▁considering ▁the ▁two - sheet ed ▁hyper bol oid ▁ ▁in ▁three - dimensional ▁M ink owski ▁space , ▁where ▁. ▁ ▁Each ▁of ▁these ▁surfaces ▁of ▁constant ▁curv ature ▁has ▁a ▁trans itive ▁Lie ▁group ▁of ▁symmet ries . ▁This ▁group ▁the or etic ▁fact ▁has ▁far - re aching ▁consequences , ▁all ▁the ▁more ▁remarkable ▁because ▁of ▁the ▁central ▁role ▁these ▁special ▁surfaces ▁play ▁in ▁the ▁geometry ▁of ▁surfaces , ▁due ▁to ▁Po inc ar é ' s ▁uniform ization ▁theorem ▁( see ▁below ). ▁ ▁Other ▁examples ▁of ▁surfaces ▁with ▁Gaussian ▁curv ature ▁ 0 ▁include ▁con es , ▁tang ent ▁develop ables , ▁and ▁more ▁generally ▁any ▁develop able ▁surface . ▁ ▁Local ▁metric ▁structure ▁▁ ▁For ▁any ▁surface ▁embedded ▁in ▁E uclidean ▁space ▁of ▁dimension ▁ 3 ▁or ▁higher , ▁it ▁is ▁possible ▁to ▁measure ▁the ▁length ▁of ▁a ▁curve ▁on ▁the ▁surface , ▁the ▁angle ▁between ▁two ▁curves ▁and ▁the ▁area ▁of ▁a ▁region ▁on ▁the ▁surface . ▁This ▁structure ▁is ▁encoded ▁inf init es im ally ▁in ▁a ▁Riemann ian ▁metric ▁on ▁the ▁surface ▁through ▁line ▁elements ▁and ▁area ▁elements . ▁Class ically ▁in ▁the ▁nin ete enth ▁and ▁early ▁tw enti eth ▁centuries ▁only ▁surfaces ▁embedded ▁in ▁ ▁were ▁considered ▁and ▁the ▁metric ▁was ▁given ▁as ▁a ▁ 2 × 2 ▁positive ▁definite ▁matrix ▁varying ▁smooth ly ▁from ▁point ▁to ▁point ▁in ▁a ▁local ▁paramet riz ation ▁of ▁the ▁surface . ▁The ▁idea ▁of ▁local ▁paramet riz ation ▁and ▁change ▁of ▁coordinate ▁was ▁later ▁formal