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]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)
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