
ritchie pravah

vibhedak bhoomiti aanee bhaumitik vishleshanachya ganiteey kshetranmadhye, rikki pravah (,), jyala kadhikadhi hamiltoncha rikki pravah asehi mhanatat, heey remanian metriksathi eka vishisht aanshik vibhedak samikaran aahe. samikaranachya ganiteey sanrachanetil aupcharik samanatemule, heey anekada ushnatechya prasaran aanee ushnatechya samikaranashi samaan asalyache mhatale jaate. tathapi, teey araikhik aahe aanee ushnatechya samikaranachya abhyasat nasalelya anek ghatana darshavate.
tyachya vyakhyet ritchie tensorchya upasthitisathi naav dilele ritchie pravah, richard hamilton yanni sadar kele hote, jyanni 1980 chya dashakat remanian bhoomitimadhye lakshavedhi naveen parinam siddha karnyasathi tyacha vapar keela. nantar vividh lekhakanni hamiltonchya paddhatinchya vistaramule bhoomitivar naveen anuprayog jhaale, jyat sayman brendle aanee richard scone yanni kelelya vibhedak goal andajanchya sankalpnasah.
ritchie pravahachya upayanchi ekatmata, william thurstenchya bhoomitikaran andajadvare andajit topological deta olakhoo shakate yaa shing-tung yauchya soochanenantar, hamiltonne 1990 chya dashakat anek parinam dile jeey andajachya sankalpanekade nirdeshit hote. 2002 aanee 2003 madhye, grigori perelmanne hamiltonchya karyakramachya kahee tantrik pailunchya naveen prakarasah, ritchie pravahabaddal anek moolbhut naveen parinam sadar kele. hamilton aanee perelman yanchya karyanna aata thursten anumanancha purava mhanoon mothya pramanavar manale jaate, jyat 1904 pasoon bhaumitik topologychya kshetrat eka suprasiddha khulee samasya asalelya poincare anumanasah eka vishesh prakaran aahe. tyanche parinam bhoomiti aanee topologychya kshetratil eka mailacha dagad manale jaatat.
ganiteey vyakhya.
gulgulit anek patanvar, eka gulgulit remanian metrik aapoaap ritchie tensor nirdharit karate. pratyek ghatkasathi, vyakhyenusar sparshikechya jagevar eka sakaratmak-nishchit antargat utpadan aahe. jarr remanian metriksache eka-parameter kutumb dile gele, tarr nantar ekhada vyutpatticha vichar karoo shakato, jee nantar pratyek vishisht moolyala aanee samamiteey dwillingi swarupas niyukt karate. remanian metrikcha ritchie tensor dekhil pratyekas samamiteey dvirupi swaroop niyukt karat asalyane, khaleel vyakhya arthpurn aahe.
ritchie tensorla anekada vibhagiy vakrateche sarasari moolya kinwa reeman vakrata tensorche bijganitiy trace manale jaate. tathapi, ritchie pravahache astitva aanee advitiyatechya vishleshanasathi, heey atyant mahattvache aahe qee ritchie tensorchi vyakhya, sthanik nirdeshankat, metrik tensorchya pahilya aanee dusya derivativescha samavesh asalelya sutradvare kelee jaau shakate. yamule richicha pravah bhaumitikdrushtya paribhashit aanshik vibhedak samikaranat hoto. sthanik samanvay sootrachya lambvartulakarteche vishleshan heey ritchie pravahachya astitvacha paya pradan karate; sambandhit parinamansathi pudhil vibhag pahaa.
eka shunyetar sankhya asoo dya. madhyantaravar ritchie pravah dilyas, aanee darmyancha vichar qaraa. mug. tarr, parameterschya yaa kshullak badalamule, ritchie pravahachya vyakhyet disanari sankhya-2 itar konatyahi shunyetar sankhyene badalali jaau shakate. yaa karanastav,-2 chaa vapar eka aniyantrit parampara mhanoon manla jaau shakato, jaree moolatah pratyek pepper aanee ritchie pravahavaril pradarshan anusaran karate. farak aevadhach aahe qee jarr −2 chee jaaga sakaratmak sankhya ghetali gelee, tarr pudhil vibhagat charcha kelele astitva pramey heey eka pramey banel jeey prarambhik detamadhun parameter mulyanmadhye mage (pudhe jaanyaivaji) sarkanara ritchie pravah tayar karate.
