Yesterday MEIR
"
submfd .
,
× EM .
¥ : IM → Des KTM.IR) ,
IR )
Cluj : 4, is a linear isomorphe isn .
Süß The rest of the proof ( of Then .
3.24 ) ; uauelg ,
we
finish the proof that 4 ×
is surjedve .
f- E LM.IR)
,
Wort oraedx ( V. n ) wihnk) - O
and Belo) Eu LU ) .
We have see a that for ye U wir
ulg) z Bdo) we how
fly) = fk) t
{nily) hily) hi : vielBdo) ) → R .
By Cor .
2.32 we can exteud hi and ni to smooth fcts.ae M
without men
ging
than to
cdlg eraed × : The keuchen
fk) + Lnihi ¥)
i
isaehe@xheudedtofce.ouMtuotcoincideswhflocdlg
arendt .
If de Der.
(CNN.IR) ,
IR) ,
thenÄnne 3.23 :
Dlf) =
dlfk)
t.fi h ;) =
§ dlui ) hik) e- ÜK) dlhi)
11 Fo
Of
Tui
k )
=
§ d)8¥!
)
0=0
,
← Loki ) #E)
→
z.5TaugatbmdagawpsfobrdmniflssupposelM.it) abstract uefd .
of dim .
K .
Then we defiue the
tangeut space
of M at x as the vector
Space :
IM Des Lola,
R) ,
R) .
Notation:
↳ (f) : =
9
;
f t f- E LMIIR) Remake:
Alternative ly ,
we could how de find IM as
the ser of equivoleuce classes of Smooth Curve s c : I → M ,
OEI
www.hc/0)=x.wherecneCz,ifx=Cslo)=Cz(0 )
and for a ( eqniv .
, any ) Chart ( Ka ) arandx In.
g)
'
lo) =
(a)
'
lo) .
The
tangent bundle of M is defined es
TM : -
4.MTXM =
Yeah} XIM
p :
TM → M naturel projektion .
For a smooth uvp between cufds . f : M → N we
defiue wesauetiuesjw.ge
Tflx , ) : = Lfk) , Ifk) (Tt
!) Ifk
) .
when If i
.
IM →
Tf ) is
giveu by
If 4) Ig) : = T.tk/.g:--klgof)=rs;lgof)
Then . 3.24 .
V
ge
CALN NR) .
One weites direct leg that :
.
.
T (h. f) = The off for h :
N → P
Cueop
betreuen
:
Tld
,
=
lof,
uebels .
•
f is a Local diffeeen . ⇐ Txf : IM →
Tee!
is a Linear is au .
tfx EM .
(u
,
U ) E A Tu : TU =p
Ilo
) - Tutu) = ulu) x IR
"
.
There ex ists a
unique topologyaeTMS.l.TV CTM is
Open and Th : TV → Tutu) is a kaueiamorpwsnv-LU.nl EA .
It is seuaed uaehoble und Hausdorff .
Moreau , Ajjü { LTU,
Tu) :
Hu ) e- it}
olefine, n
Cb - athos of Charts wie wenig in IRZK (see the cornesp .
Statement for subufd
MEIR"
) .
=) (TM , Atm) is a Gueook man ifdd of dim 2k .
Moreau , o ) for subuefd.
of IR
"
, p
: TM → Yt is Smooth
and it defilees u veao knolle of rank
k over M ,
and
Vector fields on More de find es ( Smooth ) Seehaus of p
:TM) N .
Local loordiude expressiaes for the tagen ueop
Tf of a Smooth
map f. MAN
,
which is
legale Sunday and ↳ for Mad
fields remo in valid .
Definitions / Statements about pull
back
of redd fields vie
↳ ud oliffeom . and Local flows of vector fields nleuoinvoliet
without
my charge .
3- 6 Vector fields as derivation s and the Lieb rocket
(
M
,
it ) n maifeld .
For se HIM) and f ECHMIIR)
g.f : M - IR
⇐ f) k) : =
4. f =
Ifk
defilees u smootufct ,
since g. f is tue secaud
uaupouenl
of Tfos : M - TM → TIR - IRXIR ,
while is sooo k -
Def.3.cn A derivation of the
algebra (MDR ) is a
linear ueop D : LM.IR) - OLM
.IR/s.t.Dlfg)--DH)gtfDlg)V-f,gEC0LMilR) .
Notation Der (01M .IR)) : =
{ D : Chloe .IR) → CNN.IR) : D is a
derivaten } .
