
paramavadhi compact upagroop

ganithashaasthrathil, oru topological grouppaaya "jii" yude oru maximal compact subgroup "keey" ennathu oru subgroupaanu "keey", athu subspace topologyil oru compact space atharam subgroupukalkkidayil maximalumaanu.
nuna grouppukaludeyum prathyekichu ardha-lalithamaaya nuna grouppukaludeyum vargeekaranathil paramavadhi othukkamulla upagrooppukal oru pradhana punk vahikkunnu. nuna grouppukalude paramavadhi compact upavibhagangal pothuve savisheshamalla, pakshe samyojanam vare savisheshamaan-avaa adisthaanaparamaayi savisheshamaan.
udaaharanam.
oru udaaharanam pothuvaaya leanier group jii. ell (2, aarr)-nullile oorthoogoonal grouppaaya ooo (2) enna subgroupaanu. bandhapetta oru udaaharanam sarkkil grouppaanu, athinaal (2) sl (2, r)-nullil. vyakthamayum athinaal (2) ullile jii. ell (2, aarr) kompaakt aanu, maximal allah. eathoru aantharika uthppannathinum oru anubandha oorthoogoonal grooppundennathinaal eee udaaharanangalude savisheshathayillaayma kaanaan kazhiyum, koodaathe avashya prathyekatha aantharika uthppannathinte avashya savisheshathayumaayi poruthappedunnu.
nirvachanam.
oru maximal compact subgroup ennathu compact sabgrooppukalkkidayila oru maximal subgroupaanu-oru "maximal (compact subgroup)"-kompaakt aaya oru maximal subgroup (saadhyamaaya reeding) ennathinupakaram; ithine orupakshe "kompaakt (maximal subgroup)" ennu vilikkam, pakshe eathu saahacharyathilum uddeshicha ardhamalla (vaasthavathil maximal sariyaaya subgroupukal pothuve kompaaktalla).
asthithvavum savisheshathayum.
carton-ivasava-malsev sidhaantham vaadikkunnathu, bandhippichirikkunna oro nuna group (yadhaarthathil bandhippichirikkunna oro praadeshika compact group) paramavadhi compact upavibhaagangale amgeekarikkunnu, avayellaam parasparam samyojippichirikkunnu. oru ardha lalithamaaya nuna grouppinte prathyekatha carton fixed pooyint sidhaanthathinte anantharafalamaanu, ithu oru compact group poornnamaayum bandhippichirikkunna negatteevu vakramaaya remanian manifoldil isometrikalaal pravarthikkukayaanenkil athinu oru nischitha pooyint undennu vaadikkunnu.
connectucheytha lye grouppukalude paramavadhi compact subgroupukal saadhaaranayaayi "advitheeyamalla", ennaal avaa konjugeshan vare savisheshamaan, athaayath "k", "l" ennee rund paramavadhi compact subgroupukal nalkiyaal, "g" ω "g" enna moolakam undu, athaayath "gg" −1 = "l". athinaal oru maximal compact subgroup adisthaanaparamaayi savisheshamaan, aalukal palappozhum "maximal compact subgroup" ennu samsaarikkunnu.
pothuvaaya leanier group jii. ell. ("enn, aarr")-nte udaaharanathinu, ithu "aarr" enn "-leeii eathoru" aantharika uthppannavum oru (compact) oorthoogoonal groupine (athinte isometry group) nirvachikkunnu-athu oru oorthoonoormal adisthaanam amgeekarikkunnu adisthaanathinte maattam isometry groupine classical oorthoogoonal group oo ("enn, aarr")-maayi samyojippikkunna samyojitha gadakathe nirvachikkunnu.
thelivukal.
oru yadhaartha ardha lalithamaaya nuna groupinu, oru paramavadhi compact subgroupinte nilanilppinteyum savisheshathayudeyum kaarttante thelivu 'aandu' ll kandethanakum. bandhippicha nuna grouppukalilekkum bandhippicha praadeshikamaayi othukkamulla grouppukalilekkum ulla vipuleekaranathekkurichu charcha cheyyuka.