jaree heey aanshik vibhedak samikarananchya ganiteey kshetratil manak anaupacharik paribhashecha keval eka bhag asale taree, yaa mapdandala sahasa mhanatat. tee bhautikdrushtya arthpurn sandnya nahee. kharey tarr, rincy pravahachya pramanit quantum kshetrachya saiddhantik vyakhyemadhye, renormalisation gatachya drushtine, mapdand veleaivaji lambi kinwa urjeshi sambandhit asato.
samanyikrut ritchie pravah.
samaja too eka sankshipt gulgulit manifold aahe aanee madhyantarasathi eka ritchie pravah asoo dya. pratyek remanian metriksacha khand 1 asel, asey paribhashit qaraa; heey shakya aahe kaaran teey sankshipt aahe. (adhik sarvasadharanpane, pratyek remanian metrikche khand maryadit asalyas heey shakya hoil.) nantar jyache antidirative nahise hote asey paribhashit qaraa. sakaratmak mulyanche asalyane, tyachya pratimevar eka dwibhajan aahe. aata parametersasathi paribhashit kelele remanian metricks, samadhankarak aahet
sootra _ 1
yethe scalar vakrata darshavate. yala samanyikrut ritchie pravah samikaran mhanatat. asha prakare, pramanamadhye spashtapane paribhashit badal aanee mapdand mulyanchya punarparichayasaha, ritchie pravahache samanyikrut ritchie pravahat rupantar kele jaau shakate. vareel ganana ulat karoon ulat dekhil lagoo hote.
samanyikrut ritchie pravahacha vichar karnyache prathamik kaaran mhanje teey ritchie pravahasathi pramukh abhisaran prameyanche soyiskar vidhan karnyas anumati dete. tathapi, asey karane aavashyak nahee aanee aksharshah serve karanansathi ritchie pravah tyachya pramanit swarupat vicharat ghene purese aahe. shivay, samanyikrut ritchie pravah samanyatah naun compact manifoldswar arthpurn nasato.
astitva aanee advitiyata.
sootra _ 2 heey eka gulgulit bund bahugunit asoo dya aanee sootra _ 3 heey sootra _ 2 varr konatehi gulgulit remanian metrik asoo dya. nash-moser antarnihit karya pramey vaparoon, khaleel astitva pramey darshavileh
tyane khaleel advitiyata pramey darshavileh
astitva pramey gulgulit remanian metriksache eka-parameter kutumb pradan karate. kharey tarr, asey konatehi eka-parameter kutumb dekhil parametervar sahajpane avalambun asate. tantotant, heey sangate qee sootra _ 2 vareel konatyahi gulgulit samanvay chart sootra _ 16 chya sapeksh, karya sootra _ 18 konatyahi sootra _ 19 sathee gulgulit aahe.
dennis deturkney nantar vareel parinamancha purava deela, joou tyaeivaji banach implicit function pramey vaparato. tyanche karya moolatah yvon choket-bruhatchya suprasiddha puravyachi aanee laurentzian bhoomititil einstein samikaranansathi changlya sthiticha arth lavnyachi eka sopee remanian aavrutti aahe.
hamiltonchya astitva aanee advitiyatechya prameyamule, jevha deta sootra _ 20 dile jaate, tevha prarambhik deta sootra _ 3 saha sootra _ 2 vareel 'rikki flow' baddal konihi spashtapane bolu shakato aanee tyache kamal sambhavya moolya ghenyasathi sootra _ 5 nivadu shakato, jeey anant asoo shakate. visheshtah poinker anuman aanee bhoomitikaran anuman yanchya puravyamadhye, ritchie pravahachya aksharshah serve pramukh anuprayoganmagil tattva asey aahe qee, sootra _ 11 yaa kamal mulyachya jawal yet asatana, metricks sootra _ 6 chey vartan sootra _ 2 baddal sakhol mahitee prakat aanee pratibimbit karoo shakate.
