
lagbhag jatil kaee gunaa

ganit mein, ack lagbhag jatil manifold ack chikani manifold haye joe pratyek sparshrekha sthaan parr ack chikani raikhik jatil sanrachana seey lais haye. pratyek jatil manifold lagbhag jatil manifold haye, lekin lagbhag jatil manifold hain joe jatil manifold naheen hain. lagbhag jatil sanrachanaon key symplectic jyaamiti mein mahatvapurna anuprayog hain.
yeh avadharana 1940 key dashak mein charles ehresman aur henz hoff key kaaran haye.
aupchaarik paribhasha.
m" koo ack chikna manifold honey dein. m" parr ack lagbhag jatil sanrachana "j" ack raikhik jatil sanrachana haye (yaani, ack raikhik maanchitra joe-1 taka varg kartaa haye) joe manifold key pratyek sparshrekha sthaan parr hoti haye, joe manifold parr sucharu roop seey bhinn hoti haye. dusare shabdon mein, hamaare paas ack chikna tensor kshetra "jey" degree kaa haye joe sootra _ 1 haye jaba sparshrekha bundle parr ack vector bundle samroopata sootra _ 2 key roop mein maanaa jaataa haye. lagbhag jatil sanrachana seey lais ack manifold koo lagbhag jatil manifold kahaa jaataa haye.
yadi "m" lagbhag ack jatil sanrachana koo sweekaar kartaa haye, tou yeh summ-ayami honah chaahiye. ise iss prakaar dekhaa jaa sakata haye. maan leejiye kii "m" "n"-ayami haye, aur lagbhag ack jatil sanrachana haye. aggar taba. lekin aggar "m" ack vaastavik bahugunit haye, tou ack vaastavik sankhya haye-iss prakaar "n" taba bhee honah chaahiye jaba "m" kii sanrachana lagbhag jatil hoe. koee yeh dikhaa sakata haye kii yeh abhivinyas yogya bhee honah chaahiye.
raikhik beejganit mein ack aasaan abhyaas seey pataa chalata haye kii koee bhee summ ayami sadish sthaan ack raikhik jatil sanrachana koo sweekaar kartaa haye. isaliye, ack summ ayami manifold hamesha ack-rank tensor koo "bindu kii disha mein" (joe pratyek sparshrekha sthaan parr sirf ack raikhik parivartan haye) sweekaar kartaa haye jaise kii pratyek bindu parr "pee". kewal tabhi jaba iss sthaaniya tensor koo vaishvik star parr paribhaashit karane key liye ack saath patch kiya jaa sakata haye, tou bindu kii drishti seey raikhik jatil sanrachana lagbhag jatil sanrachana pradaan karati haye, joe taba vishisht roop seey nirdhaarit kii jaati haye. iss patching kii sambhavana, aur isaliye ack manifold "aim" parr lagbhag jatil sanrachana kaa astitva sparshrekha bundle key sanrachana samooh koo ack seey dusare taka cumm karane key barabar haye. astitva kaa prashn taba vishuddh roop seey beejganitiya topological haye aur kaafee achchi tarah seey samajha jaataa haye.
udaaharan.
pratyek purnank n key liye, sapaat sthaan r2 "n" lagbhag ack jatil sanrachana koo sweekaar kartaa haye. aisee lagbhag jatil sanrachana key liye ack udaaharan haye (1 ≤ "i", "j" ≤ 2 "n"): summ "i" key liye sootra _ 3, visham "i" key liye sootra _ 4.
kewal way kshetra joe lagbhag jatil sanrachanaon koo sweekaar karate hain, way hain s2 aur s6 (). vishesh roop seey, aiss4 koo lagbhag jatil naheen diyaa jaa sakata haye. 
