Exarchat
•
Friday ,
5.
Febravrg 2021
•
Thursday ,
11 .
Februoryzoz)
|
Exau will have two parts :
Written
part ,
2 hours (
morning )
- oral exon
,
~ 112 hours
•
End of the class : 12.1 .
2021
KM ,
g) EKR
"
! genie . > ) hynoswtece.
Levi - Civita Connection und Riemann herunter :
mag
he viewed es Smooth fcl
RE
TLTM ) TÄTE , Ihk) EIMEIIR
"
"
= Ruth
and a sum we war form its deine time in
direction of onokhr weder
Jiebd SETLTM ) :
E -
2 E) =
Izsx
tu
general , 3.24) EIIR
" "
- IR
"
"
is not on
element in IM .
But we von
project (Sy ) k) orthogonallyw.r.to gerät, >
↳ Tx M L IIR
" "
= IM ⑦ M )
'
.
)
We wr.be
(4) k ) EIM for the res
utting element .
If w is a Local mit normal vf . de fand araeudx EM
,
then
µz)H=kz)H-dsy)H,vkDvk)UThe
formula show,
that PR is a vector fiele an M .
Def.6.12-IM.gl EH
"
! < ,
> )
D : TLTM) x T (TM ) → T (TM )
Lan) -
Be
is
Called the Levi Civita
connection of (
Mig) .
( or Coverieut der iv.
of (
Mg) ) .
For
ze
TITM )
,
D= <
z ,
w > and differentiehing
in direction
of 3 EITTM ) ging :
0=4.4 ,
u >
taz,
Sir>
„F-
Ins) -1
( Gauß equatiae) .
-5
Prop.6.13-IM.gl E ( R
"
! < ,
> ) hynosurtace , s.nl ,
EETLTM )
und f- E LM .IR) .
Then D : TLTM ) x TLTM) → TLTM ) has the
tollowrg
Pnohoties :
① is bi linear over IR .
②
%?
=
f- Bz and
Pty .
→
( i. e .
02 E T ( TM ① TM ) )
③ % Is
=
Ihre] ←
P
←
④ s .
gln,
e) =
gl % ,
e) tgly , Be) ( O is
Laurence
wie
g) .
Proof
① s.nl is hiliuew over IR and SOÜÄI_
, _
) , which
im
plie , the Che im
hy Gauß eq .
② (fs ) .
n
=
flsz) and g. lfz) =
Kf)y
+
Hsg)
Since I is a I ! ) teuer
( ie .
linear an CNN.IR )),
result tolles freu Geußeq .
Shows
③ Proof of Prep . 6.7 ✓
sy
g.
s - Es ,
y] # und
syueuehry of It, _ ) .
④ s -
gly,
e) =
< sie , ED + <
q ,
s.es
=
e) the . Be >
.
¥7
→
Mt
"
"
dicht) e- Fast
üH=n÷iĵ
That Suppen LM .
GM ) and ( N ,
g
"
) are
kyrersuf .
ef
"
TL,
>) .
wir Levi - Civita comedians PM and TN . Iff : (M,
5) →
(N,
GN)
is an isauetry ,
then
FIRE) =
FITZ V-s.ee TEN ) .
Proof Akg) : =
f TK) !!
FE .
• A is
Symmetrie .
FREI files
) -
talk
-
Ef ) = FEED
Pro
p,
. 613
= It's , f) =
¥77 -
BIETE
③
Prop . 6.13
③
=) Als .nl) Ala
) =D
•
EETLTN )
,
then ④ of Pop - 6.13 :
*
t.ie ) =
gut !! tz ,
tee ) tg
"
HE ,
Efre)
1
Since f is on
isauetry , GIVE) A- E) k))
=
G)„
lzlfk)) ,
elfk))) ,
i. e .
g
"
( ff , fee ) -
5h, e) of -
-
LHS of ¥) =
(f) INK ,
e) of ) =
(s -
gute,
e)) .
f
④
=
g
"
( 0,4 ,
e) of tg.ly, BEI
of Pop.
