▁of ▁a ▁surface ▁in ▁E uclidean ▁space ▁have ▁also ▁been ▁extens ively ▁studied . ▁This ▁is ▁well ▁illustrated ▁by ▁the ▁non - linear ▁E uler – L ag range ▁equations ▁in ▁the ▁calculus ▁of ▁variations : ▁although ▁E uler ▁developed ▁the ▁one ▁variable ▁equations ▁to ▁understand ▁ge odes ics , ▁defined ▁independently ▁of ▁an ▁embedding , ▁one ▁of ▁Lag range ' s ▁main ▁applications ▁of ▁the ▁two ▁variable ▁equations ▁was ▁to ▁minimal ▁surfaces , ▁a ▁concept ▁that ▁can ▁only ▁be ▁defined ▁in ▁terms ▁of ▁an ▁embedding . ▁ ▁Over view ▁ ▁Poly hed ra ▁in ▁the ▁E uclidean ▁space , ▁such ▁as ▁the ▁boundary ▁of ▁a ▁cube , ▁are ▁among ▁the ▁first ▁surfaces ▁encountered ▁in ▁geometry . ▁It ▁is ▁also ▁possible ▁to ▁define ▁smooth ▁surfaces , ▁in ▁which ▁each ▁point ▁has ▁a ▁neighborhood ▁dif fe omorphic ▁to ▁some ▁open ▁set ▁in ▁, ▁the ▁E uclidean ▁plane . ▁This ▁elabor ation ▁allows ▁calculus ▁to ▁be ▁applied ▁to ▁surfaces ▁to ▁prove ▁many ▁results . ▁ ▁Two ▁smooth ▁surfaces ▁are ▁dif fe omorphic ▁if ▁and ▁only ▁if ▁they ▁are ▁home omorphic . ▁( The ▁analog ous ▁result ▁does ▁not ▁hold ▁for ▁manif olds ▁of ▁dimension ▁greater ▁than ▁three .) ▁It ▁follows ▁that ▁closed ▁surfaces ▁are ▁class ified ▁up ▁to ▁dif fe omorphism ▁by ▁their ▁E uler ▁characteristic ▁and ▁orient ability . ▁ ▁Sm ooth ▁surfaces ▁equ ipped ▁with ▁Riemann ian ▁metrics ▁are ▁of ▁found ational ▁importance ▁in ▁differential ▁geometry . ▁A ▁Riemann ian ▁metric ▁end ows ▁a ▁surface ▁with ▁not ions ▁of ▁ge odes ic ,