abhisaran pramey.
khaleel abhisaran prameyanche sampoorn spashtikaran aanee madhye dile aahe.
trimiteey parinam yache kaaran aahe. harmonic nakashatil ushnatechya pravahachya abhisaranavaril james eals aanee joseph sampson yanchya 1964 chya yugantkari peparpasoon prerit aanee shithilpane tayar kelelya hamiltonchya puravyamadhye, samamiteey 2-tensorchya mandaniparyant kamal tattvacha vistar yasarkhya anek naveen vaishishtyancha samavesh hota. tyancha pepper (eals-sampsons) haah vibhedak bhoomitichya kshetrat sarvadhik uddhrut lekhanpaiki eka aahe. tyachya parinamache eka pradarshan aahe.
puravyachya drushtine, dvimiteey prakaranakade teen vegvegalya parinamancha sangrah mhanoon yogyaritya pahile jaate, pratyek prakaranasathi eka jyamadhye yularache vaishishtya sakaratmak, shoonya kinwa nakaratmak aahe. darshavilyapramane, nakaratmak prakaran kamal tattvadvare hatalale jaate, tarr shoonya prakaran avibhajya andajadvare hatalale jaate; sakaratmak prakaran adhik sookshm asate aanee hamiltonne pieter lii aanee shing-tung yauchya ritchie pravahachya pravahachya andajala eka abhinav "entropy andaj" saha ekatra karoon sakaratmak vakrata asalelya upkesala hatalale. hamiltonchya tantrachya vistarat, benetne sampoorn sakaratmak prakaran dakhavale. dvimiteey manifoldvaril konatahi ritchie pravah ekach anuroop vargapurata maryadit asalyane, nishchit remanian manifoldvaril scalar functionsathi aanshik vibhedak samikaran mhanoon tyachi punarrachana kelee jaau shakate. asha prakare, yaa vatavaranatil ritchie pravahacha abhyas purnapane vishleshanatmak paddhatindware dekhil keela jaau shakato; tyanusar, dvi-aayami abhisaran siddhantache paryayi gair-bhoomitiy purave aahet.
uchch-aayami prakaranala dirgh itihas aahe. hamiltonchya yashasvi nikalanantar lagechach, gerhard huskenne tyachya paddhatincha uchch parimananparyant vistar keela, jeey darshavate qee jarr jawaljawal sthir sakaratmak vakrata asel (ritchie vighatanchya kahee ghatakanchya lahanpanachya arthane), tarr samanyikrut ritchie pravah sahajpane sthir vakratemadhye ekatra yeto. uttal sanchadvare adakavanyachya drushtine kamal tattvache eka naveen sootra aadhalale, jyamule sakaratmak vakr metriksachya ritchie pravahachya abhisaranas vishisht bahuayami samanya vibhedak samikaranasathi "pinching sanch" chya astitvashi sambandhit eka samanya nikash tayar jhaala. parinami, jyaa prakaranat chaar-aayami aahe aanee sakaratmak vakrata operator aahe, teey prakaran too nikali kadhu shakala. vees varshannantar, christophe boham aanee burkhard wilking yanni "pinching sanch" tayar karanyachi eka naveen bijganitiy paddhat shodhali, jyamule hamiltonchya nikalatoon () chaar-aayamipanachi dharana door jhaali. sayman brendle aanee richard scone yanni dakhavun dile qee bund manifoldvaril ritchie pravahadvare samasthanik vakratechi sakaratmakata jatan kelee jaate; boham aanee wilkingchi paddhat lagoo karoon, teey eka naveen ritchie pravah abhisaran pramey () prapt karnyas saksham hote. tyanchya abhisaran prameyamadhye eka vishesh prakaran mhanoon vibhedak kshetra prameyache sankalp samavisht hote, jeey tyaa velee eka dirghakalin anuman hote. varr dilele abhisaran pramey huken, hamilton, boham aanee wilking aanee brendle aanee sconechya purvichya uchch-aayami abhisaran parinamanna samavisht karate.
