avakalana vyavasthala kosam samaikyata paristhitulu

ganitamlo, pakshika avakalana sameekaranaala yokka konni vyavasthalu, vaati antarleena rekhaganita mariyu beejaganita nirmaanam yokka konam nundi, avakalana roopaala vyavastha paramgaa upayogakaramgaa roopondinchabaddaayi. bhedaatmaka roopam upamaanipattuku "parimitam" chese vidhananni sadviniyogam chesukovadam, mariyu eee parimiti bahya utpannaaniki anukuulamgaa untundi anae vaastavam. konni athi-nirnayinchabadina vyavasthalaku idhi oka saadhyamaina vidhanam, udaaharanaku, integrable vyavasthala yokka vadulugaa unde jatalatho sahaa. oka fafian vyavastha 1-roopaala dwara matrame perkonabadutundi, conny eee siddhaantamlo avakalana vyavastha yokka itara rakaala udaaharanalu unnaayi. vistrutamgaa cheppalante, oka pafafian vyavastha anedi oka mruduvaina manifold meeda 1-roopaala samiti (idhi vyavasthaku "parishkaralanu" kanugonadaaniki kanugonadaaniki samanamga untundi).
suutram _ 2-dimensional manifold formula _ 3 paie avakalana 1-roopaala suutram _ 1 yokka sekarananu ichinatlayithe, oka samagra manifold anedi oka nimajjanam cheyabadina (tappanisarigaa ponduparachabadani) upamaanipadam, dheeni prati point formula _ 4 vadda sparsaka sthalam prati suutram _ 5 dwara (pullback) naasanam cheyabadutundi.
garishta samagra manifold anedi nimajjanam cheyabadina (tappanisarigaa ponduparachabadani) upamanipold
forumlapy parimiti mapp yokka kernell
suutram _ 10 yokka prati binduvu suutram _ 9 vadda suutram _ 5 dwara vistarinchi untundi. adanamgaa suutram _ 5 saralamgaa swatantramgaa unte, suutram _ 10 anedi (suutram _ 13)-dimensional.
suutram _ 3 garishta samagra manifolds dwara foliationnu angeekaristhe oka fafian vyavastha puurtigaa samagramgaa untundani chebutaaru. (folliation kramam tappakunda undavalasina avasaram ledani gamaninchandi; i.ii. folliation yokka aakulu upamanipolds ponduparachabadakapovachhu.)
samaakalyata paristhiti anedi taginanta adhika parimaanam yokka samagra upamanalu untaayani haamii ivvadaaniki suutram _ 15 paie oka sharatu.
avasaramaina mariyu taginanta paristhitulu.
oka fafian vyavastha yokka puurti samaikyataku avasaramaina mariyu taginanta paristhitulu frobenius siddhaantam dwara ivvabaddaayi. oka samskarana prakaram, ring lopala α "i" sekarana dwara beejaganitaparamgaa utpatti cheyabadina aadarsha suutram _ 16 vaerugaa moosiveyabadi unte, maroe maatalo cheppalante,
appudu vyavastha garishta samagra manifolds dwara foliationnu angeekaristundi. (nirvachanaala nundi dheeniki viruddhamgaa spashtamgaa untundi.)
samagrata laeni vyavasthaku udaaharana.
prati paphian vyavastha frobenius ardhamlo puurtigaa samagramainadi kaadu. udaaharanaku, eee krindi oka-roopaanni pariganinchandih
dθ" anedi "θ" dwara utpatti cheyabadina aadarsamlo unte, vedge utpatti yokka vakrata dwara manaku untundi.
conny pratyaksha ganana istundi
idhi r3 paie praamaanika valume roopam yokka sunna kaani gunakam. anduvalla, dvi-miteeya aakulu laevu, mariyu vyavastha puurtigaa samakalanam kaadu.
marovaipu, nirvachinchina vakrarekha kosam
appudu paina perkonna vidhamgaa θ anedi 0 gaa nirvachinchabadindi, anduvalla vakrarekha anedi sulabhamgaa parishkaram (i.ii. edainaa sunna kaani sthiraamkam "c" kosam paina perkonna pfaffian vyavastha kosam oka samagra vakrata).
anuvartanaala udaaharanalu.
remanian jyaamitiloo, oka orthogonal cofframe "θ" i ", i nuu kanugone samasyanu manam pariganinchavacchu.ii., 1-roopaala sekarana, prati binduvu vadda cottengent space yokka aadharaanni yerparustundi, suutram _ 22 thoo, ivi muusivaeyabadataayi (dθ "i" = 0, "i" = 1,2,..., enn"). paayinkaare lemma dwara, θ "i" sthaanikamgaa manifold meeda konni functionla kosam d "xi" roopaanni kaligi untundi, tadwara r "n" yokka open upasamititoe "m" yokka open upasamiti yokka isometryni andistundi. atuvanti manifoldnu sthaanikamgaa flat anie pilustaaru.
eee samasya "m" yokka cofframe bundle paie prasnaku taggutundi. manaku atuvanti closed cofframe undani anukundam
manaku maroka cofframe formula _ 24 unte, appudu rendu cofframelu orthogonal parivartana dwara sambandham kaligi untaayi.
connection 1-roopam "ω" ayithe, appudu manaku untundi.
marovaipu,
conny suutram _ 28 anedi orthogonal samuuhaaniki mourer-carton roopam. anduvalla, idhi nirmaanaatmaka sameekaranaaniki kattubadi untundi.
suutram _ 29 mariyu idhi m yokka vakrata matrameh suutram _ 30
frobenius siddhaantam yokka anuvartanam taruvaata, daani vakrata adrusyamaite matrame manifold m sthaanikamgaa chadunainadani okaru nirdhaaristaaru.
saadhaaraneekaranalu.
oka-roopaala dwara tappanisarigaa utpatti cheyani avakalana vyavasthalapai samaikyata paristhitulaku aneka saadhaaraneekaranalu unnaayi. veetilo atyanta prasiddhamainavi carton-kahler siddhaantam, idhi nijamaina vislaeshanhaatmaka avakalana vyavasthalaku matrame panichestundi, mariyu carton-kuranishi podigimpu siddhaantam. vivaraala kosam "marinta chadavadam" chudandi. newlander-neerenberg siddhaantam daadaapu sanklishtamaina nirmaanaaniki samaikyata paristhitulanu istundi.

