
upamaanipadam

ganithashaasthrathil, oru manifold "m"-nte oru upavibhagam ennathu oru upavibhagamaanu, athinu thanne manifold gadanayundu, athinaayi ulppeduthal bhoopadam chila gunangale thripthippeduthunnu. eathu propertykal aavasyamaanu ennathine aashrayich vyathyastha tharam sub maanifoldukalundu. vyathyastha ezhuthukaarkku palappozhum vyathyastha nirvachanangalund.
aupachaarika nirvachanam.
thazhepparayunnavayil ellah maanifoldukalum oru nischithathinu klaas "sii" aarr "yude deferentiable maanifoldukalaanennum ellah morphisangalum klaas" sii "aarr" yude deferentiable aanennum nammal anumaanikkunnu.
mungiya submanifoldukal.
oru manifold "emm"-nte oru immezhsdu submanifold oru immezhsshan maappinte "ezz" aanu; pothuve eee chithram oru upavibhaagamenna nilayil oru submanifold aayirikkilla, koodaathe oru immezhsshan mapp kuthivayppu polum aavasyamilla (onnu muthal onnu vare)-ithinu svayam-intersectionukal undakam.
kooduthal sookshumamaayi paranjaal, mapp oru kuthivayppu (onnu muthal onnu vare) aayirikkanamenn oraalkk aavasyappedaam, athil nammal athine oru kuthivayppu immezhsan ennu vilikkunnu, koodaathe oru topology deferential ghadanayum ulla oru immezhsdu submanifold imej subset "s" aayi nirvachikkuka, athaayath "s" oru manifold aanu, koodaathe "f" ulppeduthal oru diphiomorphisamaanu ithu "n" leeii topology maathramaan, ithu pothuve subset topologyumaayi yojikkilla pothuve, upavibhagam "s" ennathu upavibhaga topologyil "m" nte upavibhaagamalla.
eathenkilum injective immersion nalkiyaal "m"-ll "n"-nte chithram oru defiomorphisamaayathinaal oru immezhsdu submanifoldinte ghadana advitheeyamaayi nalkaam. mungiya submanifoldukal krithyamaayi kuthivayppu nimajjanangalude chithrangalaanennu ithu pinthudarunnu.
oru mungiya submanifoldile submanifold topology "m" ll ninnu paarambaryamaayi labhicha subspace topology aayirikkanamennilla. pothuve, ithu subspace topologyekkaal mikachathaayirikkum (i.iii. kooduthal ooppan settukal undu).
nuna upagrooppukal swaabhaavikamaayi muzhukiya upamaanivrithangalaaya nunagrooppukalude sidhaanthathilaanu mungiya upamaanivrithangal sambhavikkunnath. frobinius sidhaantham theliyikkaan sariyaaya sandarbham nalkunna elakalekkurichulla padanatthilum avaa kaanapedunnu.
ulchertha submanifoldukal.
ulchertha submanifold (regular submanifold ennum vilikkunnu), oru immezhsdu submanifoldaanu, athinaayi inclusion mapp oru topological embedding aanu. athaayath, "s"-leeii submanifold topology subspace topologic thulyamaanu.
m" ll oru manifold "n" ulcherthirikkunnavil, chithram "f" ("n") swaabhaavikamaayum ulchertha oru submanifoldinte gadanayaanu. athaayath, ulchertha submanifoldukal krithyamaayi ulchertha chithrangalaanu.
palappozhum upayogapradamaaya oru ulchertha submanifoldinu oru aantharika nirvachanamund. m" oru "n"-dimensional manifold ayirikkatte, koodaathe "k" oru poornnasamkhyayaayirikkate. m"-nte "k"-dimensional embaded submanifold ennathu oro pooyintinum "p" adangiya oru chart nilanilkkunna oru upavibhagamaanu, athaayath "k"-dimensional playininte "φ" ("u")-maayi intersection. eee jodikal "s"-leeii deferential gadanaykku oru atlas roopappeduthunnu.
alexanderude sidhaanthavum jordan-shoyenflies sidhaanthavum sugamamaya ulcherkkalukalude nalla udaaharanangalaan.
mattu vyathiyanangal.
saahithyathil upayogikkunna upamaanivrithangalude mattu chila vyathiyaanangalund. oru vrithiyulla submanifold ennathu muzhuvan manifoldinteyum athirthiyumaayi yojikkunna oru maanifoldaanu. sharp (1997) oru ulchertha submanifoldinum mungiya submanifoldinum idayil evideyengilum sthithi cheiyunna oru tharam submanifoldine nirvachikkunnu.
pala rachayithaakkalum topological sub maanifoldukalum nirvachikkunnu. iva "sii" aarr "sub maanifoldukalkku thulyamaanu. ulchertha topological submanifold, ulcherkkal vipuleekarikkunna oro pooyintilum oru praadeshika chaarttinte nilanilppinte ardhathil sthiramaayirikkanamennil. wilde aarkkukalum wilde nots udaaharanangalil ulppedunnu.
swathukkal.
m"-nte eathenkilum submanifold "s" nalkiyaal, "s"-leeii oru pooyint "p"-lekkulla taanjant space swaabhaavikamaayum taanjant spacinte "m"-leeii "p"-lekkulla oru leanier subspace aayi kanakkakkaam. ulppeduthal bhoopadam oru nimajjanavum kuthivayppum nalkunnuvenna vasthuthayil ninnaan ithu pinthudarunnathu.
s" ennathu "m"-nte oru mungiya submanifoldaanennu karuthuka. ulppeduthal bhoopadam adachittundenkil "s" yadhaarthathil "m"-nte ulchertha oru upamaanipadhathiyaanu. neremarichu, "s" oru ulchertha submanifold aanenkil athu oru clossed subset aanenkil inclusion mapp adaykkum. ulppeduthal bhoopadam "i": "s" → "m" sariyaaya bhoopadam aanenkil maathram adaykkum (i.iii. compact settukalude vipareeta chithrangal kompaakt aanu). i" adachittundenkil "s" neey "m" yude adacha ulchertha submanifold ennu vilikkunnu. clossed embaded submanifoldukal submanifoldukalude ettavum mikacha klaas roopappeduthunnu.
yadhaartha coordinate spacinte upamaanipadhathikal.
chila "n"-kalkk, yadhaartha coordinate space r "n"-nte ulchertha submanifoldukalaayi smooth maanifoldukale chilappol "nirvachikkunnu". eee kaazhchappaad saadhaarana, amoortha sameepanathin thulyamaanu, kaaranam, vitni embedding sidhaantham anusarichu, randaamathe kanakkakkavunna minusamarnna (amoortha) "m"-manifold r2 "m" ll sugamamayi ulcherkkaan kazhiyum.