parinam.
teen aanee tyahoon adhik parimananmadhil parinam darshavtat qee dilelya prakarache metrik swikaranari konatihi gulgulit bund bahugunit sakaratmak vakrateche avkash swaroop asane aavashyak aahe. hii antaral rupe mukhyatve ellie carton aanee itaranchya karyadvare samajali jaat asalyane, ekhadi vyakti pudhilpramane parinam kadhu shakateh
tyamule jarr konee thet heey dakhavu shakat asel qee konatehi gulgulit bund kelele saral-jodalele trimiteey manifold sakaratmak ritchie vakrateche gulgulit remanion metrik swikarte, tarr poinkere andaj lagech yeil. tathapi, sadhya goshti samajoon ghetalya jaat asalyane, haah parinam ultyaivaji keval poinker anumanancha (kshullak) parinam mhanoon olakhala jaato.
sambhavya vistar.
donpeksha mothe asalyas, anek bund-aayami gulgulit manifolds astitvat aahet jyat sthir vakrateche konatehi gulgulit remanian metricks naheet. tyamule vareel abhisaran prameyatoon vakratechya paristhiti sahajpane qamee karanyachi aasha konihi karoo shakat nahee. vakratechya paristhitichi jaaga kahee paryayanni ghene shakya aahe, parantu complexes projective spesasarkhya compact manifoldsche astitva, jyamadhye naun-negative vakrata operatorche metrik (fubini-studdy metrik) aahe parantu sthir vakrateche metrik nahee, yamule yaa paristhitila kitee dhakalale jaau shakate heey aspasht hote. tyachpramane, nakaratmak vakr remanian metriksasathi samaan abhisaran parinam tayar karanyachi shakyata bund remanian manifoldschya astitvamule guntaguntichi aahe, jyanchi vakrata aniyantritpane sthiratechya jawal aahe aanee tarihi sthir vakrateche konatehi metricks swikarat nahee.
lii-yau vishamata.
pieter lii aanee shing-tung yau yanni remanian manifoldsvaril parabolic differential samikaranansathi suru kelelya tantracha vapar kelyane khaleel "lii-yau asamanata" siddha jhaali.
 khaleel paryayi lii-yau asamanata darshavili.
yaa donhi ullekhaneey vishamata yaa poinker anuman aanee bhoomitikaran anumananchya puravyasathi atyant mahattvachya aahet. perelmanchya lii-yau asamanatechya ujavya bajuchya sandnya tyachya "qamee lambichya" karyatmak vyakhyela prerit karatat, jyache vishleshan tyachya "noncolepsing pramey" kadey netat. naun-collapsing pramey hamiltonchya compactness prameyala (hamilton 1995) "singularity models" tayar karnyasathi lagoo karnyas anumati deto, jeey naveen trimiteey manifoldswar ritchie flow aahet. hamilton-aay. vhee. eae. chya andajanusar, yaa naveen ritchie pravahanmadhye nakaratmak nasaleli vakrata asate. tyanantar hamiltonchi lii-yau asamanata lagoo kelee jaau shakate, jenekarun scalar vakrata, pratyek tappyavar, veleche naun-decressing (naun-negative) karya aahe. haah eka shaktishali parinam aahe jyamule pudhil anek yuktivad houu shakatat. shevati, perelman darshavito qee tyache konatehi ekatva model lakshanrahitpane sampoorn pravan sachchidra hott asalelya rikki solitonsarkhe aahe, jeey purnapane vargikrut kelele aahet; mageel vibhag pahaa.
hamiltonchya lii-yau asamanatebaddalachya tapashilansathi pahaa; vareel donhi asamanatanche varnan asaleli pustake aanee tyateel vivaran.
