l

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

duck Xue Ets't Xi war ni Xu

in

Eggen C11 w XixD Xi Xi is Xue

A I Egg tÉÉiÉaixD Xviii Mix

I É E.inwICxnxIXiYxiiixn

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