▁distance , ▁angle , ▁and ▁area . ▁An ▁important ▁class ▁of ▁such ▁surfaces ▁are ▁the ▁develop able ▁surfaces : ▁surfaces ▁that ▁can ▁be ▁fl atten ed ▁to ▁a ▁plane ▁without ▁stretch ing ; ▁examples ▁include ▁the ▁cyl inder ▁and ▁the ▁cone . ▁ ▁In ▁addition , ▁there ▁are ▁properties ▁of ▁surfaces ▁which ▁depend ▁on ▁an ▁embedding ▁of ▁the ▁surface ▁into ▁E uclidean ▁space . ▁These ▁surfaces ▁are ▁the ▁subject ▁of ▁extr insic ▁geometry . ▁They ▁include ▁Min imal ▁surfaces ▁are ▁surfaces ▁that ▁minim ize ▁the ▁surface ▁area ▁for ▁given ▁boundary ▁conditions ; ▁examples ▁include ▁soap ▁films ▁stretch ed ▁across ▁a ▁wire ▁frame , ▁c aten oid s ▁and ▁hel ico ids . ▁Ru led ▁surfaces ▁are ▁surfaces ▁that ▁have ▁at ▁least ▁one ▁straight ▁line ▁running ▁through ▁every ▁point ; ▁examples ▁include ▁the ▁cyl inder ▁and ▁the ▁hyper bol oid ▁of ▁one ▁sheet . ▁ ▁Any ▁- dimensional ▁complex ▁manifold ▁is , ▁at ▁the ▁same ▁time , ▁a ▁- dimensional ▁real ▁manifold . ▁Thus ▁any ▁complex ▁one - man if old ▁( also ▁called ▁a ▁Riemann ▁surface ) ▁is ▁a ▁smooth ▁orient ed ▁surface ▁with ▁an ▁associated ▁complex ▁structure . ▁Every ▁closed ▁surface ▁adm its ▁complex ▁structures . ▁Any ▁complex ▁algebraic ▁curve ▁or ▁real ▁algebraic ▁surface ▁is ▁also ▁a ▁smooth ▁surface , ▁possibly ▁with ▁singular ities . ▁ ▁Complex ▁structures ▁on ▁a ▁closed ▁orient ed ▁surface ▁correspond ▁to ▁conform al ▁equivalence ▁classes ▁of ▁Riemann ian ▁metrics ▁on ▁the ▁surface . ▁One ▁version ▁of ▁the ▁uniform ization ▁theorem ▁( due ▁to ▁Po inc ar é