acing =" 0 " ▁cell padding =" 5 " ▁| the ▁ge odes ics ▁through ▁ 0 ▁cut ▁the ▁circles ▁cent red ▁at ▁ 0 ▁orth og on ally . ▁| } ▁ ▁T aking ▁polar ▁coordinates ▁, ▁it ▁follows ▁that ▁the ▁metric ▁has ▁the ▁form ▁. ▁In ▁ge odes ic ▁coordinates , ▁it ▁is ▁easy ▁to ▁check ▁that ▁the ▁ge odes ics ▁through ▁zero ▁minim ize ▁length . ▁The ▁topology ▁on ▁the ▁Riemann ian ▁manifold ▁is ▁then ▁given ▁by ▁a ▁distance ▁function ▁, ▁namely ▁the ▁inf imum ▁of ▁the ▁lengths ▁of ▁piece wise ▁smooth ▁paths ▁between ▁ ▁and ▁. ▁This ▁distance ▁is ▁real ised ▁locally ▁by ▁ge odes ics , ▁so ▁that ▁in ▁normal ▁coordinates ▁. ▁If ▁the ▁radius ▁ ▁is ▁taken ▁small ▁enough , ▁a ▁slight ▁sh ar pen ing ▁of ▁the ▁G auss ▁lemma ▁shows ▁that ▁the ▁image ▁ ▁of ▁the ▁disc ▁ ▁under ▁the ▁exponential ▁map ▁is ▁ge odes ically ▁convex , ▁i . e . ▁any ▁two ▁points ▁in ▁ ▁are ▁joined ▁by ▁a ▁unique ▁ge odes ic ▁lying ▁entirely ▁inside ▁. ▁ ▁The or ema ▁E greg ium ▁ ▁T aking ▁ ▁and ▁ ▁coordinates ▁of ▁a ▁surface ▁in ▁ ▁corresponding ▁to ▁, ▁the ▁power ▁series ▁expansion ▁of ▁the ▁metric ▁is ▁given ▁in ▁normal ▁coordinates ▁ ▁as ▁ ▁This ▁extraordinary ▁result ▁— ▁G auss ' s ▁The or ema ▁E greg ium ▁— ▁shows ▁that ▁the ▁Gaussian ▁curv ature ▁of ▁a ▁surface ▁can ▁be ▁computed ▁sole ly ▁in ▁terms ▁of ▁the ▁metric ▁and ▁is ▁thus ▁an ▁intr