la etty 2ydyldxldz - math.toronto.edu
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a Theflow of X is flt x y z Etty tx tty z
Then Lxw ly Ft'tw adz
leo tetsu test dexoft Ad toft
la Ettyexpected dxldZ t t dyldz
Xy 2y dxldZ Hey dyldz
b lx dw Lt 2ydyldxldz
29 dux donde date dude dz x dyed
29 dxldZ ydude
daw d ay y dz
2xydxldZ x 35 dyldz
LxW lxdw dixW 2xytzy dxtdz Rey dyedZ
C Ss caw f du f3yd
rdx 3 2daddy
3 f xyz daddy
3S r rdrdo 3
2 aFMppose
RAW Auto
complete win to a coframe W'w w uh nearP
let Xi XD be thedual basis
Jenforcis 1,2 0 2 Aw Aw Xi Xi Xi Xi
RCXi.li wwwxd ncxiitilwhwgD
NCXi Xj n Xj xi 2RCXiixj
R Xi Xi o forcis 3 in R annihilates D
E Suppose R annihilatesDThen RAW Aw Xi Xie Xis Xiy where
Ee 2 Xiao Xian W Xiong W Xian
foreach rest at least two of oct ok 0131and04 are
in 31 in If 013 rally is in 3 in then W Xia 0
or w Xian o respectively If not then both radioed e 31 in
andso I XianXian 0 Soeach component inthe sumvanishes implying
NAW'M w2 0
b Suppose naw two
Asabore Let w w own be a completion ofthecoframe on nhbd Up AP
Then I E fist winds for foie c o
0 RAWAwa Eefi wit w Aw nut
ince winesaww 3Licien are linearly independent wehave fi 0
for 4523So 1 4 fi with A f wi w'dÉfrjw
h ItEricUp
Let Sr bea Partitionofunity subordinate to Up
Define B EenSpiti for i 112
Then foranyGEM
w AB Iq WAB q Wig 1 Scottie Wiang
Wig 1 Scott Wiang
Icp W A Efi Wp twinfajawipsp
sa WAPp warp q
SCP Nq Iq
so Ya WAp w'dBe everywhere
C If du ha na datdatAw O
Then Disinolutive
Let Vid be a flat Chat neerp sit 3,1 ign is alocal
frameJD on
Then Since W Zi o for 0 3 in wehave
W W Epi dx t W Byz dx9 dx ga da
Similarly we W Epi dx eat Bye dx
hi dx t hadx asneeded
3a let Uid and V 4 be charts s t
Yofo d x nm x ith
Such Charts exist by thesubmersion theorem
Then Zane i gym is a local commuting frame J D
herp on on f r So D is asmooth rankm n involutive
distribution In fact UNF V y is a flatchartYAnearP
b
forPEM VieiracTPM IxpRp Vi Vn Up Xp Vir vn
f Wfan Xp Vir iVa
Wfor
frfrFApur Fapun
D
so 1 1 0
ILx2 Hdr didLx f'd wD inthe sameway asabove
C N RI de
1g l o for anti non 0 21 lag de
If I doesnt contain i then 12g d D
so OI gontaining
RI 12g dad
Igntaini RI É di i di n Adi's A dx
Ernani RI C't dinAdin rain
21 0 forany I containing i
2 Earn RI de
13,2 0 brittle im 0 2255 deIfn
OFF so for i Mtl im RICH im RI xi xn.o no
Define WIE CCIR by WI x xh RICHmix or no
and let w WIdxI Er IR
Then FAW E WI of f deIchi in
since Fisthecanonicalsubmersion WIof x exm WI x in Recti.it oi i0
Also F'da da
Iggy.mnWIFE Fdaso Ftw E
EIeuan
RICH No o dxfy.am
LILL n
RICH th d CA nm Rex am
Away everywhere
d o F wt w
EIshan
W of WI of de
WI of WI of WI we since fis surjective
4 a observethat we lx din Adt so by definition
definesthe boundary orientation on 2M
In Lx did nda f dix din ndam
Smd Ee xi dinnah di
ISmwnet f din ndthM
Cnn Vol m asneeded
b Suppose Flom is the identity Since we rush dw o
Soo f f'd wM
Smd Ftw
fan it ftw
Sam Foi w
since Foi Flom IdSon W
Socontradiction
5a am Axs U 13 5
I fomented withrespect toorientedwithrespectto theopposite nonstandard
Thestandardorientationonsorientation onSh
b Since werism duro
So o Jnf dw
fdf w
M
Is i ftw fan it Ftw
where is Axs Cs M
in Bash com
f.sn io wtfsnlfocitw
Ssnfo wxfsnftwfsnfo w fsnf w
asneeded
so ff.tw Senf w
Sn
4Recall from problem 2211 in theball that
theantipodalmap is orientation reversing if niseren
So if Wis non vanishing topform specified byThe standardorientation Then f w is a non vanishing
topform specified by theopposite orientation
so Ssn w so
and fsnfitwa f.sn w Lo
So fsnf.tw fsnftwifniseven
6a the
y w x xa X warmthw Xin CM xa
i
XCalXi i Xa
bystokesb The linear isclear
F win Sawada fern Jadwin
fdwan
In radarF R W so Fissymmetric
C The S admits a global frame Xi 42,13 Proven earlier
forPEM let Yep Zep E U be local frameforAm
nhbd UpAPThen Yip fi Xi tf tr tf Xz on 0
Zip FfXi 92 2 93 3 on U
where fi gie Cr UpConstruct aglobal frameD D using partitionJunity
Y EspYeaZ ESPZep
d fate as is false
let flag 45 1
Let X Ex which isnot tangent tos
but X f 2645 1 2x so on S
e Try n fur new fatwa
Suppose RAW 0 and let 0,0 beachart So 2Ridx w wid
Assumewho that W 0
Theno RAW now dxidwi Kiwi Kiwi deild
Riwj njwi oforcij ti.in since dxinda Kittenarelinearlyindependent
ni It wi2 1 w mu
indepenterdythechat
So Nsfw for some FE CCM
f The letpes
Since X N Y wehavethat Tap Xp Yup Thp Xp
Since IT Tof Tap Ap Tap XPHof p Xp
Tlap FapXp
Recall that IT is alocal diffeomorphism so tap isinjective
Aep X p Fa pXpA FAX
7
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in
Eggen C11 w XixD Xi Xi is Xue
A I Egg tÉÉiÉaixD Xviii Mix
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t.EEwCCxEIBTFIii xii xu
et e it Eg w x oh Xi HindiXue
imposition
et Gift wanni xuLxi wChi hiii Kati
Itemfi altar Mi nm Lxiwt
asneeded