sanrachana (ehresman aur hopp). aiss2 key mamle mein, lagbhag jatil sanrachana reeman kshetra parr ack imandar jatil sanrachana seey aatee haye. 6-kshetra, aiss6, jaba ikai maanak kaalpanik octonion key samooh key roop mein maanaa jaataa haye, tou octonion gunan seey lagbhag jatil sanrachana viraasat mein milati haye; yeh sawal kii kyaa isaki ack jatil sanrachana haye, jise heinz hopp key baad "hopp samasya" key roop mein jaanaa jaataa haye.
lagbhag jatil manifolds kii vibhedak topology.
jaise ack sadish sthaan parr ack jatil sanrachana "v" "v" mein "c" key apghatan kii anumati detee haye aur "v"-("j" key aayatan sthaan kramshah + "i" aur-"i" key anuroop), tou "m" parr ack lagbhag jatil sanrachana jatil sparshrekha bundle "tm" c (joe pratyek bindu parr jatil sparshrekha sthaanon kaa sadish bundle haye) koo "tm" + aur "tm"-mein apghatan kii anumati detee haye. tm" + key ack khand koo prakaar (1,0) kaa sadish kshetra kahaa jaataa haye, jabaki "tm"-kaa ack khand prakaar (0,1) kaa sadish kshetra haye. iss prakaar "jey" jatil sparshrekha bundle key (1,0)-vaahak kshetron parr "aaee" seey gunaa karane key liye aur (0,1)-vaahak kshetron parr − "aaee" seey gunaa karane key liye mell khaata haye.
jaise humm cottangent bundle kii baahari shaktiyon seey vibhedak roopon kaa nirmaan karate hain, vaise hii humm jatil cottangent bundle kii baahari shaktiyon kaa nirmaan curr sakate hain (joe jatil sparshrekha bundle key dohare sthaanon key bundle key liye canonical roop seey isomorphic haye). lagbhag jatil sanrachana "aar"-roopon key pratyek sthaan key apghatan koo prerit karati haye.
dusare shabdon mein, pratyek ω "r" ("m") c ack apghatan koo ω ("p", "q") ("m") key yog mein sweekaar kartaa haye, jisamen "r" = "p" + "q" hotaa haye.
kissy bhee pratyaksh yog key saath, ack vihit prakshepan π "p", "q" ω "r" ("m") c seey ω ("p", "q") taka haye. hamaare paas baahari vyutpann "d" bhee haye joe ω "r" ("m") c seey ω "r" + 1 ("m") c taka maanchitrit kartaa haye. iss prakaar humm lagbhag jatil sanrachana kaa upayog baahari vyutpann kii kriya koo nishchit prakaar key roopon mein parishkrit karane key liye curr sakate hain.
taaki sootra _ 8 ack aisaa maanchitra haye joe prakaar key samroop bhaag koo ack seey badhaata haye (prakaar key roop ("pee", "kyu") koo prakaar key roop ("pee" + 1 ", kyu") mein letaa haye), aur sootra _ 9 ack aisaa maanchitra haye joe prakaar key samroopata virodhi bhaag koo ack seey badhaata haye. inn prachalakon koo dolbiolt prachaalak kahaa jaataa haye.
choonki sabhi anumaanon kaa yog pehchaan maanchitra honah chaahiye, isaliye humm dhyaan dete hain kii baahari vyutpann likhaa jaa sakata haye.
lagbhag jatil sanrachanaon koo ekikrit karna.
pratyek jatil kaee gunaa apane aap mein lagbhag ack jatil kaee gunaa haye. sthaaniya samroop nirdeshank sootra mein _ 11 maanchitron koo paribhaashit curr sakata haye.