ardha lalithamaaya grouppukale sambandhichidatholam, compact allatha ardha lalithamaaya lye groupinte oru compact riyal roopathinteyum anubandha carton vighadanathinteyum anantharafalamaanu nilanilpp. g"/"k" enna remanian samamiti sthalathinu negatteevu vakrathayum undu enna vasthuthaye aashrayichaan savisheshathayude thelivu. 
kaarttante sthira pooyint sidhaantham. g"/"k" enna eathu binduvileyum exponential maappinte derivative thripthippeduthunnuvennu kaanichu |dii exp "x"| ≤ |x|. ithu soochippikkunnathu "g"/"k" ennathu oru hudmard space aanu, i.iii. oru euclidean sthalathe samaantharagraaminte durbalamaaya roopathe thripthippeduthunna oru samboornna metrik space. bruhet-titts fixed pooyint sidhaanthathil ninnu savisheshatha kanakkakkaam. vaasthavathil, oru hudmard sthalathe eathoru parimithamaaya adacha settum savisheshamaaya ettavum cheria adacha panthil adangiyirikkunnu, athinte madhyabhaagathe athinte sarkka sentar ennu vilikkunnu. prathyekichum isometrikal upayogichu pravartthikkunna oru compact group athinte oro bhramanapathathinteyum chuttalav nischayikkanam.
ardha lalithamaaya grouppukalude savisheshathayude thelivu.
 ardha lalithamaaya grouppukalude pothuvaaya prashnathe jii. ell ("enn", aarr) ennathumaayi bandhappettirikkunnu. positteev cemetric maatriksukalude spaceaanu anubandha cemetric space. eee sthalathinte praadhamika gunangale aasrayikkunna savisheshathayude nerittulla thelivu nalkiyirikkunnu.
soothravaakyam _ 1 carton involsion σ ulla oru yadhaartha ardha lalithamaaya beejaganithamayirikkatte. athinaal σ-yude nischitha pooyint upavibhagam maximal compact upavibhagamaaya "k" aanu, koodaathe oru aijanspace vighadanamundu.
ivide soothravaakyam _ 3, "k" yude lye aljibra, + 1 aijanspace aanu. carton vighadanam nalkunnu
b" ("x", "y") = tr (adx) (ady) nalkiyirikkunna soothravaakyathile kola roopamaanu "b" ennaanengil,
formula _ 1-leeii oru yadhaartha aantharika ulppannamaanu. adjayint praathinidhyathinu keezhil, eee aantharika uthppannam samrakshikkunna "jii" yude upavibhagamaanu "keey".
h" ennathu "g" yude mattoru compact upavibhaagamaanengil, haar alavumaayi bandhappett "h" nuu mukalilulla aantharika uthppannam sharashari cheiyunnathu "h" nuu keezhil oru aantharika uthppannam invariyant nalkunnu. p" ll "p" ulla "p" enna oopparettarmaar positteev cemetric oopparettarmaaraanu. eee puthiya aantharika uthppaadanam ingane ezhuthaam
ivide "s" ennathu formula _ 1-leeii oru positteev cemetric oopparettaraanu, 
h" ll "h" ennathinu ad ("h") "t" s "ad" h "=" s "(aantharika ulppannavumaayi bandhappett transposukal kanakkakkunnu). koodaathe, "jii" yill "x" ennathinu,
athinaal "echu" inn "echu",
formula _ 12-ll "x"-nuu nirvachikkuka
e" i "ennathu" s "-nuu" s "-nuu" s "= λ" i "e" i "ulla eejanvektarukalude oorthoonoormal adisthaanamaanengil,
athinaal "f" karsanamaayi positteev aayirikkukayum φ aayi pravanatha kaanikkukayum cheiyunnu |"x"| φ vare pravanatha kaanikkunnu. vaasthavathil eee maanadandam samamiti oopparettarmaaraaya ad "x"-leeii oopparettar maanadandatthinu thulyamaanu, koodaathe oro poojyamallaatha eejan moolyavum athinte negatteevu aayi sambhavikkunnu, kaaranam i ad "x" compact riyal phoom formula _ 15-leeii "scue-adjayint oopparettar" aanu.
