Different
Mini
Geometry
case
Day 3
,
-
film
N }
That If
:
pony th
The
NAI
swoth &
"
,
ksif
,}
boils dam to
is not the
case
,
.
ft 1 )
o
by
Dye
.
.
a
i
the
=
quetr
f Rn
;
Chat
B
fl
×
-
Use the
let Matnxyk )
be the
nice
nt
,
teal cordials
.
set
f
real
04
prove
Then
,
a
H
> 0
r
,
can
symmetric
nxn
Matnalk
flay z)=zt
,
regw rates
are
OH
{A
=
,
rrithl Wu
a
}
of
f yet
,
f ( r)
'
'
f
radius
rr
an
mtias uhdn
ID
=
B
a
manifold
.
RY
be identified
can
,
A A's
:
A"÷
with
Define
by
AAT
the
identity
metix,
so
the chdbp
to
B
shot
,
Matnxnlk )
→
TI
Symmln
)
th } ntetht TA Mata .IR )
sujectk
B
,
4
f YI ) & CETISKNH
→
dfa
Nhtnnlk )
all
AE OH
tht
.
& Tpfymmk )
CA
)
AHA
HCA
KCA
st.
a
lot
) dfqx )
=
for
idnlilndwl Metnnlk )
an be
.
E
' lo
=
=
A
n
-
.
0
.
m
theorem
the sphere
so
,
nxnmtic
be identified with
Symmln )
→
M
f ( ID the inwxingef
for A
:
apply the hnsefmhh
&
f
diners in
.
metrics which
nxn
'
:
dfq Ta
To
is
submitted of M
smooth
a
.
A
=
ftlq )
R gin by
→
shtht the ottogalgpp
to
fi
Then
fi
'
5
Then all
Symmln ) be the set of
In turn let
off
then
,
submitted
Exit
=
"
cnsidriy
-
xD
,
in
I dink
of diners in
theorem
.
.
rgww
a
by
see
can
,
is
this
smooth
a
mnifld
sub
of
write
you
IR by
→
smooth
as
EN
he
I claim
,
&
( fox )
'
a
6)
'
lo )
=
dfalk
=
C
on
to
.
be
see
ikdtrd Y synuk )
this
,
let
Then
=
.
adthafdlt ) NAT )
=
210 )
a
'
lost +2%12 lost
y
=
t CA At
k AATCT + ECAAT
=
kct
=
=
+
(
t
since
t
(
AAT In & C
=
symmelm
.
.
,
}
.
thefe 04
till In )
=
,
B
.
smooth
A similar gnat stars tht UH
sit
the
,
Nnthtwehmetayut
Df A
reefer
.
field X
on
a
× M →TM
4
Note :
UEIR
:
{
A new field X
an
by
:
V
=
ykxtx
-
he
1122
an
coordinate
a
Ex :}
M
B
ab
Hf) Ip )
-
M
tht A oh
M B
→
&
swath
B
"
smooth manifold
so
=
f
is
PCM
chat in
the ksis
of
he
V
.
'
-
p
I U 6)
,
everywhere
at
:
5
Aside
Then
fr
Nn
,
as
pe
Ian
the
5
,
a
smooth
Zailp )
=
nhu
Fxi
,
each ai
:
an
→
R
.
.
\
-
an
€*
1
q
5
the Hairy Ball Theory
tht
no
9
g
a
a theorem 1 add
,
.
\
Its
.
) by
toil cards
in
manifold
bullet M many
target
then XIP)
,
.
ni dinesh
an
sdkflte
Mm
→
nh÷l
:
associated to
Tpm
Eailp ) ¥Flp
f
⇐
tie
xtx By 't ' Asean
.
=
,
Tf
nhd
a
-
:
}
Hp) ETPM
,
CTM )
mp
a
as
.
vector fields
smooth
a
)
n
unity minus
nxn
HI!
'
diners in
of
talk abut
man also
,
:
map
.
reeks
told
mi
smooth
( or
em
2+5 ti
tat
B
r
s
ok
no
,
)
rear field
on
5
can be
mvnishiy
.
wit
we
define
on
qhetrnins :
53
think of
Tps
=
{ atsitcjtde
3
as
the
:
a
nhthmpoprdialr top
qmtrnhs
3 mntlforthonrlmelvfdds
X
,
Y Z
,
on
5
as
Hbo
Far exgde
,
.
:
1Es3&T±s3={ aitbjtck }
X (p)
ah pEs3,
Far
Ayhtbtthseaetapt
4 p ip )
,
If
:
p
=jp
pot p
p=
pzjt Bk
pot P , it
So this B striking different from the 5
Df
:
be
lit X
a
M A
on
.
the irtepd any
,
coadhts the ktgnl
,
So you
,
Then
d :
=
alt
B the ihtbeek
t SE )
-
-
E
B
f
the
5pA) to
.
( at
an
→
F t,
) EQD
tattle
-
Halts )
so
as
Now
,
he
the
see
space
fer QE
let La
:
tht
04
oh
→
f
2 lo )
)
st
a
.
E)
,
nxn
'
Oh ) be
a
-
Sheu
) , wht
,
M
o
)
5
I
-
symmetn
does
sm
'
d '
so
by Lal A)
Then
B
t
H
I
=
D
minus
Tuan )
if
the ants
in
w
'
a
ek
,
.