udaharane.
sthir vakrata aanee einstein metricks.
sootra _ 48 heey eka remanian manifold asoo dya jeey einstein aahe, mhanje eka sankhya sootra aahe _ 49 asey sootra _ 50. mug sootra _ 51 heey sootra _ 52 saha eka ritchie pravah aahe, tevhapasoon
jarr sootra _ 2 bund asel, tarr vareel hamiltonchya advitiyatechya prameyanusar, prarambhik deta sootra _ 55 asalela haah ekmev rikki pravah aahe. visheshtah, ekhadyala asey disateh
pratyek prakaranat, sootra _ 11 chya vegvegalya moolyanna niyukt kelele remanian metricks keval sthir praman ghatkadvare bhinn asalyane, ekhadyala asey disu shakate qee samanyikrut ritchie pravah sootra _ 72 sarvkal astitvat aahe aanee sootra _ 73 madhye sthir aahe; visheshtah, teey sootra _ 74 mhanoon sahajpane (tyachya sthir mulyaparyant) ekatra yete.
einstein sthitimadhye sthir vakrateche eka vishesh prakaran aahe; mhanoon golakachi vishisht udaharane (tyachya manak metricks) aanee hyperbolic space vareel vishesh prakarane mhanoon disatat.
ritchie solitans.
ritchie soliton heey ritchie pravah aahet jeey tyancha aakar badaloo shakatat parantu tyancha aakar bhinnarupannusar badalat nahee.
eka pravanata sakshamikaran karananya ritchie solitonmadhye eka gulgulit remanion manifold ("m", "g") aanee "f" ω "c" φ ("m") asey asatat qee
jarr 'm' heey bund tri-aayami gulgulit bahugunit asel, tarr 'm' vareel 'ricsi' pravahachi maryadit-velchi ekatmata, sampoorn pravanata sankuchit hott asalelya 'ricsi' solitonver (bahudha 'm' peksha bhinn asalelya antarnihit bahugunitanvar) aadharit aahe heey dakhavane hii 'm' chee eka pramukh kamgiri hotee. 2008 madhye, huai-dong kaao, bing-long cheen aanee zee-ping jhu yanni yaa solitonche vargikaran purn kele, jeey darshavitatah
konatyahi uchch parimananmadhye ritchie solitonchya pravanata sankuchit honyabaddal adyap changli samaj nahee.
eksamanta aanee bhoomitikaranashi sambandh.
hamiltonche ritchie pravahavaril pahile kaam tyach velee prakashit jhaale jevha william thurstenchya bhoomitikaran andajanusar, joou trimiteey gulgulit manifoldschya topological vargikaranashi sambandhit aahe. hamiltonchi kalpana eka prakarche nonlinear diffusion samikaran paribhashit karanyachi hotee jeey metrikmadhil aniyamitata suralit karel. thurstonne yogya pramanit rupe aadhich olakhali hotee; thursten model bhoomiti mhanoon olakhalya jaananya shakyatanmadhye, tri-kshetra ess3, tri-aayami euclidean space eae3, tri-aayami hyperbolic space etch3, jeey sajatiy aanee samasthanik aahet aanee thode adhik videshi remanion manifolds, jeey samaroop aahet parantu samasthanik naheet, yancha samavesh hoto. (hii yadee tri-aayami vastavik algebranchya nauu varganmadhye kelelya biyanchi vargikaranashi jawalun sambandhit aahe, parantu samaan nahee.)
hamilton heey siddha karnyat yashasvi jhaale qee konatehi gulgulit bund teen-manifold jeey "sakaratmak" ritchie vakrateche metrik swikarte, teey dekhil eka adviteey thursten bhoomiti swikarte, mhanje golakar metrik, jeey kharokhar ritchie pravahachya khalee eka aakarshak nishchit bindupramane karya karate, jeey khand jatan karnyasathi punha samanya kele jaate. (aniyamit ritchie pravahachya antargat, manifold maryadit velet eka binduparyant kosalate.) tathapi, vakratevaril nirbandhatmak gruhitkamule heey sampoorn bhoomitikaran anuman siddha karat nahee.
