ength . ▁ ▁The ▁topology ▁defined ▁by ▁this ▁metric ▁is ▁equivalent ▁to ▁the ▁usual ▁E uclidean ▁topology , ▁although ▁as ▁a ▁metric ▁space ▁ ▁is ▁complete . ▁ ▁A ▁hyper bol ic ▁triangle ▁is ▁a ▁ge odes ic ▁triangle ▁for ▁this ▁metric : ▁any ▁three ▁points ▁in ▁ ▁are ▁vertices ▁of ▁a ▁hyper bol ic ▁triangle . ▁If ▁the ▁sides ▁have ▁length ▁, ▁, ▁ ▁with ▁corresponding ▁angles ▁, ▁, ▁, ▁then ▁the ▁hyper bol ic ▁cos ine ▁rule ▁states ▁that ▁ ▁The ▁area ▁of ▁the ▁hyper bol ic ▁triangle ▁is ▁given ▁by ▁. ▁ ▁The ▁unit ▁disk ▁and ▁the ▁upper ▁half - plane ▁ ▁are ▁conform ally ▁equivalent ▁by ▁the ▁M ö b ius ▁transformations ▁ ▁Under ▁this ▁correspond ence ▁the ▁action ▁of ▁ ▁by ▁M ö b ius ▁transformations ▁on ▁ ▁corresponds ▁to ▁that ▁of ▁ ▁on ▁. ▁The ▁metric ▁on ▁ ▁becomes ▁ ▁Since ▁lines ▁or ▁circles ▁are ▁preserved ▁under ▁M ö b ius ▁transformations , ▁ge odes ics ▁are ▁again ▁described ▁by ▁lines ▁or ▁circles ▁orthogonal ▁to ▁the ▁real ▁axis . ▁ ▁The ▁unit ▁disk ▁with ▁the ▁Po inc ar é ▁metric ▁is ▁the ▁unique ▁simply ▁connected ▁orient ed ▁ 2 - dimensional ▁Riemann ian ▁manifold ▁with ▁constant ▁curv ature ▁− 1 . ▁Any ▁orient ed ▁closed ▁surface ▁ ▁with ▁this ▁property ▁has ▁ ▁as ▁its ▁universal ▁covering ▁space . ▁Its ▁fundamental ▁group ▁can ▁be ▁identified ▁with ▁a ▁t ors ion - free ▁con comp act ▁subgroup ▁ ▁of ▁, ▁in ▁such ▁a ▁way ▁that ▁ ▁In ▁this ▁case