
moodu bhougolika siddhaantaalu

avakalana jyaamitiloo moodu bhooganitam yokka siddhaantam, dheenini lysternic-sneerelman siddhaantam anie kooda pilustaaru, oka golam yokka topologytho prati remanian manifold kaneesam moodu saadhaarana closed geodesics kaligi undani perkondi (i.ii. moodu embeded geodesic circles). phalitamgaa convex polyheadron paie quasigodesics kuu mariyu reversible finsler 2-golaala yokka closed geodesics kuu kooda vistarinchavacchu. siddhaantam padunainadihi prati remanian 2-golamlo anantamaina aneka vibhinnamaina closed geodesics unnappatikee, vaatilo moodu matrame sweeya-vibhajanalu undavani haamii ivvabaddaayi. udaaharanaku, deerghavruttaakaarapu moodu pradhana akshaala podavulu vibhinnamgaa, conny okadaanikokati taginanta daggaragaa unte, deerghavruttaakaara phalitamgaa, appudu deerghavruttaakaaramlo moodu saadhaarana closed geodesics matrame untaayi.
charitra mariyu rujuvu.
oka geodesic, oka remanian uparitalampai, daani prati binduvu vadda sthaanikamgaa neerugaa unde vakrarekha. udaaharanaku, euclidean samatalamlo bhougolika rekhalu rekhalu, mariyu oka golam yokka uparitalampai bhougolika rekhalu goppa vruttaalu. rendu paayintla madhya uparitalamlo atichinna maargam ellappuduu geodesic, conny itara geodesics kooda undavacchu. oka geodesic daani prarambha binduvu mariyu prarambha disaku tirigi vasthe adhi closed geodesic anie cheppabadutundi; alaa cheyadamlo adhi aneka saarlu daatavacchu. moodu bhooganitam yokka siddhaantam prakaram, golaaniki homomorphic uparitalaala kosam, kaneesam moodu sweeya-crossing kaani closed geodesics unnaayi. udaaharanaku, moodu kante ekkuva undavacchu, golam kooda anantamgaa anekam kaligi untundi.
eee phalitam samudra navigation yokka ganitam nundi udbhavinchindi, ikkada bhoomi yokka uparitalaanni deerghavruttaakaaramtho khachitamgaa roopondinchavacchu, mariyu deerghavruttaakaarampai geodesics adhyayanam nundi, noukalu prayaaninchadaaniki atichinna margalu. mukhyamgaa, daadaapu-golaakaara tribhujaakaara deerghavruttaakaaramlo kevalam moodu saadhaarana closed geodesics, daani bhoomadhyarekhalu matrame untaayi. 1905loe, henry paayinkaare oka golaaniki samaanamaina prati mruduvaina uparitalamlo kaneesam moodu saadhaarana closed geodesics unnayani oohinchaaru, mariyu 1929loe lazar leusternic mariyu lev schnirelman eee oohaku rujuvunu prachurinchaaru; rujuvu yokka saadhaarana topological vaadana sarainadi ayinappatikee, idhi oka vaikalya phalitaanni upayoginchindi, adhi taruvaata lopabhooyishtamgaa kanugonabadindi.
aneka mandhi rachayitalu eee antaraaniki asantruptikaramaina parishkaralanu pratipaadinchaaru. 1980laloe grayson dwara, vakrata pravahanni tagginchadam dwara vishwavyaaptamgaa aamodinchabadina parishkaram andinchabadindi.
saadhaaraneekaranalu.
siddhaantam yokka balopetam cheyabadina samskarana prakaram, sthalaakrutigaa oka golam ayina edainaa remanian uparitalampai, moodu saadhaarana closed geodesics tappanisarigaa unnaayi, veeti podavu uparitalam yokka vyaasaaniki atyanta anulomaanupaatamlo untundi.
oka mruduvaina sthalavignaana golamlo garishtamgaa "l" podavutho unna closed geodesics sankhya "l"/log "l" nishpattilo perugutundi, conny atuvanti geodesics annee saralamgaa undavani haamii ivvalemu.
compact hyperbolic reeman uparitalaalapai, anantamaina aneka saadhaarana closed geodesics unnaayi, conny icchina podavutho parimitamainavi matrame unnaayi. avi selberg gta function dwara vislaeshanhaatmakamgaa encode cheyabadataayi. saralamaina closed geodesics sankhya yokka vruddhi retunu, vaati podavu yokka vidhiga, marium mirzakhani parisoedhimchaaru.
rendu-golamlo edainaa reversible finsler metrik kosam moodu saadhaarana closed geodesics uniki kooda kaligi untundi.
naan smooth metricks.
convex polyheadra vanti pratichota mruduvugaa laeni konni uparitalaalapai geodesicsnu nirvachinchadam kooda saadhyame. oka kumbhakara polyheadron yokka uparitalam polyheadron yokka sheershaala vadda tappa sthaanikamgaa euclidean ayina metriknu kaligi untundi, mariyu idhi polyheadron yokka prati mukhamlo sarala rekha vibhaagaalanu anusaristhe mariyu adhi daate prati polyheadron anchuku addamgaa neerugaa unte, sheershaalanu nivaarinche vakrarekha geodesic avutundi. konni polyheadraalu saadhaarana closed geodesics kaligi unnappatikee (udaaharanaku, saadhaarana tetrahedron mariyu dispenoidlu anantamgaa aneka closed geodesics kaligi untaayi, annee saadhaaranam) itarulu alaa cheyaru. mukhyamgaa, convex polyheadron yokka saadhaarana closed geodesic tappanisarigaa sheershaala yokka mottam koneeya lopaanni vibhajistundi, mariyu daadaapu anni polyhedralaku atuvanti dvibhaagaalu undavu.
ayinappatikee, moodu bhougolika siddhaantaalanu quasigodesics, polyheadra yokka sheershaala vadda tappa geodesic vakrarekhalanu parigananaloki teesukovadam dwara convex polyhedraku vistarinchavacchu mariyu avi daatutunna prati sheersham vadda rendu vaipula kante takkuva koonaalanu kaligi untaayi. kumbhakara polyhedraku moodu geodesics siddhaantam yokka oka samskarana, anni polyheadraalu kaneesam moodu saadhaarana closed quasigodesics kaligi unnayani perkondi; dheenini polyhedronnu mruduvaina uparitalam dwara anchanaa veyadam dwara mariyu moodu geodesics siddhaantaanni eee uparitalaaniki vartimpajeyadam dwara niroopinchavacchu. eee quasigodesics loe deeninainaa bahupadi samayamlo nirminchavacchaa anedi bahiranga samasya.