▁the ▁sphere ▁and ▁the ▁E uclidean ▁plane , ▁the ▁only ▁possible ▁examples ▁are ▁the ▁sphere ▁itself ▁and ▁t ori ▁obtained ▁as ▁quot ients ▁of ▁ ▁by ▁discrete ▁rank ▁ 2 ▁sub groups . ▁For ▁closed ▁surfaces ▁of ▁genus ▁, ▁the ▁mod uli ▁space ▁of ▁Riemann ▁surfaces ▁obtained ▁as ▁ ▁var ies ▁over ▁all ▁such ▁sub groups , ▁has ▁real ▁dimension ▁. ▁By ▁Po inc ar é ' s ▁uniform ization ▁theorem , ▁any ▁orient able ▁closed ▁ 2 - man if old ▁is ▁conform ally ▁equivalent ▁to ▁a ▁surface ▁of ▁constant ▁curv ature ▁ 0 , ▁+ 1 ▁or ▁– 1 . ▁In ▁other ▁words , ▁by ▁multip lying ▁the ▁metric ▁by ▁a ▁positive ▁scaling ▁factor , ▁the ▁Gaussian ▁curv ature ▁can ▁be ▁made ▁to ▁take ▁exactly ▁one ▁of ▁these ▁values ▁( the ▁sign ▁of ▁the ▁E uler ▁characteristic ▁of ▁). ▁ ▁E uclidean ▁geometry ▁ ▁In ▁the ▁case ▁of ▁the ▁E uclidean ▁plane , ▁the ▁symmetry ▁group ▁is ▁the ▁E uclidean ▁motion ▁group , ▁the ▁sem id irect ▁product ▁of ▁the ▁two ▁dimensional ▁group ▁of ▁transl ations ▁by ▁the ▁group ▁of ▁rot ations . ▁Ge odes ics ▁are ▁straight ▁lines ▁and ▁the ▁geometry ▁is ▁encoded ▁in ▁the ▁elementary ▁formulas ▁of ▁trig on ometry , ▁such ▁as ▁the ▁cos ine ▁rule ▁for ▁a ▁triangle ▁with ▁sides ▁, ▁, ▁ ▁and ▁angles ▁, ▁, ▁: ▁ ▁Fl at ▁t ori ▁can ▁be ▁obtained ▁by ▁taking ▁the ▁quot ient ▁of ▁ ▁by ▁a ▁lattice , ▁i . e . ▁a ▁free ▁Ab elian ▁subgroup ▁of