kharey tarr, ekonisavya shatkatil bhoomiticha vijay haah eksamantechya siddhantacha purava hota, gulgulit done-manifoldschya samaan topological vargikaran, jithe hamiltonne dakhavun dile qee ritchie pravah kharokhar nakaratmak vakr done-manifoldmadhye dvi-aayami bahu-chhidrit torressmadhye vikasit hoto joou hyperbolic plainsathi sthanik patalivar samamiteey aahe. haah vishay vishleshan, sankhya siddhant, gatishil pranali, ganiteey bhautikshastra aanee agadi vishvashastratil mahattvachya vishayanshi jawalun sambandhit aahe.
lakshat ghya qee 'eksamanta' haah shabd bhoomititil aniyamitata eka prakare suralit door karnyacha sallah deto, tarr 'bhoomitikaran' haah shabd bhoomitila gulgulit anek patanvar thevanyacha sallah deto. "yethe 'bhoomiti' chaa vapar clainchya bhoomitichya kalpanepramanech tantotant keela jaat aahe (adhik tapashilansathi bhoomitikaran anuman pahaa). visheshtah, bhoomitikaranacha parinam hii samasthanik nasaleli bhoomiti asoo shakate. sthir vakratechya prakaranansah bahutek prakarananmadhye, bhoomiti adviteey asate. vastavik aanee guntaguntichya formulationmadhil parasparkriya hii yaa kshetratil eka mahattvachi sankalpana aahe. visheshtah, eksamantechya anek charcha vastavik done-mankeaivaji jatil vakrambaddal bolatat.
ekatmata.
hamiltonne dakhavun dile qee eka sankshipt remanian manifold nehmich alpakalin ritchie pravah dravan swikarato. nantar sheene alpakalin astitvachya parinamache samanyikaran karoon, anek patinni baddh vakrata purn kelee. tathapi, sarvasadharanpane, ritchie pravah samikaranachya atyant gair-rekheey swarupamule, ekatmata maryadit velet tayar hote. hii ekatmata vakrata ekatmata aahet, jyacha arth assa aahe qee jasjashi ekhadi vyakti ekvachan velechya jawal jaate tasatase vakrata tensor sootra _ 81 chey manak ekatvachya pradeshat anantteparyant udate. ekatmatechya serve sambhavya bhoomiti samajoon ghene hii ritchie pravahatil eka moolbhut samasya aahe. jevha yashasvi hote, tevha yamule anek patanchya topologychi antardrushti houu shakate. udaharanarth, 3dii ritchie pravahat vikasit houu shakananya ekvachani pradeshanchya bhoomitiche vishleshan karane, haah perelmanchya poincare aanee bhoomitikaran andajanchya puravyatil mahattvapoorn ghatak aahe.
ekatmatechya maryada udavoon dene.
ekatmatechya nirmiticha abhyas karnyasathi, itar naun-linier differential samikarananchya abhyasapramane, sphot maryadancha vichar karane upayukt aahe. antardnyanane sangayache tarr, vel aanee jaaga punha vaparoon ritchie pravahachya ekeri pradeshat zoom hoto. kahee vishisht gruhitkannusar, pravahat vaadh karoon, maryadit rikki pravah sutrakade cull asato, jyala ekatmata model mhanatat. ekatmateche namune heey pracheen ritchie pravah aahet, i.eae. teey anant bhootkalat vistarle jaau shakatat. ritchie pravahatil sambhavya ekatmata model samajoon ghene haah eka sakriy sanshodhan prayatna aahe.
