gromov seema

ganitat, δ-hyperbolic jaagechi (visheshtah hyperbolic gut) gromov seema hii hyperbolic jagechya seema kshetrache samanyikaran karnari eka amoort sankalpana aahe. sankalpanatmakdrushtya, gromov seema haah anantatechya serve binduncha sanch aahe. udaharanarth, vastavik reshechi gromov seema done bindu aahet, jeey sakaratmak aanee nakaratmak anantateshi sambandhit aahet.
vyakhya.
bhooganitiy aanee yogya δ-hyperbolic jagechya gromov simechya anek samatulya vyakhya aahet. sarvat samanya vaparanpaiki eka mhanje bhooganitiy kirananche samatulya varg.
utpatti mhanoon hyperbolic metrik space formula _ 2 chey kahee bindu sootra _ 1 nivada. bhooganitiy kiran haah samamiteey sutradvare dilela marg aahe, jase qee pratyek khand sootra 4 haah sootra 1 teey sootra 6 paryantchya sarvat qamee lambicha marg aahe.
jarr sthir sootra _ 8 asel tarr done bhooganiti sootra _ 7 samatulya mhanoon paribhashit kele jaatat, jase qee serve sootra _ 10 sathee sootra _ 9. sootra _ 11 chaa samatulya varg sootra _ 12 darshavila jaato.
bhooganitiy aanee yogya hyperbolic metrik space sootra _ 2 chee gromov seema heey sanch sootra aahe _ 14 heey sootra _ 15 madheel bhooganitiy kiran aahe.
topology.
teen gunanche gromov utpadan vaparane upayukt aahe. metrik jaaget teen gunanche sootra _ 16 chey gromov utpadan aahe
sootra _ 17. vrukshamadhye (aalekh siddhant), heey sootra _ 18 teey sootra _ 19 aanee sootra _ 20 paryantche marg vegale honyapurvi kitee kaal ekatra rahatat heey mojate. hyperbolic jaaga zadasarkhya asalyane, gromov utpadan heey mojate qee sootra _ 18 teey sootra _ 19 aanee sootra _ 20 heey vegale honyapurvi kitee kaal geodesic jawal rahatat.
gromov simevar bindu sootra _ 24 dile tarr, aapan sanch sootra _ 25 paribhashit karato, tethe sootra _ 27 aanee sootra _ 28 saha bhooganitiy kiran sootra _ 7 aahet. heey khule sanch gromov simechya topologiesathi aadhar tayar karatat.
heey khule sanch heey keval bhooganitiy kirananche sanch aahet jeey vichalan honyapurvi _ 29 yaa antarachya sutraparyantchya eka nishchit bhooganitiy kiranache anusaran karatat.
hii topology gromovchya seames eka sankshipt mapanaksham jaaga banavate.
atibadhya gatachya tokanchi sankhya mhanje gromov simechya ghatakanchi sankhya.
gromov simeche gunadharm.
gromov simeche anek mahattvache gunadharm aahet. gut siddhantat sarvat varanvar vaparlya jaanarya gunadharmanpaiki eka khalilpramane aaheh jarr gut sootra _ 30 heey δ-hyperbolic jaaget bhaumitikdrushtya karya karat asel, tarr sootra _ 30 heey hyperbolic gut aahe aanee sootra _ 30 aanee sootra _ 2 madhye homomorphic gromov seema aahet.
sarvat mahattvachya gunadharmanpaiki eka mhanje teey ardh-samamiti aparivartaniy aahe; mhanje, jarr done hyperbolic metrik jaaga ardh-samamiteey asateel, tarr tyanchyamadhil ardh-samamiti tyanchya simanmadhye samarupata dete. heey mahatvache aahe kaaran sankshipt jaganche homomorphism heey jaganchya ardh-samamitipeksha samajoon ghene khoop sope aahe.
samanyikaran.
manjar (0) jaagechi drushya seema.
sampoorn manjar (0) jagesathi, 'x' chee drushya seema, δ-hyperbolic jagechya gromov simepramane, asymptotic geodesic kiranancha samatulya varg asato. tathapi, gromov utpadanacha vapar tyavar topology paribhashit karnyasathi keela jaau shakat nahee. udaharanarth, sapat pratalachya babatit, viruddha dishene naan naan bindupasoon bindupasoon konatyahi done bhooganitiy kirananmadhye tyaa binduchya sandarbhat anant gromov utpadan asel. tyaeivaji drushya seema shankuchya topologyne sampann aahe. x' madhye 'o' haah bindu nishchit qaraa. konatyahi seema bindula "oo" pasoon utsarjit utsarjit adviteey bhooganitiy kiranandware darshavile jaau shakate. oo' pasoon jaaree honare kiran sootra _ 11 aanee sakaratmak sankhya 'tii'> 0 aanee 'aar'> 0 dile tarr, seema bindu sutravar shejarcha aadhar _ 12 formchya sanchadvare deela jaato.
varr paribhashit kelyapramane shanku topology "oo" chya nivadipasoon swatantra aahe.
jarr "x" yogya asel, tarr shankuchya topologies drushya seema sankshipt aahe. jevha 'x' heey manjar (0) aanee yogya bhooganitiy δ-hyperbolic jaaga donhi asate, tevha shankuchya topologycha samkram gromov simechya topologieshy hoto.
tofancha andaj.
tofancha andaj anantatevar 2-goal asalelya gatanchya vargikaranashi sambandhit aaheh
tofancha andajah anantatevar 2-goal asalela pratyek gromov hyperbolic gut hyperbolic 3-spacever bhaumitikdrushtya karya karato.
yaa andajache anuroop heey 1-golansathi kharey aanee 2 peksha mothya parimananchya serve golansathi chukiche asalyache dnyat aahe.

