 lai aljibranu vivarinchandi 
 ganitasastramlo ,  sadarana lai groupe anede konnect cheyabadina naane - abelian lai groupe  G,  idi naane ‌ trivial konnect cheyabadina sadarana vupa samuuhaalanu kaligi undadhu .  sadarana lai algibras mariyu remannian symmetric spaces ‌ l jaabitaanu chadavadanike sadarana lai samuhal jaabitaanu upayoeginchavachchu .

 vaasthava sankhyala commutative lai samuuhamtoe kalisi ,
\mathbb {R} ,  mariyu unit - magnituude complexes sankhyalu , U(1) ( unit circle ),  sadarana lai groupulu aatomic  " black ‌ lanu "  andistaayi ,  ivi aapareshan dwara konnect cheyabadina anni  ( parimitha - dimensionally )  lai groupulanu tayyaru chestaayi .  samuham podiginpu .  saadharanamga edurkone anek lai groupulu saralamainavi ledha  ' daggaraka '  untai :  udaharanaku , " pratyeka rekheeya samuham " SL(n) n  by  n  maathrikalu  1 ki samaanamaina determinant ‌ thoo anni  n > 1 ki sulabham .

 sadarana lai samuhal yokka modati vargikarana wilhelm killingche cheyabadindhi mariyu yea pania taruvaata elae cartan dwara paripuurnam cheyabadindhi .  chivari vargeekarananu tarachugaa killing - cartan vargeekaranagaa suchistaru .
 duradrushtavashaattu ,  sadarana abaddhala samuuhaaniki vishvavyaaptanga aamoodhinchabadina nirvachanam ledhu .  pratyekinchi ,  idi allappuduu abaddha samuhamga nirvachinchabadadu ,  idi ooka viyukta samuham vale saadhaaranamainadi .  sadarana abaddhala samuuhaanni konnect cheyala ledha adi naane - trivial senter ‌ nu kaligi undataniki anumatinchabadutundaa ledha aney dhaanipai rachayitalu vibhedistunnaaru .
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\mathbb {R}  ooka sadarana abaddha samuham .

 athantha sadarana nirvachanam aemitante ,  lai groupe anusandaaninchabhadi vunte ,  abelian kanidi mariyu prathi clogead konnect cheyabadina sadarana vupa samuham gurthimpu ledha motham samuhamga vunte adi sulabham .  pratyekinchi ,  sadarana samuuhaalu kanni alpamaina kendraanni kaligi undataniki anumatinchabadataayi ,  conei 
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\mathbb {R}  sulabham kadhu .

 yea kathanamlo trivial senter ‌ thoo anusandhaaninchabadina sadarana abaddha samuuhaalu jaabithaa cheyabaddaayi .  ivi telisina tarwata ,  naane - trivial senter unnavi yea krindhi vidhamgaa jaabithaa cheeyadam sulabham .  trivial senter ‌ thoo unna edaina sadarana abaddhala samuham yuunivarsal cover ‌ nu kaligi umtumdi ,  deeni kendram sadarana lai samuham yokka praadhimika samuham .  naane - trivial senter ‌ thoo sambandhitha sadarana lai groupulanu senter ‌ loni vupa samuham dwara yea yuunivarsal cover ‌ ku gunakaalugaa pomdavacchu .

 pratyaamnaayaalu 
 lai correspondences nundi saralamaina abaddhala samuuhaaniki samaanamaina nirvachanam umtumdi :  konnect cheyabadina abadham samuham dani lai beejaganitam saralamgaa vunte adi sulabham .  ooka mukhyamaina saankethika amsham aemitante ,  ooka sadarana lai samuham vivikta sadarana vupa samuuhaalanu kaligi vumdavacchu .  yea kaaranamgaa ,  ooka sadarana abaddha samuham yokka nirvachanam ooka nairoopya samuham vale saralamaina lai samuham yokka nirvachanaaniki samaanam kadhu .

 simply lai groupullo anek classically lai groupulu unnayi ,  ivi felix klein yokka erlangen prograamme yokka ardhamlo golakara jyaamiti ,  projective jyaamiti mariyu sambandhitha jyaamiti choose samooha - siddhaamta aadharaanni andistaayi .  suparichitamaina jyaamitiki anugunamga laeni anek asaadaaranhamaina avakasalu kudaa unnayani sadarana lai samuhal vargikarana samayamlo idi udbhavinchindi .  yea asaadaaranhamaina samuuhaalu anek pratyeka udaaharanalu mariyu ganitasastramloni itara shaakhalu ,  alaage samakaaleena saiddhaantika bhautikasaastramlo configuration ‌ laku kaaranamavutaayi .

 vyatireka udaharanaga ,  sadarana sarala samuham saadhaaranamainadi kadhu ledha semicimple kadhu .  endhukante gurthimpu yokka gunijaalu naane ‌ trivial sadarana vupa samuuhaanni yerparustayi ,  tadwara nirvachananni tappinchukuntundi .  samaanamgaa ,  sambandhitha lai beejaganitam ksheeninchina killing rupaanni kaligi umtumdi ,  endhukante beejaganitam yokka sunnaa muulakaaniki gurthimpu map yokka gunijaalu .  anevalla ,  sambandhitha lai beejaganitam kudaa saadhaaranamainadi ledha semi simply kadhu .  maroka prathi - udaaharanha samaana parimaanamlo pratyeka orthogonal samuuhaalu .  viitiki matrhuka umtumdi 
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- neenu madyalo ,  mariyu yea moolakam gurthimpu moolakamtho path - konnect cheyabadindhi ,  kabaadi yea samuuhaalu nirvachananni tappinchukuntaayi .  yea rendoo taggimpu samuuhaalu .