khalee, aapan sphot prakriyeche adhik tapshilvar rekhatan karatoh sootra _ 83 haah eka ritchie pravah asoo dya joou sootra _ 84 mhanoon ekatmata vikasit karato. sootra _ 85 haah antaralkalatil binduncha kram asoo dya, jase qee 
sootra _ 87 mhanoon. mug parabolically rescale kelelya metriksacha vichar keela jaato
parabolic vistarantargat ritchie pravah samikaranachya samamitimule, metricks sootra _ 89 dekhil ritchie pravah samikaranache upay aahet. asha paristhitit
aay.eae. kalbahya sootra _ 91 paryant, vakratechi kamal sankhya sootra _ 92 varr prapt hote, tyanantar ritchie pravahachya sutracha bindu kram _ 93 nantarchya kalat sahajpane maryadit pracheen ritchie pravah sootra _ 94 madhye villin hoto. lakshat ghya qee sarvasadharan sutramadhye _ 95 heey sootra _ 2 peksha bhinn nahee.
prakar I aanee prakar II ekatmata.
hamilton haah ritchie pravahatil prakar I aanee prakar II madheel ekatmata yaat farak karato. visheshtah, eka mhanato qee eka rikki pravah sootra _ 97, ekatmatecha samna kartana vel sootra _ 5 haah prakar i aahe jarr
anyatha ekatmata hii prakar II chee asate. heey dnyat aahe qee prakar I ekatmatechya sphot maryada mhanje pravanata sachchidra hott asalele ritchie soliton aahet. prakar II chya babatit, ekatmata model sthir ritchie soliton asane aavashyak aahe qaa haah eka khula prashna aahe-aataparyant serve dnyat udaharane aahet.
3dii ritchie pravahatil ekatmata.
3demadhye ritchie pravah ekatmatechya sambhavya sphot maryada changlya prakare samajoon ghetalya jaatat. hamilton, perelman aanee brendle yanni nuktyach kelelya kamamule, kamal vakratechya bindunvar udalyane khaleel teen ekatmata modelspacky eka tayar hotoh
pahile done ekatva model prakar I ekatvatoon udbhavtat, tarr shevatche prakar II ekatvatoon udbhavtat.
4 dii ritchie pravahatil ekatmata.
chaar parimananmadhye sambhavya ekatmatebaddal farach qamee mahitee aahe, yashivay shakyata teen parimananpeksha khoop jaast aahet. aajparyant khaleel ekatmata models dnyat aahet
lakshat ghya qee pahili teen udaharane 3 dii ekatmata modelschee samanyikaran aahet. phic sankochak swayam-chhed kramank-1 saha antarbhut golakachya kosalanyache model tayar karato.
prasaracha sambandh.
ritchie pravah paribhashit karnare utkranti samikaran heey kharokhar eka prakarche araikhik prasar samikaran qaa aahe heey pahanyasathi, aapan (vastavik) done-manifoldschya vishesh prakaranacha adhik tapshilvar vichar karoo shakato. done-manifoldvaril konatahi metrik tensor haah exponential isothermal coordinate chartchya sandarbhat lihila jaau shakato.
aaplya remanian too-manifoldsathi rikki tensor aanee laplas-beltramy operatorchi ganana karnyacha sarvat sopa marg mhanje ellie cartonchi vibhedak swaroop paddhat vaparane. cofframe kshetra ghya
jenekarun metrik tensor hoil
pudhe, aniyantrit gulgulit karya sootra _ 107 dile tarr, bahya vyutpattichi ganana qaraa.
hodge duheri ghya
aanakhi eka bahya vyutpann ghya
(jethe aamhi bahya utpadanacha prati-vinimay gunadharm vaparla). mhanje,
aanakhi eka hodge duheri dene
jeey laplas/beltramy operatorsathi ichchit abhivyakti dete
vakrata tensorchi ganana karnyasathi, aapan aaplya cofframechi nirmiti karananya covactor kshetranche bahya vyutpann ghetoh
yaa abhivyaktinmadhun, aapan ekmev swatantra firki jodani eka-swaroop vachoo shakato.