(jaise π/2 kaa ghadee kii vipareet disha mein ghoornan) yaa
yeh aasaani seey jaanch curr sakate hain kii yeh naksha lagbhag ack jatil sanrachana koo paribhaashit kartaa haye. iss prakaar ack bahugunit parr koee bhee jatil sanrachana lagbhag ack jatil sanrachana pradaan karati haye, jise jatil sanrachana dwara 'prerit' kahaa jaataa haye, aur jatil sanrachana koo lagbhag jatil sanrachana key saath 'sangat' kahaa jaataa haye.
iss prashn kaa uttar haye kii kyaa lagbhag jatil sanrachana kaa tatparya ack jatil sanrachana key astitva seey haye, yeh bahut cumm tuchchh haye, aur samanya roop seey sach naheen haye. ack manmaane lagbhag jatil manifold parr koee bhee hamesha nirdeshank paa sakata haye jisake liye lagbhag jatil sanrachana kissy bhee diye gaye bindu "pee" parr uparokt vihit roop letee haye. samanya taur parr, halanki, nirdeshank khojana sambhav naheen haye taaki "jey" "pee" key poore pados parr vihit roop leey. aise nirdeshank, yadi way maujood hain, tou unhein 'jey key liye sthaaniya samroop nirdeshank' kahaa jaataa haye. yadi "m" pratyek bindu key aspas "j" key liye sthaaniya samroop nirdeshank koo sweekaar kartaa haye tou yeye patch ack saath "m" key liye ack samroop atlus banaate hain joe ise ack jatil sanrachana detaa haye, joe isake alawa "j" koo prerit kartaa haye. "jey" koo taba 'samakalan yogya' kahaa jaataa haye. yadi "jey" ack jatil sanrachana dwara prerit haye, tou yeh ack advitiya jatil sanrachana dwara prerit haye.
m" key pratyek sparshrekha sthaan parr koee bhee raikhik maanchitra "a" diyaa gayaa haye; i.ee.a" shreni (1,1) kaa ack tensor kshetra haye, phir nizenhuis tensor shreni (1,2) kaa ack tensor kshetra haye joe isake dwara diyaa jaataa haye.
yaa, lagbhag jatil sanrachana "a = j" key samanya mamle key liye, jaise kii sootra _ 15,
daaeen or kii vyaktigat abhivyaktiyaan chikani sadish kshetron "x" aur "y" key chayan parr nirbhar karati hain, lekin baain or vaastav mein kewal "x" aur "y" key bindu-vaar moolyon parr nirbhar karati hain, yahee kaaran haye kii "n" a "ack tensor haye. yeh ghatak sootra seey bhee spasht haye
froliture-nizenhuis koshthak key sandarbh mein, joe vector kshetron key jhooth koshthak koo samanya banaata haye, nizenhuis tensor "naa" ["a", "a"] kaa sirf aadhaa haye.
neulander-nierenberg pramey bataata haye kii ack lagbhag jatil sanrachana "jey" taba samakalan yogya haye jaba "enje" = 0. sangat jatil sanrachana advitiya haye, jaisa kii oopar charcha kii gaee haye. choonki ack samakalan yogya lagbhag jatil sanrachana kaa astitva ack jatil sanrachana key astitva key barabar haye, isaliye ise kabhi-kabhi ack jatil sanrachana kii paribhasha key roop mein liya jaataa haye.
kaee anya mandand hain joe nizenhuis tensor key gayab honey key barabar hain, aur isaliye joe lagbhag jatil sanrachana kii samakalan kshamata kii jaanch key liye tareeke pradaan karate hain (aur vaastav mein inmein seey pratyek saahitya mein paaya jaa sakata haye):
inmein seey kissy bhee sthiti kaa tatparya ack advitiya sangat jatil sanrachana kaa astitva haye.
lagbhag jatil sanrachana kaa astitva ack sthalvaigyaanik prashn haye aur iska uttar denaa apekshaakrit aasaan haye, jaisa kii oopar charcha kii gaee haye. doosari or, ack samakalan yogya lagbhag jatil sanrachana kaa astitva ack bahut adhik kathin vishleshanatmak prashn haye. udaaharan key liye, yeh abhee bhee gyaat naheen haye kii kyaa aiss6 antatah asatyaapit daavon key lambe itihas key bavjood ack samaakalit lagbhag jatil sanrachana koo sweekaar kartaa haye. sahajata key mudde mahatvapurna hain. vaastavik-vishleshanatmak "jey" key liye, neulander-nirenberg pramey frobanius pramey seey anusaran kartaa haye; "see" φ (aur cumm chikani) "jey" key liye, vishleshan kii aavashyakta hoti haye (niyamitata parikalpana key kamjor honey key saath adhik kathin takneekon key saath).
