
ekarupata (anek putt)

gulgulit manifoldschya siddhantat, sarvasamaveshakata haah avibhajya vakrancha sanch aahe joou manifoldver paribhashit kelelya adrushya naan honanya vector kshetradvare paribhashit keela jaato. 
sarvasadharan sapekshatemadhye sarvasamatv hii eka mahattvachi sankalpana aahe aanee remanian bhoomitichya kahee bhaganmadhye dekhil tee mahattvachi aahet.
preranadayi udaharan.
eksandhatechi kalpana kadachit vyakhyepeksha udaharan deuun adhik changlya prakare spasht kelee jaate. gulgulit manifold r2 chaa vichar qaraa. vector kshetranna "first aurder linier partial differential operators" mhanoon nirdisht kele jaau shakate, jase qee 
yaa prakaranat, heey 'first aurder linier ordinary differential equations' chya pranalishi sambandhit aahet.
jethe bindu kahee (dummy) parameterchya sandarbhat vyutpann darshavito. asha pranalinche upay mhanje "parameters vakranchi kutumbe", yaa prakaranat
yaa kutumbala anekada 'vakranchi ekarupata' kinwa thodakyat fakt 'ekarupata' mhanatat.
yaa vishisht udaharanamadhye done "ekatmata" asatat, jethe sadish kshetra nahise hote. heey 'pravah' chey nishchit bindu aahet. (pravah haah bhinnarupancha eka-aayami gut aahe; pravah haah eka-aayami khote gut r dware krutee paribhashit karato, jyache sthanik patalivar changle bhaumitik gunadharm asatat.) hii done ekatmata done vakranaivaji done "bindun" shhee sambandhit aahet. yaa udaharanat, itar avibhajya vakr serve sadhya bund vakr aahet. anek pravah yapeksha barech klisht aahet. ekatmatechya upasthitimule udbhavananya guntagunt talhanyasathi, sahasa vahak kshetra 'adrushya' asane aavashyak asate.
jarr aapan adhik ganiteey rachana jodali tarr aapali susangatata naveen mahatva prapt karoo shakate.
remanian manifoldsmadhil ekarupata.
udaharanarth, jarr aapan remanion metrik tensor jodoon aapale 'smooth manifold' heey 'remanion manifold' madhye banavale, tarr reshechya ghatakane paribhashit kelele samaja.
aapali susangatata hii eka 'bhooganitiy susangatata' banoo shakate. kharey tarr, aadhichya vibhagatil udaharanat, aapale vakr eka samanya goal golavar (uttar dhruwala khalee utaravun) bhooganiti hotat. tyaeivaji jarr aapan pramanit euclidean metrik sootra _ 6 jodale asate, tarr aapale vakr vartule banale asate, parantu bhooganiti nasate.
aaplya pahilya udaharanashi sambandhit remanian bhooganitiy ekarupateche eka manoranjak udaharan mhanje pii3 vareel clifford ekarupata, jee hopp bundle kinwa "hopp fibration" madhye dekhil olakhali jaate. avibhajya vakr kinwa tantu anukrame kahee "jodidarpane jodaleli" mothee vartule aahet, unit normchya dilelya unit quaterniondware davya gunakarakhali unit norm quaterniencechya jaaget kaksha.
laurentzian manifoldsmadhil sarvasamatv.
laurentzian manifoldmadhye, jase qee samanya sapekshatetil spacetime modelmadhye (jeey sahasa einstein kshetra samikaranache achook kinwa andaje uttar asel), jarr sparshika vahak sarvatra anukrame timelike, null kinwa spacelike asateel tarr sarvasamatvanna 'timelike', 'null' kinwa 'spacelike' mhanatat. jarr sparshika sadish kshetra sootra _ 7 madhye adrushya hott asalele sahparivartan vyutpann, sootra _ 8 asel tarr samarupatela 'bhooganakiy samarupata' mhanatat.