
palamadangu thakarnnuveezhunnu

remanian jyamithiyil, thakarunnatho thakarnnatho aaya manifold ennathu oru "n"-dimensional manifold "m" aanu, athu remanian alavukalude oru shreni "g" i "amgeekarikkunnu, athaayath" i "ananthathayilekku pookumbool manifold oru" k "-dimensional spacinu aduthaanu, avide gromov-hosdorff doorathil" k "<" n ". saadhaaranayaayi ("m", "g" "i") ennathinte sekshanal vakrathakalil chila niyanthranangalund. ettavum lalithamaaya udaaharanam oru paranna maanifoldaanu, athinte metrik 1/"i" kondu punarnirmikkaan kazhiyum, angane manifold oru poyintinodu aduthaanu, pakshe ellah "i" num athinte vakrata 0 aayi thudarunnu.
udaaharanagal.
pothuve paranjaal rund tharatthilulla thakarchayunda
(1) aadyathe tharam vakrathaye orupole parimithappedutthumbol oru thakarchayaanu, formula _ 1 ennu parayuka.
formula _ 2 ennathu formula _ 3 dimensional remanian maanifoldukalude oru sreniyaayirikkatte, ivide formula _ 4 "i" th maanifoldinte sekshanal vakrathaye soochippikkunnu. jeff cheeger, kenchi fukaaya, mikhaayil gromov ennivar theliyicha oru sidhaanthamundu, athu ingane prasthaavikkunnu formula _ 1 umm formula _ 7 umm aanenkil, formula _ 2 oru "n"-ghadanaye amgeekarikkunnu, formula _ 9 minifold "m" nte injectivity radiusine soochippikkunnu. ekadesham paranjaal "enn"-ghadana ennathu oru nilmaanifoldinte praadeshika pravarthanamaanu, ithu cheegar gromov avatharippicha oru eph-gadanayude saamaanyavalkkaranamaanu. eee sidhaantham yathaakramam torres pravartthanavum parimithamaaya vyaasa kesukalum maathram kaikaaryam cheiyunna cheeger-gromov, fukaaya ennivayude muun sidhaanthangale saamaanyavalkkarichu.
(2) randaamathe tharam, vakrathayude thaazhnna paridhi maathram nilanirtthumbol thakarukayaanu, formula _ 10 ennu parayuka. 
ithu eathaandu negatteevu allatha valanja manifold kesumaayi adutha bandhamullathaan, ithu negatteevu allatha valanja maanifoldukaleyum eathaandu paranna maanifoldukaleyum saamaanyavalkkarikkunnu. formula _ 12, formula _ 13 thudangiya alavukalude formulayude oru kramam amgeekarikkukayaanenkil oru manifold eathaandu negatteevu allatha valanjathaanennu parayappedunnu. eee takarunna saahacharyathil vakrata thaazhe parimithappedutthumbol eathaandu negatteevu allatha valanja manifold vahikkunna punk vakrata parimithappedutthiya kesil eathaandu paranna manifold kalikkunnathinu thulyamaanu.
vakrata thaazhe ninnu maathram parimithappedutthumbol, formula _ 14 ennu vilikkunna paridhi idam oru alexandrove space aanu. paridhi sthalathinte saadhaarana bhaagath, formula 17 mathiyaya valuppamullappol, fiber eathaandu negatteevu allatha valanja maanifoldaayirikkumbol formula 14 lake praadeshikamaayi oru nissara fibration phoom formula _ 15 undennu yaamaguchi theliyichu. ivide saadhaarana ardham formula _ 18-strainer aaram thaazhe ninnu oru positteev samkhyayaal allengil ekadesham paranjaal, euclidean speysilekku praadeshikamaayi adachirikkunna sthalam kondu ekeekrithamaayi parimithappedutthiyirikku ennaanu.
formula _ 14-nte oru eaka binduvil enthaanu sambhavikkunnath? eee chodyathinu pothuve oru utharavumilla. ennaal alavukal 3 ll, sheo yamaguchiyum eee tharatthilulla thakarchayude poornna vargeekaranam palamadangu nalkunnu. oru trimana manifold formula _ 22 formula _ 23-neey thripthippeduthukayaanel inipparayunnavayil onnu sariyaanennu avar theliyichu (i) "m" ennathu oru graph manifold aanu allengil (ii) formula _ 22-nuu formula _ 21-nekkaal vyaasakkuravum parimithamaaya adisthaana grouppumaanu.