6.13
=
g
"
HRE ,
Ic ) +
g
"
HE , t.lu)
Summe
chiuy
this identity freu ¥) gives
:
O =
gut Als
.ie
) , t.ee/tgMlfy ,
Als ,
e
)) Hin ,
l
C- TITN).
Ä gml Akg), f.e) =
gMHK.D.tl/=-g4t;,AlqeDI--g#TALe,n))--gMltKs)AE)--grethe), tz)
=
gml
tee , Als . ) )
gm ( Als ,
g) ,
IT)
= 0
Since
any tagebuecher at ×
of M war he waren
es
f- e k) for en orrnopr.EE TEN ) ,
this in
plie,
Als ,
g) =
0
by non degeneres
of g
"
.
D .
Heine ,
we see the Levi - Civita Connection
of (
Mg)
is iutriuzic und so are all
quontities and ebjects
Which can he written internes
ef g and T .
For 4,4 ,
CE TLTM ) we home :
¥) s.ly .
e) yfs
.
e) Esp
] e
= O
-
-
LHS
has a
part tangential ho te and a port drkeog.to M
Land hatte med to wenig in oder Leer ¥ ) to kded .
•
nie =
Ye
III.
e)v =
Ye
-
glhlz),
e) w .
s
-
4. e) =
s.ge µ
.
gun), e) w gkk),
e) Lk)
-
-
-
-
-
g
( KK ) , e) u -
-
9L Lk ) , Be ) DT
•
Ihre] :C =
!;]
GILLES,
e) v .
=) ¥ ) is
Aquivalent to
• E. Ge Ke
-
E.)
=
Äh ,
e) Us ) -
IKRK)
- =
: RK .
g) IC) . 8•
-
gl! 4g) , e) w t
g ( Bells),
e) wtgkks.is )
-
=D
⇐
Lltisy] ) =
94g) -
Gus)
Mainareli Egnatia.
Def-6.IN ( M ,
g) c- ( IR
""
,
< ,
> ) kynosurfoce ,
Levi - Civita
Connection
of ( M .
g) .
The Riemann Curve here ( tensor ) of µ ,
g) is
the l ! ) -
teurer
gieren by :
R : T (TM) × TITM) → LLTLTM ),
TLTM ) )
ka) ins Lem Randle) ) .
Frau the formula Rls .
g) =
Pelz,
} ) .
Heule
,
R E T ( FFM ⑦ End HM ) ) = TLITTM ④
Tff)
TY ( Theoreme
egregiuue )
Suppose (M , g) EUR3 ,
< ,
> ) is a Surface ,
x EM
,
Es ,
q
} EIN
ortlwhormdhosisoftxM.lw.r.bg ) .
Then
Kk ) =
go.lt?lsy)lz) , s ) .
In particulier ,
the Gauß Gruene is intriusre .
If r =
as
.
tbz ,
T =
es
tdz one
ohitreiy weder )
ef IM
the Rtr, t ) in tue her , 44g? it
9mm by
Ledbe
) Enn) ö
"
) .
Proof .
RINK) Ikea)
Us ) - Ils) Hq)
=
gun) ,
g) Us ) -
g
( Lk), g) Hq)
g. LR Kindle) ,
3) =
g.
Kkk ) g. Netz)
g.
lustig) g. Kkk) RHS
= de Terminal
of L in the basis { 424 = Kk) .
Freu
go.lt?k.rdy,z)--0followsRlsy)ly)--=
Kk)
s .
und Rts) ) = Nas)
s = KEI
.
-
By skew -
syunnehry
and
hiliueuhy one IR this im
p lieg
tue termin fuer general
T and T .
D .
Siguifiuaece of the Riemann anruehre
of ( M.
g) E ( IRI )
t
(M .
g) is
buddy isometrie • and
any paul + e- M to
be
open Seebad OJIIR
"
, g) ⇐ R =
°
( RIO KE
)
h pwk neuer , a
swtoee.in/R3is6cdIyiseuebrc
↳ an
gen siebst of IRZ ⇐ ,
Kk ) = O FAM .