Tutor
① M mfd .
, 4.
ze
HIM) .
①
④ TS.ro =
0 ⇐
Flieg =
z
lwheueuerdetined )
⇐
FLI.FI?=FLf.FLIlwheuenudefiuud).
Recall :
Isin] k) =
¥+1 IgG) KAM '
④ ⑤
t.rs#4tyG)=qk) is a laestert Gerne
defruedOh
intervenierend 0
0 =
¥!#4) Ek ) -
Esnik)
① LEIGH) TIETZ
#IN )
T
FEI
¥.
=
: !¥:
.int?I..:.F+:*
=
Ist 4¥44 -
EHE.IE?KH✓
-
E)
'
¥+0
tu
FEIG) is
unstarr and Sauce
(F)Jk) -
µ)
,
we love
¥44) =
pk)
wkeueuecdefived .
2-
Themis
FIEL! =
ftp.FLIEIFLF-FEI.FG.FI
-
⇐
iF|
FEE
⇐
Er;
Why k) ? :
T
%!!!:O - .FI/kI--I#II.zlFEk))
- P -
ETHIK)
.
-
Rj) :
More geuerally , Suppen f- : M → N mop
between cufds
and S c- KLM ) ,
ye
# ( M ) one f- related li e
.
) .
f.FI?e=FLIofV ( in
particulier ,
if f- es udiffean .
,
= a then
FLI =
EINE
.fm#aof.H-.EIkxt)--T!,slFIkD---
-
=
qfflt-h.sk)) )
tust
#In) is an
integral cure of q trage1- k) .
=) flthk) ) - ( Ff! f) ( x ) .
③ f . M → N vuop
,
g EHIM ) f- rel.to Tt HIN)
q c- HIN) f - nel .
to
µ ← ¥ ( N ) .
Then
They] is then f- rel . to IT ,
Ü ] .
Prof h E ( N ,
IR ) , hof E IM .IR ) .
k) =
silhof) If
! ) .
h =
Tf,
h = k)
and the Save
of Course for if
and j .
=
=
s.net) ) als
)
=
s.kz?IoI)-q.Ks?nIof)-- i) of =
-
-
of
= IET .
II. h ) of
( Tf Es , h = II. äh of
Tf Es ,
y
] =
TEE ].
f
%
GLLUIIR) AEGLK.IR )
& ) : G- Llu.IR) → G- LIu.IR) , ht : Ella,
B)→ GLK.IR).
are
bijed, aus wih inverse
¥,
and
£ .
Smooth kess of there mops fellows from Smooth ness
of
µ : Ella, RIXELIu.IR ) → GLK.IR) _
( !)!! ( A. B) EEHn.IR#LkdB
Note that JA , FA one vesrrctieus to G- Llu
,
IR) of linear mops
MalIR) → Mal R) .
TB ! (B ,
x ) =
(AB
,
Ax )
TB & ( B ,
X ) =
(BA ,
+ A) .
⑥ Carrie the
taugen wwp of µ
:
Ellen IR) x Ella ,
IR )
→ GLIn.IR) .
T.n.BY
G- Uhr ) × C- Llu , IR ) =
TAGLIn.IR/xTBGLln.lR)
/ elements into
of ten from .
(( A. B) ,
IX. Y ) ) X. ye Mal IR) .
( lt) =
( GH)
, cz (t) ) Cs ( t ) =
Attx AB
#
txßtt
-14+4
↳ (t ) =
BTTY
(Attx) LBTTY)
-
Tea,
M ( ( A. B) 144) ) =
(AB
,
1¥44) , KH) )
-
=
LAB ,
XB + AY )
=
Tief t Ta FBX
② XE Mulk )
K
=
T.dELlu.IR)
↳ LB ) = ( B ,
BX ) smook neuer
fields an
ELK.IR/RxlB)--lB,FB)
A vector fidel see on GLK.IR ) is called left -
innerer
(
resp
.
right
invariant
) ,
if I; s.
