
6. 3. 2 .
The Levi-
Civita
"
Connection of a
Riem .
nfd .
Fundamental
Theorem in Riem .
geometry :
Thailand Suppe ( M .
g) is a Riemann- uefd .
Then there ex ists a
Unique Torsion free
affine
connection D s .
h .
¥) { .
glp , e) =
g ( By , e) tgly , Be ) i
-
( ⑦ war he also written a ,
Tg =D
,
because
Connection an 5T Mineure is
gun
by Kg ) 4. e) =
s-gk.CI -
9194 , e) -94, )
Proof Assuue Sun a Connection exists .
D= s
glue
) glkq.ee)
-
gly, Be)
0--4 .
gleis) ggfs)
-
gle.gs )
D=
e.gl?q)-gl0es.y) gls.kz
) .
Addi
wg
the fish two and Substrat
regten last
identity
und
us.mg
twsiee freenon
to a
place teens
'
Ye
+
Es
by Ehe]
,
Get
! 2 by -
Ey,
e ] and
Gz
-
% by
-
2
!zt
Es ,
y ] .
im
pli es Koszul formula
GHz,
e) =
Els
-
gle.ee/ty-gl4s)-e.gkz)ItglEniheI-9Ej!!)
#*
)
\= :
yls.dk) .
RHS just iuvdve ,
g
and I ,
] and so
hey waeng
.
Y 9 im plies that is
unique if it ex ists .
Kath %
,
e) - o te % Be
) .
(** ) heute und ho paar extreme :
Fer
4,4 E TLTM ) , y Key) defilees a 1- formen M
(Check this ! ) .
Defilee Be a ,
the unique vl.si .
g ( By ,
_ ) -
Hsg)
One weites directtty that this delius on affine fan-
hee
bemühe au M sdisfyuy 09=0 .
Check the remo:
weg aoims !
• .
Def.6.27-LM.gl Riem . mhd . w
① The affine Connection of Them .
6.26 is called the
Levi - Civita Connection of (M, g) .
II one Called
tue Christoffel
Symbols af LM.
g) .
② The Riemann: en awake (er Riemann teuer ) of
(M .
g) is the Curve here
of its Levi Civita
connection .
③ Eeodesks
of (M ,
g) air are
the
geodesrcs ofthe Levi Guide
connection .
Prop.6.28-suprovlM.gr ) is a Ri au .
uf weh Levi-Gute
Laune Chien P and c : IT M a G- Gerne .
① If 4.
more
now fields
day
a
,
then
d gland =
glas .
e) tgls.kz)dt
_
In particulier ,
to if a is a
ge
des:c
,
then
% glatt) ,
dt) ) = 0 ft
,
Lance • Hält)
11g
Igldltl.CH/)T is konstant in t .
② It tto.tn] EI ,
tun tue isauorüsn PEEK) :
I II
(
detem.by O
) is
orthogonal w-r.bg .
Then
③ Surreal ( Un ) is a dort. 3 Christoffel Jubels
of wir .
↳ ( Un) are
give by z
E ELFE +
Ii -
Eier
whee lgij ) oe tue
coelficietsefgw-r.to ( Ku)
and ( g
"
) the inverse uudrix .
Pri
① Follow, freue k ) in Thun .
6.26 .
② Follow , freu ① ,
since
gls,
2) is Cad hast hier
weder Leeds s .
q parallel
day
c .
③ heut s -
%.
,
g- ¥ and E %
in
the Koszul for the Levi - Gv .
kann .
(in
proof of
Then .
6.26),