ature . ▁Thus ▁a ▁closed ▁Riemann ian ▁ 2 - man if old ▁of ▁non - pos itive ▁curv ature ▁can ▁never ▁be ▁embedded ▁is omet r ically ▁in ▁; ▁however , ▁as ▁Adri ano ▁G ars ia ▁showed ▁using ▁the ▁Bel tr ami ▁equation ▁for ▁qu as icon form al ▁ma ppings , ▁this ▁is ▁always ▁possible ▁for ▁some ▁conform ally ▁equivalent ▁metric . ▁ ▁Sur faces ▁of ▁constant ▁curv ature ▁The ▁simply ▁connected ▁surfaces ▁of ▁constant ▁curv ature ▁ 0 , ▁+ 1 ▁and ▁– 1 ▁are ▁the ▁E uclidean ▁plane , ▁the ▁unit ▁sphere ▁in ▁, ▁and ▁the ▁hyper bol ic ▁plane . ▁Each ▁of ▁these ▁has ▁a ▁trans itive ▁three - dimensional ▁Lie ▁group ▁of ▁orientation ▁pres erving ▁is omet ries ▁, ▁which ▁can ▁be ▁used ▁to ▁study ▁their ▁geometry . ▁Each ▁of ▁the ▁two ▁non - comp act ▁surfaces ▁can ▁be ▁identified ▁with ▁the ▁quot ient ▁ ▁where ▁ ▁is ▁a ▁maximal ▁compact ▁subgroup ▁of ▁. ▁Here ▁ ▁is ▁is omorphic ▁to ▁. ▁Any ▁other ▁closed ▁Riemann ian ▁ 2 - man if old ▁ ▁of ▁constant ▁Gaussian ▁curv ature , ▁after ▁scaling ▁the ▁metric ▁by ▁a ▁constant ▁factor ▁if ▁necessary , ▁will ▁have ▁one ▁of ▁these ▁three ▁surfaces ▁as ▁its ▁universal ▁covering ▁space . ▁In ▁the ▁orient able ▁case , ▁the ▁fundamental ▁group ▁ ▁of ▁ ▁can ▁be ▁identified ▁with ▁a ▁t ors ion - free ▁uniform ▁subgroup ▁of ▁ ▁and ▁ ▁can ▁then ▁be ▁identified ▁with ▁the ▁double ▁cos et ▁space ▁. ▁In ▁the ▁case ▁of