samaroop aavaran

ganitamadhye, 's' yaa aadhar yojnevar 'x' yaa yojnevar 'g' yaa gut yojaneche krutee sootra _ 1 dile tarr 'x' vareel samaroop aavaran 'f' heey sootra _ 2-modulesche aavaran aanee sootra _ 3-modulesche samarupata aahe.
jeey kokikalachi aut purn karateh gunakarasathi "m" lihine,
vyakhyevar tippanya.
dandachya patalivar, kokikal sthitee mhanate qee samarupata sootra _ 6 heey rachana sootra _ 7; i sarkhech aahe.eae., samooh krutichi sahsambandhata. gatache ekak olakh mhanoon karya karate hii aut dekhil eka parinam aaheh sootra 9 midvanyasathi donhi bajunna sootra 8 lagoo qaraa aanee mhanoon sootra 10 hii olakh aahe.
lakshat ghya qee sootra _ 11 hii eka atirikt mahitee aahe; tee 'x' varr 'f' sheefvar 'g' chya krutichi 'lift' aahe. yavyatirikt, jevha "g" haah jodalela bijganitiy gut asato, "f" haah eka ulat karata yenyajoga aavaran asato aanee "x" qamee hoto, tevha kokikalachi sthitee svayanchalit asateh konatehi samarupata sootra _ 12 aapoaap kokikal sthitiche samadhan karate (hii vastusthiti ch chya puravyachya shevati nondavali jaate. 1, §3., mumfordchya "bhaumitik aparivartaniy siddhantacha prastav 1.5.")
jarr 'jee' chee kriya mukt asel, tarr tourserschya bajoone utaralyamule, sampariman aavaranachi kalpana bhagfal 'ex'/'jee' varr aavaranat sopee hote.
yonadachya shabdalekhananusar, sutrala samaroop aavaranachi rachana dene _ 2-module "f" heey sootra _ 14 varr ringjasathi gut samarupata denya sarkhech aahe,
sadhya katadyanchya sandarbhat vishamatapoorn katadyanchi vyakhya dekhil aahe. vaikalpikaritya, susangat aavarananchya shreneet, samaja, samaroop aavaran hii samaroop aavaran mhanoon paribhashit kelee jaau shakate.
resheey resheche bundle.
ulat karata yenyajogya aavarnavar kinwa reshechya bandalvar asalelya vishamatapoorn aavaranachya sanrachanela rekhiyakaran dekhil mhanatat.
x' haah bijganitiyadrushtya bund kelelya kshetravar sampoorn prakar asoo dya, joou jodalelya 'g' yaa qamee qamee gatadware aanee 'l' yaa ulat karata yenyajogya aavarnadvare karya karato. jarr "x" samanya asel, tarr "l" chey kahee tensor power sootra _ 16 resheey aahe.
tasech, jarr "l" khoop puresa aanee rekheey asel, tarr "x" pasoon sootra _ 17 paryant eka "g"-rekheey bund visarjan aahe, jenekarun sootra _ 18 rekheey aahe aanee "l" vareel rekhiyakaran sootra _ 18 dware prerit kele jaate.
tensor utpadane aanee rekheey ulatya aavarananche ulat punha naisargik paddhatine rekheey kele jaatat. asha prakare, "x" yojnevaril rekheey ulatya aavarananche samarupata varg abelian gut tayar karatat. x" chya picard gatat eka samarupata aahe jee rekhiyaran visarat jaate; heey samarupata samanyatah injection kinwa surjective nasate aanee tyache curnell kshullak reshechya bandalchya rekhiyakaranachya samarupata vargansah olakhale jaau shakate. 
jyaa prakarasathi bahutek reshanche bundle resheey nasatat, tyaa prakarache udaharan 2,16 chey udaharan pahaa.
vishamatapoorn aavarananchya vibhaganvar duheri krutee.
bijganitiy gut "g" aanee "g"-samatulya aavaran "f" heey "k" kshetravaril "x" varr dile tarr sootra _ 20 heey jaagtik vibhaganchi jaaga asoo dya. tyanantar teey 'jee'-modulechi rachana swikarte; i.eae., "v" heey khalilpramane "g" chey resheey pratinidhitva aahe. gut krutisathi sootra _ 21 lihine, pratyek 'jee' sathee 'jee' madhye aanee 'vhee' madhye 'vhee' sathee, chala
jethe sootra _ 23 aanee sootra _ 24 heey samarupata aahe jeey samaroop-aavaran sanrachanedvare "f" varr dile jaate. kosayakal sthitee nantar heey sunishchit karate qee sootra _ 25 heey eka samooh samarupata aahe (i.eae.sootra _ 26 heey pratinidhitva aahe.)
udaharaneh sootra _ 27 aanee sootra _ 28 ghya, jee swatahvar 'jee' chee kriya aahe. mug sootra _ 29, sootra _ 30 aanee
sootra _ 26 mhanje 'jee' chey daway niyamit pratinidhitva aahe.
varr paribhashit kelele pratinidhitva sootra _ 26 heey eka tarksangat pratinidhitva aaheh 'v' madheel pratyek vector 'v' sathee, 'v' chey eka maryadit-aayami 'g'-upghatak aahe jyamadhye 'v' aahe.
samparivartanshil vector bundle.
vector bandalsathi vyakhya sopee aahe (i.eae., sthanik patalivar mukt asalelya sthir shrenichya shifshi sambandhit vividhata). jarr 'jee' tantumaypane karya karat asel tarr aapan mhanato qee bijganitiy gut 'jee' dware karya kelele 'ex' yaa bijganitiy prakaravaril vector bundle 'eae' heey 'samaroop' aahehi.sootra _ 34 heey vector spacesche "rekheey" samarupata aahe. aahe shabdant, equivarient vector bundle hii eka jodee aahe jyat vector bundle aanee krutee sootra _ 35 heey sootra _ 36 paryant uchalane asate jenekarun prakshepan sootra _ 37 equivarient asel.
naun-equivarient settingpramanech, equivarient vector bandalcha equivarient vaishishtyapurn varg paribhashit keela jaau shakato.

