Download - Jon - Sharif
←attire✓ Jon : Rp'
= Real projective plane .
Rp'=R%→v → Coordinate charts : R'= # ii. xD
L fi, u
rt : auto si.
.- Ea. sina.tt a -- E. p.es,
Mr : art . ⇐ I 5,3=1 a'= nat p 's.azW W
Us : asf . E- an's 5!,=÷ a"
- I; p"
-- E
Minus : a'=L P'=L ⇒ F- I ga÷l . ) !''
Ts'
/'II ¥
=µp.
÷ If = to → Riots not orientable .
I
Definition: let M be an orientable manifold of diner. .
We can define an n - form W I W # no one all M .
Uv
, ah' ¥me form
.^h⇒dnhdn: - Ian has > o:o-
defined on VI
I 1
on Ninh w = hoax 'D df÷, gn÷. - fg÷.
.
dat 's - din-
dnihdniin drain= E" " in dnlidnin di
'
←
on Vin Uz W -
- hcxcx 's) §n÷. . . Finn, E" " "
dxi.ndxw.TW⇐ h'ex 's 1197,11 dnindni . dm " so
- -
then we extend this . to Mr and then we confine ou IM.
W is positive over all d . It is ' called a volume form .
EXAMPLE : our 82 W = Dino dandy o IO En
w> o over all E- { North pole ]- { South pole } .
-
I : of,
Ads -- GandoI
J A.all ⇒ I @tas.ds%
as s7
C
.io#iy.EioEiIoojooo.%ea-
µa=↳dmi÷E€µ££p& . §.tw ⇐ If hxsdntdn? - dah .
f. cus I
pen
%siEnj¥
.¥ : ! .eu:1/eco.e,s...odo9ijbeMddOeIo,tMCfEfo,2nJ"""
"
"
ten -
- t.eu . !!" - H: ÷:*,I
I
: Uni paracompact if any poitfrl can he
conned by a finite numb f dats I
-Def : partition f Unity : { Ei! : M- R are continuous faiths
on M .
I = 2-Ei-
j) o E Eilp) E l Ez
e'is C- icp, -_ o'f p f- Mi . . .%⇒-
iii) § Evp = I t pent
{ fw I fmfw⇐ ED = E f fue .
Hi
#
⇐ Ms. S'
: •y•\um- sun ÷
.
Us -- E- Ls) . xiixi.
Un : Xii :÷.
xi -- Ii
.
Us : xi -- 7¥ xi -- I.÷
No = Cno X! = Pino Gil=
SinceGe
{ me Simone Tao q-cotoasp
Nz= Since Sony XI = Cotoz Sing
in Un ( Ot . ) Xi - ate ay XI - ate since
zti-X.tw/ht--cotezei9enUsCo--tn)Xi=tamEzbpXI--tanEzSinf2- = Xitixi = tangent
in UµnUg : 2-+= ↳z-
on S'
: we had W [email protected]' )a(ftp.edxieff.dxi)off, 9¥W = Sino ) I dxindxi
.
¥. :L.
W = Sino 1-&
✓1%7*1
dxihdxi.
HEE Fie't . I.÷÷e.ee -↳ e.Fiat ones. .
:*÷÷, since
at.ae/=ttzl#ez3atEt ' }
/
J - - I 8÷ ⇒ w-tsinoe.gg.
dxindxi'
w = -4 @ in E)" dxindxi
'
dis't exit. ate
' + at 'E=÷q - w.it#;IydxindxIty*We = W on UN .
)
2- Joe E"I 1-6 In=
Za Xtc -Xa E = X, - i -Xu d 2- ad I = - zidxindxz .
ZE = xiexi - W+=£z¥dZ -
W,
is. always non -zero 8 welldefined on Un -
to find hi, we use change f coordinates :
For simplicity . -2=5 5 -- ¥ z -- f. E --f.
W-
= Ends di =
on Us : w_=j÷d-I
¥i÷÷⇒*.
I
