▁given ▁point ▁, ▁there ▁is ▁a ▁ge odes ic ▁ ▁defined ▁on ▁( − 2 , 2 ) ▁with ▁ ▁and ▁. ▁Moreover , ▁if ▁, ▁then ▁. ▁The ▁exponential ▁map ▁is ▁defined ▁by ▁ ▁( 1 ) ▁and ▁gives ▁a ▁dif fe omorphism ▁between ▁a ▁disc ▁ ▁and ▁a ▁neighbourhood ▁of ▁; ▁more ▁generally ▁the ▁map ▁sending ▁ ▁to ▁ ▁gives ▁a ▁local ▁dif fe omorphism ▁onto ▁a ▁neighbourhood ▁of ▁. ▁The ▁exponential ▁map ▁gives ▁ge odes ic ▁normal ▁coordinates ▁near ▁. ▁ ▁Comput ation ▁of ▁normal ▁coordinates ▁There ▁is ▁a ▁standard ▁technique ▁( see ▁for ▁example ▁) ▁for ▁computing ▁the ▁change ▁of ▁variables ▁to ▁normal ▁coordinates ▁, ▁ ▁at ▁a ▁point ▁as ▁a ▁formal ▁Taylor ▁series ▁expansion . ▁If ▁the ▁coordinates ▁, ▁ ▁at ▁( 0 , 0 ) ▁are ▁locally ▁orthogonal , ▁write ▁ ▁where ▁, ▁ ▁are ▁quadratic ▁and ▁, ▁ ▁cub ic ▁hom ogeneous ▁polynomials ▁in ▁ ▁and ▁. ▁If ▁ ▁and ▁ ▁are ▁fixed , ▁ ▁and ▁ ▁can ▁be ▁considered ▁as ▁formal ▁power ▁series ▁solutions ▁of ▁the ▁E uler ▁equations : ▁this ▁uniqu ely ▁determ ines ▁, ▁, ▁, ▁, ▁ ▁and ▁. ▁ ▁G auss ' s ▁lemma ▁ ▁In ▁these ▁coordinates ▁the ▁matrix ▁ ▁satisfies ▁ ▁and ▁the ▁lines ▁ ▁are ▁ge odes ics ▁through ▁ 0 . ▁E uler ' s ▁equations ▁imply ▁the ▁matrix ▁equation ▁, ▁a ▁key ▁result , ▁usually ▁called ▁the ▁G auss ▁lemma . ▁Ge omet r ically ▁it ▁states ▁that ▁{ | ▁border =" 1 " ▁cell sp