jithe aapan jodanichya samamiteey-virodhi gunadharmancha (sootra _ 117) fayda ghetala aahe. aanakhi eka bahya vyutpann ghya
yamule vakrata done-swarupat yete
jyatoon aapan reeman tensorcha ekmev resheey swatantra ghatak vachoo shakato
mhanje
jyatoon ritchie tensorche ekmev shunyetar ghatak aahet
yavarun, aaplyala samanvay kobasisachya sandarbhat ghatak aadhaltat, mhanje
parantu metrik tensor dekhil karn aahe,
aanee kahee prathamik hatalaninantar, aaplyala ritchie pravahasathi eka mohak abhivyakti milateh
heey serve dnyat prasar samikarananpaiki, ushnatechya samikaranasarkhech aahe.
jethe aata sootra _ 127 heey euclidean pratalavaril nehamiche laplacian aahe.
vachak aakshep gheu shakato qee ushnateche samikaran heey nishchitach eka resheey aanshik vibhedak samikaran aahe-jethe pii madhye vachan dilele "nonalinearity" aahe.dii.eae. ritchie pravah paribhashit karane?
yache uttar asey aahe qee nonalinearity pravesh karate kaaran laplas-beltramy operator tyach function p varr avalambun asato jeey aapan metrik paribhashit karnyasathi vaparat hoto. parantu lakshat ghya qee sapat euclidean pratal heey sootra 128 gheuun dile jaate. mhanoon jarr sootra 129 parimanane lahan asel tarr aapan sapat pratalachya bhoomitipasun lahan vichalananchi vyakhya karnyacha vichar karoo shakato aanee jarr aapan ghatankachya gananet fakt pahilya kramankachya atty rakhoon thevalya tarr aaplya dvimiteey jawaljawal sapat remanion manifoldvaril ritchie pravah nehamiche dvimiteey ushnateche samikaran banate. hii ganana suchavate qee, jyapramane (ushnatechya samikarananusar) garam platemadhil aniyamit tapman vitaran kalantarane adhik eksandh hote, tyachpramane (ritchie pravahanusar) jawaljawal sapat remanion manifold dekhil tyach prakare sapat hoil jyaa prakare ushnata anant sapat platemadhye "anantteparyant" vahoon nellie jaau shakate. parantu jarr aapali garam plate aakarane maryadit asel aanee ushnata vahoon nenyasathi konatihi seema nasel, tarr aapan tapman "ekroop" honyachi apeksha karoo shakato, parantu spashtapane aapan teey shunyavar aananyachi apeksha karoo shakat nahee. tyach prakare, aamhi ashee apeksha karato qee vikrut goal golavar lagoo kelela ritchie pravah kalantarane bhoomiti purn karnyacha cull darshavel, parantu tyache sapat euclidean bhoomitimadhye rupantar karnar nahee.
alikadil ghadamodi.
1981 pasoon ritchie pravahacha sakhol abhyas keela jaat aahe. kahee alikadil kamanmadhye ritchie pravahachya antargat uchch-aayami remanian manifolds nemke kase vikasit hotat aanee visheshtah, konatya prakarchya parametric ekatmata tayar houu shakatat yaa prashnavar laksh kendrit kele aahe. udaharanarth, ritchie pravahachya upayancha eka vishisht varg darshavito qee, pravah kahee vaishishtyapurn vel sootra _ 131 chya jawal pohochat asatana, vishisht topological gunadharm (sakaratmak yular vaishishtya) asalelya vikasit hott asalelya sutravar, 36-aayami metrik remanion manifoldver, galyatil ekatmata tayar hoil. kahee prakarananmadhye, asha galyatil pinch ritchie soliton navachya manifold tayar kartil.
tri-aayami manifoldsathi, parelmanne manifoldver shastrakriya vaparoon ekatmata kashee pudhe neavi heey dakhavale.
kahler metricks heey ritchie pravahachya antargat kahler rahatat aanee mhanoon yaa settingmadhye ritchie pravahacha dekhil abhyas keela gela aahe, jithe tyala kahler-ritchie pravah mhanatat.
sandarbh.
lokpriy ganiteey prekshakansathi lekh.
sanshodhan lekh.