=
s V-AEGLK.IR)
( resp .
if fis s
V-AEGLK.IR))
$
(Äh) (B) =
¥)
-
'
↳ LAB ) =
Tt
#
(AB
,
ABX)
-
-
=
(B. BX) ↳(
B )
¥+13) =
Tz.
? ( BA ) =
Tft s
( BA , XBA]
=
( B
,
XB ) =
KIB)
-
Reim In Tech any
left inv
.
weder Leid ( resp .
right
iv.
uf)
is
of the form ↳ for Save XE MDR) ( resp.
Rx for HMM)
Suppen ge # ( Gllu , B) / is Left inner;D
.
( fis ) (B) =
4) HABE Ella, R ) .
"
(TI )
-
'
s HB) s ( AB ) =
Ist*
3lb)|
"
particulier ,
SLA) =
Td) Hd)
slld ) =
(Id ,
x ) =L LA )
Ekd )
= ×
.
Simi Lab ,
for night iuueiet neue find .
EY ? c : I → G- Un .IR ) is on
Integral are
of Lx
through
BE GLK.IR) ,
↳
if It =
↳ KH) ) =
alt ) ×
( lo) =
B
,
ofjßetx = Bett × = alt) X
dt ) < Bett.
d
e-=
Etf;
-7For Rx :
(
'
lt) =
Re ( dt) ) - Xclt )
[ ( o) =D
=) alt) =
g
#
ß Integral Curie
@ Rx
through
B Integral
Circus olefilud Kt
- ↳ und Rx one beendete .
⑨ [ Lx .
Rea ] = 0 their flows waermte .
Y
FLY .
FLY"
( B ) =
FI
"
Ie"
B ) - e
"
Betx
- =
TÄTELB ) )
Reiner
Note that statements ⑥ ④
one vaud for
my
matrix group GEGLIu.IR) ( ie.
Liesubgr .
of ELln.IR)) .
.
§ A E E m )
¥ ↳
←
-
.
.
Äh ↳ , Rx for
my
X UdK
) EIDE .
( Td Ethik ) ^ -
Lx
<
*„j
[ ↳ , ey]
" "
! Ü
'
[ , ] loeeuukotor is Liebrecht .
=
[lx , Ly] Hd).
③ Cousine n Veda fields an IR
" "
4,71--1×7 . .
,
x
"
, z ; . .
,
>
x
) .
Xj =
¥,
t
of ka) ¥.
( Einstein sumatras
Convention
; härteren
j -
1 , → n
There
never fields poirwrse weggehen.
Und'
Iff! 477¥ ).
[ A. xm] = T# * IKEA! .
Gut alüg]
¥
-
-
II.
¥ EE
! -49¥ # -
a
:3!:
t-FE.s-l-ts.ro?-ooe..:EE)Ee.---
Hehe .
Egli
7 a
coordiwde dort ü :
Ü → ülu ) - ÜXÜEIRYIRK
↳
edhg arad
any goal Ko , } ) EU S! "
Xj ) ü
=
¥, 5=1 . . . .
in .
¥1
For each a e- Ü ,
u
'
IW , sag ) integrales uf .
of
< Eine . .
.
Eis Lxnlü
,
. _
, Ihr> .
"
ni
"
=
Un+ „
=
a is an
egnatia for E).
Leon Le Watter
Imprint fd .
The
ivhegnd '
unufd - is in a
heights huvof ↳ in
as the
gneph of a Id . f : VT HR"
wie tlxo ) zo
.
de
Tagen) Space of gr
lt) is
gun by
kultur Ty" , grlf ) = T4 Im
II.y ) 4)
4k) = (x
,
f- k ) .
4 : V -
Rutte
TG ) is
spannend by Iv, (4) =
% t
¥
for
j -
1
,
- -
in
=
¥ t d
! ¥ .
-
④
×
:* .
s →¥
=
f) →
¥ = flog
ttany
=
Im-
\
¥ +
± EF -
9- + a
IF
¥ +2¥
= f-
Sing _
flog ftatycosy
#
=
- Sing
= fsiny
( It
6g f) .
YI ta
§! =
f-
los
y tony ( ttlogf )
- = -7ns , " ,
] !