]rrnrnersfon, Ernb_eddrng arrol ny Whr{n4/+ ErnbedA;nX Theo.crn

,() Irnmersran,

bef.lr Le+ M/ N be

'funch'on f 'M -r Nd;4ferenilabte rnanifolds, Thcn dn Tggerrg.., ls a

whore derivafiv.r- is .r*rJwhere inl ediue.di{ferenf ialc

'TExarn p\c

r)

sAn irnmerslon 13 u map thcr{.

br$, ar il * T4,pN in;eclive

2) ErnbEddiag r

!Sl3 , A rnap {, rrt *n, ts calaol lePer. i{ the- preimage o'p e'vortd carnpact

set in N is comPact in M'

An ir,rnmcrSion that i6, in3e ctive- arrd propar is called an gglg4$fl/ ,.t n .- t.F \

/ff.tof ,a, dxt ernAead;ng {' lil -: N orn (rnme-r'sion , whrch rnotPs iA \

\hr*.0*orph;call6 onto i"ts \rna6e. (+(u) cs) I

lr:poSl"l j ' \* {', ttr -+N \s an embrda['nf r rhan 4(n) is a subrnon'{old ,l N

---

osr

oQ"*.o \U2'

not \alechva

is [n3+clve on tangant spcrc€s ] -{,hoLt is

for aL[ Pin[s P e M.

Li)

*(-f--+---\F.> ^

-tro

e

\

-. 0n irnrne,rsion o4 g{

. bqt the \rnmercfon is

. oot an ernbeddiag . Consider the c,r,t"rvc P. GTr,n) -r fL'deflled 11 p Ltl = ( sin 1t , stnt ). p is cn inlech've- smoo*h irnmersiofl

[bu.out. p'L+) neuer vanishes)

. it ?s not an enbedd';ng because ite,inaage

ie cornpaut whilg its dorno"v' is not'

6,/ u,liiii-s atr it, \/ \tmir poMts ,5o i+ is \I a closed s"bset cl llt'' II s*,u if is bcvnded qs

I

\ ,rff rbg'the- Heine-Bre) , I\ thc'rerrr ,it is q cor\^fac+ ,

\ subsc/ o{ fu P\alt l[1 ' /

@

*if When M ilsel{ is a urnpact rnanifold,

{or compacl manif olds , er6 bedd.r'ags ,,re

3) trr. Whi{ner6 Ern bedd.^ag Theo.errr

---a N is propet. Thus

frnme,rsiaos..

e-v,i.ty fi^? f , r"r

Jr,rSi- one- to _one

. Le+ /t(

su,f,fi'cizrtlg

*Tha BurnP

be o Lomrrdct rnanifo ld. Tnen there is an

lorge n,

Tun r{ion

eytrbe dd"nX M -+ rrf tor

c(r\r

Thera

? {xe t1lfr , [lrlln<r ,

fs o. f-{',rnchon Y on

lrr[, t lx*l <r\

ffm with follorotn6 propertr'es:

[r) *>.olii) {=,{ on

ttii) {=-O on

illt1t-m\ c(z)

Proo{ ,f ulrt , t?o

e J(+) '= IL o , t(o

is c*,

'ttrts frrncnbn is srnooth at t-*O,

(fo.t' dorivafi,,ra o{ tne {unchon on the rightiS zero at thr's porut)

€)

-0 Lt)att)t= 'ir

{tt\ +{ ('t-t)

Tar *>. I /

For { (o t

{(t-t)={t+l-o

0=).s)

gLt),g [+) ",

l,o

h[+),= gkte)1(l-+)Fcr t 2.L t

For t s '.L

$0, ht+)'i!

Fcr -l gt

X tl-t) ' o , ht{)= o

zt (t+z) =o , htt) ' 0

supporrteel lns(dc L''']

gl /htt)=L'

(ur,hJ wi-fh

I

3 chort (ul,ha) ,

L

/rA= Un;'(cc,r)\=lt

(uto , hu) as above

@

4 -\hq Ulhitnu4 EmbtdAr'n6 Thscrf/^ Co'*t'awA :

Ro4, Foc

hr(P\--0

^.4* r-

i M comPac{ +

erto.rg point ? e M we can {ind a chart

dnd h, Lur) = ct:)

[,-'( ctr)) ond M= \-J h;'(c(t))PEM

:) {; e C-( r,a )

13; c h,-'(c({)) ano( M- 0 6;. (f*i+e union ol sqb,o,^e,s)i=(

. * Define {; e L* ( M, tt*)

* g;,= Y;-' ( t) , whete {*,'= { c h; on (,t;

IY;=O olh.arwise

f{i (x)hl (x)f; (x)=./

lofr, x €U;

o'lherwise

+ 1 e c" (M / t0utrn+tt )

ftr)= (f,r(v),(a[r) . /fn(r),v.tx))

+ For x € 0i =) {l =h i t f, Is irnme_rsivra on B; :)

I is irnrncrsive D^ A\

. 16 € Bt , thc-n {; (g) = hl (g) * hi (") =.f ;(x).

, 1/B; , the,n {i tg) + I ='P; (x)

+ iJ qn in.;ective \rnrnersion A ----> lp-L(-*^). Sincn M is compac-i ,

+ banX on inlecfiue inrmersion fs an ennbe ddril .

Lernrrq '

(a) An irnrnusion f t M --+W , whrdn is a horneomorphr56 o[' U on {(ra),is on crnbadd'r'n6.

(b) frcorrtrnuous inlectir,"ernap {,U-€W w\+hcompa6* Mis a horneororphi,:

M ov*r {tul

o

of

Proo{ ,

[a)For points p€ Mn and, f(p) €*^*t, f is i4 the $,m c!,

(xi ,- ,x^\ (yn,--..r\rn,Or- ,o) (x)

(x) {ollows the Loca.t Tnmusion Theo,ern (u.,hi.h stor{ea +ha* f is locallg

€qui va)eat Lo tho carronfral irvmersion neor x ) .

{(u) cv

vt c lB**l'

tl'c fLI x1.--7 (x,o)

M

\,t

l^

.l + tu) is ope^ in {(ra) + { (u) = 4 (M) nvo + Vc oPen

.f L ({trq) nv nv" ) = ( nr**lo}) n L (v nv,)

+ 4(ra) ts submoni{old ".f W , P ,t q horneorrrc.ph;srua lrcvt

'+ f ti on e.rnbeddinX ('on Propositicn 4)

inW

A^ -+{(lt)

($

\dJu

'aBpwrrl! o 1Uo utstldtou/o"u/oq 13 tl t4t

dods JJ'roPsnnP v oluo a:ods

ltvdwol ,o ur.rorol- ,o+rr0tq Io tr t

)

'snonuJ+uo-t tt ,-,1 (-

(w),i v1 uado tt (n), +

)jroPsnaH (w)f, c Pasolr 5i (vY),|

.sno^uJ+uo, Pulo 1:oduro: s] (nfw)f,= (n)t\(wld {F

J,ro: s.1 W a) u1s 1: oduo: puo pasop 5t Ir\ \W (= t.rado aq W , n {"'i <l-

' aarp afiq puo sno^vr+uo" s.t (wl I c- W 1 J G(q)