athinaal "f" nuu aagolathalathil "y" enna minimum undu. eee minimum savisheshamaan, kaaranam "z" mattoru aayirunnengil
ivide formula _ 12-leeii "x" ennathu carton vighadanathaal nirvachikkappedunnu
f" "i" ennathu anubandha yadhaartha eejan moolyangalaaya "x"-nte eejanvektarukalude oorthoonoormal adisthaanamaanengil, appol
valathuvashath exponensialukalude positteev kompineshan aayathinal, "x" ≤0 aanenkil yadhaartha moolyamulla function "g" karsanamaayi convex aanu, athinaal ithinu oru advitheeya minimum undu. maruvashathu, ithinu "t" = 0, "t" = 1 ennivayil local minima undu, athinaal "x" = 0 umm "p" = exp "y" umm savisheshamaaya aagola minimum aanu. nirmmaanatthiloode
"f" ("" x "") = "" f "" (σ ("" h "") "" xh "" −1) "" h "" ll "" h "" ennathinu, athinaal "" h "" ll "" h "" ennathinu "" p "" = σ ("" h "") "" ph "" −1 ". athinaal σ ("h") = "Php" −1. thalphalamaayi, "g" = exp "y"/2 aanenkil, "ghg" −1 ennathu σ kondu nischayikkappedunnu, athinaal "k" ll sthithicheyyunnu.
aaplikkeshanukal.
praathinidhya sidhaantham.
jii" othukkamullathallaathpol praathinidhya sidhaanthathil paramavadhi othukkamulla upavibhagangal oru adisthaana punk vahikkunnu. anganeyaanengil oru maximal compact subgroup "keey" ennathu oru compact lye grouppaanu (oru lye grouppinte clossed subgroup oru lye grooppaayathinaal), athinu sidhaantham eluppamaanu.
g", "k" ennivayude praathinidhya sidhaanthangalumaayi bandhapetta pravartthanangal "g" ll ninnu "k" lake praathinidhyangale parimithappedutthukayum "k" ll ninnu "g" lake praathinidhyangale prerippikkukayum cheiyunnu, iva nannayi manassilaakkappedunnu; avayude sidhaanthathil golaakrithiyilulla pravartthanangal ulppedunnu.
topology.
nuna grouppukalude beejaganita topology pradhaanamaayum oru maximal compact subgroup "keey" aanu vahikkunnath. krithyamaayi paranjaal, oru connectucheytha lye group ennathu oru maximal kompaakt "k"-nteyum euclidean space-"g" = "k" × r "d"-nteyum topological ulppannamaanu (oru group saidhaanthika uthppannamallengilum)-athinaal prathyekichum "k" ennathu "g"-nte oru roopabhedam pinvalikkalaan, ithu homotopy thulyamaanu, athinaal avaykku ore homotopy grouppukalundu. vaasthavathil, inclusion formula _ 20 umm differmation retraction formula _ 21 umm homotopy thulyathayaanu.
pothuvaaya rekheeya groupine sambandhichidatholam, eee vighadanam qr vighadanamaanu, koodaathe roopabhedam pinvalikkal gram-shmidt prakriyayaan. oru pothuvaaya ardha lalithamaaya nuna groupinu, "g" ennathinte "g" = "qaan" enna ivasava vighadanamaanu vighadanam, athil "k" oru kararulla upavibhagamaaya "an" ulla oru uthppannathil sambhavikkunnu.