Hpffbrthskn
z,
,,
) uhez
tete
=
,
53
a
{
)
,
.
Monday
an
pot ip , ,
z
,
=
,
pstipg
,
.
called
the Hpf mp
dent ftp.QD
an
,
fields
met
ishg
=
the
'
'
ham
H XKHH fall
/ the Sphe
5→9P
.
0
=
is a
.
tagat vat
,r
i.e. ,
theft
.
AHKHT
lost ,
fr
×
Et
mp
RB +13 R
-
4,3 of
eh
, ,
,
take
tp , Po
pteitz eitz D= eittz
)
p Hz a)
.
pop ,
an
one
exactly
are
tisnt
get
,
integral
a
tfh
we
=
-
.
be
we
.
.
is
a
aphx the
point
here
.
last
=
=
,
,
sky
→
5pH)
al
knew
we
'
=
then
p tpoi Bit Rk )
a
'
a
Now different
.
F
lost
210 )
Since
so
.
-
21012%5+2 ' lo ) a lost
=
kp
=
pzjt Bk
it
,
the mnitld of orthogal metros from
are
a
th }
by albpsy each
QD ,
Consider
0
see
a
In psklr
to
Tzhp
are
,
the votr field X
of
example
In
at
ane
.
next
12 the
but :
uf
Ztp)
tat they're orthomoml at oh point
cheek
s.mil
an
you
(
=
,
And
to
ip Hp)
=
=
I
to
=
21
'
6)
ite :
,
of
Shen symaetn matrix Thebe
,
-
.
he
an
think
f
TIN
.
look like ?
=
QA
then
.
D=
QD
cheek
((dLa)[
this if
't
ya oh
believe
it
!
)
Sine
A
,
tam , if
In
Da
new field X ,
Lieb
Df A
( As
argue
a
by 53
:
,
013 )
.
Mae
The
.
The
.
.
Df
:
gulf
.
,
For the
v.
to wit
quieten
f othool
retries
,
anthe Lie
tht the
so
on
d) then the
04
grp
6×6
mp
fixed
the Lie
.
wit qutmin
a
Barthol
,
,
se
Da
→
,
grp
6
gin
,
}
-
:
Hpyl
=
Hp " Kyl
ad
8
.
by
↳
,
"
Htgh
B
smooth
.
.
E) , SUHR ) SLK e) OH Sold Uk) Suk )
,
,
,
,
,
6
a
an
,
field X
on
f. X
53 gin
is called
by
( why ?)
lie gp G
ae
dying
left innit if
-
Hpk ip,
notice
tht
EG Lglhtgh
frg
y
,
dLgX=X
,
&
BK)
=
hg
,
respeelmf
.
VGEG & Balled right that if dRgX=X FGEG
.
?
where
Spltk last tistlp
(dRg)p×#ldRDp
=lRq°Dko)=¥l⇐d5pHq)=F¥⇐
.
=
so
X
B
r
.3ht
.
Kwit
.
.
,
)
is
is
Qb for
,
skw symuelm
uhe
both smooth
a
GUN R ) GUN
:
RP3± So b)
talk
an
mess
field
=
Q Da
=
right hwiatveetr fields
mnifld
differentiable
a
9
X ( Q)
{ 1 !} ) }
grp
r '3ht
is also
if
-
exgdyf
are
×'
left involved
a
as
=
RIP base
Left &
ant
1 mean
to Q
with
be
G
.
be repented by
an
dat
f TAOH
ay
so
( syn ( Ulm
=
Hamby
Oh )
exapkf
Yit on 53
its swjedne &
,
Isetan )
on
an
& thy
ay mtiogrps
B
sane
is
whih
group
a
mukptuth
,
try Th
A vgtr
hj
is
the pooht
She
,
01D
oytt with
QD
a
the pnshtf
She
.
.
6
Lie grip
:
fr
B
the veetr tells X ,
Also,
,
QD
=
To
→
arbityuyfr field X
an
tdrk
a)
at
,
1dL
Here
TIOH
:
1dL
last fist ) Pg
ipq
=
X ( pp
,
.
.
§
'
.
For
A
X
so
In genet
,
a
,
left
Tf X
left
B
is
-
.
in
wit
XIA a)
,
.
Wright tmwint rear field X
LI
AQ D=
=
Bopbtf
& GEG then Xlg ) =(dLg)eXk )
,
detuned
by
its rhe
Heft
He
ihttg
ef
6!
.
left muwit vat fed X XD Cd
ETEG then
Xdm0tDsmbyXdlQ5QD.htiithffwAEdny@LnDaXb1Ql-ldLaDo.Q
LD ev
Caverly if
,
So
we can
coHalm
to He
befetht TIQD
saw
f kfthwit
identity
TEG
osisb
betty
rear fields
an
6 ( ok
sym ah
-
.
called
the Lie
the
left ihmnwt
algebra
f 6)
with the
.
so
-
.
=
by
-
a
the nxnshw symmetric mtnb
f
bijutm anespmbe w/ the Shu
.
define
,
identify the
tamgt space
We
neon
v
X.
nxn
.
,
'
menus with the bisecting
,
in
hg
new Adds
an
04
are
in