▁they ▁are ▁described ▁using ▁partial ▁differential ▁equations ▁from ▁the ▁calculus ▁of ▁variations . ▁The ▁differential ▁geometry ▁of ▁surfaces ▁revol ves ▁around ▁the ▁study ▁of ▁ge odes ics . ▁It ▁is ▁still ▁an ▁open ▁question ▁whether ▁every ▁Riemann ian ▁metric ▁on ▁a ▁ 2 - dimensional ▁local ▁chart ▁ar ises ▁from ▁an ▁embedding ▁in ▁ 3 - dimensional ▁E uclidean ▁space : ▁the ▁theory ▁of ▁ge odes ics ▁has ▁been ▁used ▁to ▁show ▁this ▁is ▁true ▁in ▁the ▁important ▁case ▁when ▁the ▁components ▁of ▁the ▁metric ▁are ▁analyt ic . ▁ ▁Ge odes ics ▁ ▁Given ▁a ▁piece wise ▁smooth ▁path ▁ ▁in ▁the ▁chart ▁for ▁ ▁in ▁, ▁its ▁length ▁is ▁defined ▁by ▁ ▁and ▁energy ▁by ▁ ▁The ▁length ▁is ▁independent ▁of ▁the ▁paramet riz ation ▁of ▁a ▁path . ▁By ▁the ▁E uler – L ag range ▁equations , ▁if ▁ ▁is ▁a ▁path ▁minim ising ▁length , ▁paramet riz ed ▁by ▁ar cl ength , ▁it ▁must ▁satisfy ▁the ▁E uler ▁equations ▁ ▁where ▁the ▁Christ off el ▁symbols ▁ ▁are ▁given ▁by ▁ ▁where ▁, ▁, ▁ ▁and ▁ ▁is ▁the ▁inverse ▁matrix ▁to ▁. ▁A ▁path ▁satisfying ▁the ▁E uler ▁equations ▁is ▁called ▁a ▁ge odes ic . ▁By ▁the ▁C auch y – Sch war z ▁inequality ▁a ▁path ▁minim ising ▁energy ▁is ▁just ▁a ▁ge odes ic ▁param etr ised ▁by ▁arc ▁length ; ▁and , ▁for ▁any ▁ge odes ic , ▁the ▁parameter ▁ ▁is ▁proportional ▁to ▁ar cl ength . ▁ ▁Ge odes ic