▁curv ature ▁ ▁The ▁ge odes ic ▁curv ature ▁ ▁at ▁a ▁point ▁of ▁a ▁curve ▁, ▁param etr ised ▁by ▁arc ▁length , ▁on ▁an ▁orient ed ▁surface ▁is ▁defined ▁to ▁be ▁ ▁where ▁ ▁is ▁the ▁" pr incipal " ▁unit ▁normal ▁to ▁the ▁curve ▁in ▁the ▁surface , ▁constructed ▁by ▁rot ating ▁the ▁unit ▁tang ent ▁vector ▁ ▁through ▁an ▁angle ▁of ▁+ 9 0 ° . ▁ ▁The ▁ge odes ic ▁curv ature ▁at ▁a ▁point ▁is ▁an ▁intr insic ▁invariant ▁depending ▁only ▁on ▁the ▁metric ▁near ▁the ▁point . ▁A ▁unit ▁speed ▁curve ▁on ▁a ▁surface ▁is ▁a ▁ge odes ic ▁if ▁and ▁only ▁if ▁its ▁ge odes ic ▁curv ature ▁van ishes ▁at ▁all ▁points ▁on ▁the ▁curve . ▁A ▁unit ▁speed ▁curve ▁ ▁in ▁an ▁embedded ▁surface ▁is ▁a ▁ge odes ic ▁if ▁and ▁only ▁if ▁its ▁acceleration ▁vector ▁ ▁is ▁normal ▁to ▁the ▁surface . ▁ ▁The ▁ge odes ic ▁curv ature ▁measures ▁in ▁a ▁precise ▁way ▁how ▁far ▁a ▁curve ▁on ▁the ▁surface ▁is ▁from ▁being ▁a ▁ge odes ic . ▁ ▁Is ometric ▁embedding ▁problem ▁A ▁result ▁of ▁ ▁and ▁ ▁shows ▁that ▁every ▁metric ▁structure ▁on ▁a ▁surface ▁ar ises ▁from ▁a ▁local ▁embedding ▁in ▁. ▁A part ▁from ▁some ▁special ▁cases , ▁whether ▁this ▁is ▁possible ▁in ▁ ▁remains ▁an ▁open ▁question , ▁the ▁so - called ▁" W ey l ▁problem ". ▁In ▁ 1 9 2 6 ▁Maurice ▁Jan et ▁proved ▁that ▁it ▁is ▁always ▁possible ▁locally ▁if ▁,