SI pukback bundleo) NCM

submanifold.E-Mvectbundlewemay.iestnt " Em NEh Via

defn 2 :

tuiiopenwerofMHtkGMlgpiu.nu. → Gecki 叫tansitīmsm{ Ui ⼆二 Uxn N : penwwofNgj= gxsluinuiVia defu I : Elw = E ' ( N ) 在

suficestoshowthatEINBasmoothmanifld.recall.lip EVCMT 彐 U : coordnatenbdofU.in M

d-ixmjsuchth.atUn N = 9 x " = x " 芝 _ _ =xmioSInfuet.Nneedsnotbec.su

bmanifldI) inp-E-Miavectwbundle.N-IYiasmwthmapcn.cmdtnmthciroutpw_LTE-Nthepull-l.aek bundk dwnsims )

Its

nnkisthesameasthatfER-E-NotethatfpEN.FR/p-Efcp,

不

Nn_f.in Let TE = { cp.es ENXEI fpies 3i

E-M~natuuyngarditasE-NXMrestrict.tnG =ftp.fcpj/pENf(embeddmgNcsNxM)

中凹deswibethepull-backbundkviadeh2exa-f.lv

→ M cnstantmapPhs q

。

⼆二 ) f E =

NXRKSI.subbundleandquotientbundlesubbundleofE.in

tuitiveymeansthatthereisavectwsubspac.ecofthesamedimensimjfeachEpdelE-M.asubbundleofE.is

a vectwFwithFts E so that . diagmm commutes

T) La . L : embeddngM · Fpts Ep Qineih

G. :injectivelqu-weufomEplFV-pEM.W.tlitpconstituteavectr bundk ?

wearegmgtocmshucttviadefn2.detalsmoothisecti.nofEisasmwthmaps.M-Esuchth.atT.siI.us/*姊 Namely.fpEM.thereisavectwsip.inEpfnirhmdthgvgsmoothlyinp_l.cn

lucal tivializatim Elu ⼆二 Ux Rk is equìvalenttoklocaly defned CP.ee ) = Sk

smoothsectnns.whichformabassfreachEppf.bydefinitim …i-ncallthmlocaltnializngs-cctonsrmt.avectorbundle.is

tivialfandalyifitadmitsglobaltnializiugsections.MGEnaturallybythezerosect.in?

Back t F d E - he 以⼆二⼦子

Fhiux Rki) wp EM 彐 U : neighbwhnod t P |

Elu ⼆二 Ux ReuxRKEFlu-isEI.EU x 代

⼼心 到 系 nticcogjlislinearlyindependent.in 是ii) Bylineanalgesne.IS wk-n.iweb.EE pYEUSUchthat.luSilp , -.- .co Sup ) , w -,,- , We 3fwmabasisofEpTakeasmwthextensim.fi

Wkti,

-_-

,wegoverU.Denote.itby 95 … -_- , Ej Fw instauceextudthem(iii)Bylookingat.IR tj mstantggE.lu

Eux Rl

SC.si/--.ciSk,in.---,Tesfnrmlocaltnivialingsectu.fEorersomeneighborhrodofplsh.in k UJ

iv) Hene,

⼆二opencouerluxb.andexiElu.EU/nxlResuchthatQ:Flux-Uxx(lRkxfoS)

Itfltws that gj = { 我 彎 | _ekOk

ekdifgjsatisfiesthecoycle andition稔 撮 捻捻 ⿃鳥 斲 嚠 t

*

exami.ie/recap__f:N-Msmwthmap(nofurtherassumptim)fdiT.N-TcpM~zfi.TN

→ FTM mwphism of bundlesorerMbLN

Iffdpisinjectivewpksfisanimmers.in )TNcanberega-dedasasubbundleoffTYItuscase.tk

qwtientbundleiscdledthemrmalbundkoffwlinM.rtl.ateron.wewillintoduetheinnerpwdutfuvectorbund.us

,andwewillseethatthequotient.mstuctionisthesame.ae wrmal " iutheusual sense§ 亞dgebmiccmstuctims.lpseudosttFcrvectwbundlesoverM.ca

nperformlinearalgebmiconstuct.vnUpt 叫togetanothervectorbundleItiseasiertdoitviadefn2.tl

ereares.me examplesl) dual bundle

-

ow unvAVznns Vz⇒ Elv ⼆二 以 RK-suxRKEE.lu

* AT * 4 , 以 l-nlx.g.ir )

uslii-Vz.lusposeīn

temsofdualbasis~Elv-EVxlRKo-UXRKEEYulx.fi?wY-lcx,w)

DefmeEtbythetunsitnsf-tifwhichshlsatisfiesthecocycle.cm

ditim

2) tensor-recallV.to : vectrspae⑦ VDW abstuctly : ⼆二 4 : Vx W → VDW bilinear

suchthatanybiliuearmapFiVXW-Zfactasthwughe.nlamey Vx W_w VDWH

⇐\, z

① explicitly.fi :basisfwV.fwjsbasisf~WVDWhasbasisfliowj.d.inVDW = dim

V.dimwju-vz.wn-wzlinearmapsnsuxowz-svzDW.AE= Eak Ūk , Bwj Ebfoi ⼀一 LADB ) ( Vi Dwp

Sbasisfwk Gbasishrwe ⼆二点 Ǜbf 倀 咤 1

Ordenythebasīssuitably , ADB =|

1 週⼀一 … nj啊 -.-

la-Say.dimu-midmkeachlthastlesamesizeasB.SE,

F : vec.twbuudleswerMDefmeEBFbythetansit.ms 戀 = gip 9点

1 吐出 Sma 9点 9点 satisgthecogdemdtiBsodoesgEDFur3Jothers@E.F

:

vectwbundleswerMEEDFisdefnedsythetmnsit.ims 8與 fiw 0

rank E = K → f E : rank = 淡 )8是

NE = MXIR.AE ⼆二ENE: rank = I,

_

1 8您 ⼆二 detl 9刏

Notethattakngdetpueseruesthecoycleondit.in

① Homl E,F ) ⼆二 FDEX lìnput E

, output FJ

1

re-TM~stnlTMTTM-flTMJ.itssetonsanedifferentiall-f.ms

m M