This is a
Vector Space in the daraus way Twins
The
map ¥ : { ↳ (tm } f)
defiueg
• linear isomorphem HLM) -7 Der ( CNN.IR)) .
Proof : fi-s.fr linear CM.IR ) → (MK) ✓
s.lt g) G) =
s; Hg ) =
# f) gk ) + f) (sig ) -
Ks.
f)gtfls.gl/t
So 4 Leos image in Der (CALM.IR) .
Heule , 4 defilees o
uop
HIM ) → Der IM.IR) ) and
evident
leg it is linear ,
since If is Liner tx EM and f- EGMR) .
hi : If 9 # 0
,
than I x EM s.tn .
EY
) # 0
By The 3.24
,
we know that 7 ft ④ IM .IR ) s) .
×
↳ af =
4. f) k ) # 0 .
Sui ,
Let D C- Der Lehre ,
IR)) .
Frag XEM
f- Im Dlf) k ) is a de ! voten at × .
Heule , by Then .
3.24
7 ! KEIM sie .
DA) k) = sif .
Rennie, tu show that
" '→
Sx de feines a veaarfiehd ,
i. e it reeuoiu) to show
smooth ness .
Fix * EM and a most ( Ku ) wih × EU .
As in the proof of Them .
3.24 ,
we
mag
exteud ni ( in
.
.
.
,
k )
to a a smooth fch üi : M → IR that coincide wie hier Saeue
gen weigern .
VCU of x .
Then Dlüi) : M → IR is a zuodutd and
Sy
Elsjü
(
y
) tyev ( see proof of
i Then .
3. 2h ) .
K
Heule .
SIE
{ Dlüi )
),
# is a suoohveaotieed
I -
1
am V .
-
D
Recall that for a Chart ( Un) ,
%,
.
f = ¥
Oni
equeds tue i - the portid deriv. of Local Wadi nahe expressiv
f-out of f . This im
pures
did for ey It HIM ) wie
SIE Esi # we how ⇐ f)
Iü ? if .
L.emma3.27-SMEHIMIveddtieedsonamfd.tl .
Then f m (s .
(g. f)) g.
(g. f) defilees a derivation
of CALM , R ) T
Proof f. g
E IM .IR) sie
.
Hg ) ) = -
Heft!
+ ftp.gD-f.K.FI/gt/n.f)lsg)
. tfs .
f) Ing) tfn.gg)
Def.3.IM uefd .
For two vector fields sitze HIM ) the
hie brocket of S and
q
is
the Unique
weder fields Is , RßE HIM)
si .
Es , yf .
f =
s .
( y
.
f) µ
(s . f) ttfECMM.IR) .
P-rop.3.IM mfd .
,
g.
q ,
e E HIM ) .
① [sind = Elis]
und Es , Iz , e) + [z ,
[e. D) + LT , IhreD=
= 0
( Jacobi ildaetg).
② Es ,
fq] = f Esp] t ( s f)
2
and [ fs , g) = f- Es ,
y
] Chef)
s .
Proof .
① Skew -
syuueety
Wand Jacobi iaeeuhy
follows frau
mind less Compilations .
② f. g
E IM.IR ) Is , fz] .
g
= . . L
Kfz).
g) G) =
ftp.ig-lfln.gl ) k) .
"
s .
4g ) = s .
(flag) ) = t
-4.9¥,
Ife) .
f.g) =
ftp.ls.g )) ¥)
=) Es , ftp.g = ¥) ¥)
=
ftp.g/tkf)R.g .
=) Es ,
fn] = f- ES, d rd .
f) R ✓ .
Prop.3.30_M.Nmfds.fiM-Naloaddiffean@phism.tOf-
*
Es
,
q] =
It's it) ts , y
EH IN ) .
In particulier ,
if U ⇐ MAN is on
open subset , Erik]
)j
[
Stu ,
( i : N I ilo ) EN drifte
^
indessen is di Mean .
euho its
image ; i 's =
Yo
.
Hance ,
Stu
= 0
Impuls 55,27/0=0 .
②
¥!!)) k )
,
= [
sind k ) ts.ge#lM),xeM .
£
IMMUN
③ Suppen ( U , n ) is a Wort an M and Size HIM) with
%
Esi
# and
ME Eni ¥ ,
then
Kindle
=
Ästhet# ,
wwe
Earth Ists ¥! ei
%) .