sangat teen.
maan leejiye kii "m" ack symplectic roop "ω", ack remanian metrick "g", aur lagbhag jatil sanrachana "j" seey susajjit haye. choonki "ω" aur "g" gair-janitra hain, pratyek ack bundle samroopata "tm → t * m" koo prerit kartaa haye, jahaan pahala maanchitra, jise "φ" ω "(" u ") =" i "u" ω "=" ω "(" u ",") dwara darshaaya gayaa haye aur doosara, jise "φ" g "kahaa gayaa haye," g "key liye anuroop sanchaalan dwara diyaa gayaa haye. iss baat koo samajhane key saath, teen sanrachanaaen ("jee", "ω", "jey") ack sangat tiguna banaati hain jaba pratyek sanrachana koo dou anya dwara nimnanusar nirdisht kiya jaa sakata haih
inn samikaranon mein seey pratyek mein, daahine haath kii or kii dou sanrachanaon koo sangat kahaa jaataa haye jaba sambandhit nirmaan nirdisht prakaar kii sanrachana pradaan kartaa haye. udaaharan key liye, "ω" aur "j" sangat hain yadi aur kewal tabhi jaba "ω" (ω, "j") ack remanian metrick haye. m" parr bundle jisake khand "ω" key anukool lagbhag jatil sanrachanaaen hain, unamen sankara reshe hotey hainh sparshrekha tantuon parr jatil sanrachanaaen symplectic roopon key pratibandh key saath sangat hoti hain.
symplectic roop "ω" key praathamik gunon kaa upayog karate hue, koee yeh dikhaa sakata haye kii ack sangat lagbhag jatil sanrachana "j" remanian metrick "ω" ("u", "jv") key liye lagbhag ack kahler sanrachana haye. yadi "jey" samakalan yogya haye, tou ("aim", "ω", "jey") ack kahler manifold haye.
yeye teen ekaatmak samooh kii 3 mein seey 2 sampatti seey sambandhit hain.
lagbhag jatil sanrachana.
nigel hitchin ney manifold "aim" parr ack saamaanyikrit lagbhag jatil sanrachana kii dhaarana pesh kii, jise unake chhatron marco gualterry aur gill kawalkanti key doctorate shodh prabandhon mein vistaar seey bataaya gayaa thaa. ack sadharan lagbhag jatil sanrachana jatil sparshrekha bundle "tii. aim" key pratyek fiber key ardh-ayami upasthaan kaa ack vikalp haye. ack saamaanyikrit lagbhag jatil sanrachana jatil sparshrekha aur sambhaajak bandalon key pratyaksh yog key pratyek fiber key ardh-ayami samasthaanik upasthaan kaa ack vikalp haye. donon hii mamalon mein ack vyakti yeh maang kartaa haye kii upa-bundle aur usake jatil sanyugma kaa pratyaksh yog mool bundle koo utpann karey.
ack lagbhag jatil sanrachana ack jatil sanrachana mein ekikrit hoti haye yadi ardh-ayami upasthaan jhooth koshthak key tahat bund haye. ack saamaanyikrit lagbhag jatil sanrachana ack saamaanyikrit jatil sanrachana mein ekikrit hoti haye yadi upa-sthaan koo corrent bracket key tahat bund kiya jaataa haye. yadi isake alawa yeh ardh-ayami sthaan kaheen bhee gayab naheen honey vaale shuddh spinor kaa vinashkarta haye tou "m" ack saamaanyikrit kailabi-yau manifold haye.