) ▁states ▁that ▁any ▁Riemann ian ▁metric ▁on ▁an ▁orient ed , ▁closed ▁surface ▁is ▁conform ally ▁equivalent ▁to ▁an ▁essentially ▁unique ▁metric ▁of ▁constant ▁curv ature . ▁This ▁provides ▁a ▁starting ▁point ▁for ▁one ▁of ▁the ▁approaches ▁to ▁Te ich m üller ▁theory , ▁which ▁provides ▁a ▁fin er ▁classification ▁of ▁Riemann ▁surfaces ▁than ▁the ▁topological ▁one ▁by ▁E uler ▁characteristic ▁alone . ▁ ▁The ▁uniform ization ▁theorem ▁states ▁that ▁every ▁smooth ▁Riemann ▁surface ▁ ▁is ▁conform ally ▁equivalent ▁to ▁a ▁surface ▁having ▁constant ▁curv ature , ▁and ▁the ▁constant ▁may ▁be ▁taken ▁to ▁be ▁ 1 , ▁ 0 , ▁or ▁− 1 . ▁A ▁surface ▁of ▁constant ▁curv ature ▁ 1 ▁is ▁locally ▁is ometric ▁to ▁the ▁sphere , ▁which ▁means ▁that ▁every ▁point ▁on ▁the ▁surface ▁has ▁an ▁open ▁neighborhood ▁that ▁is ▁is ometric ▁to ▁an ▁open ▁set ▁on ▁the ▁unit ▁sphere ▁in ▁ ▁with ▁its ▁intr insic ▁Riemann ian ▁metric . ▁Like wise , ▁a ▁surface ▁of ▁constant ▁curv ature ▁ 0 ▁is ▁locally ▁is ometric ▁to ▁the ▁E uclidean ▁plane , ▁and ▁a ▁surface ▁of ▁constant ▁curv ature ▁− 1 ▁is ▁locally ▁is ometric ▁to ▁the ▁hyper bol ic ▁plane . ▁ ▁Constant ▁curv ature ▁surfaces ▁are ▁the ▁two - dimensional ▁real ization ▁of ▁what ▁are ▁known ▁as ▁space ▁forms . ▁These ▁are ▁often ▁studied ▁from ▁the ▁point ▁of ▁view ▁of ▁Felix ▁Klein ' s ▁Er lang en ▁programme , ▁by ▁means ▁of ▁smooth ▁transformation ▁groups . ▁Any ▁connected ▁surface ▁with ▁a ▁three - dimensional ▁group ▁of