▁the ▁surface ▁is ▁then ▁given ▁by ▁the ▁second ▁order ▁deviation ▁of ▁the ▁metric ▁at ▁the ▁point ▁from ▁the ▁E uclidean ▁metric . ▁In ▁particular ▁the ▁Gaussian ▁curv ature ▁is ▁an ▁invariant ▁of ▁the ▁metric , ▁G auss ' s ▁celebrated ▁The or ema ▁E greg ium . ▁A ▁convenient ▁way ▁to ▁understand ▁the ▁curv ature ▁comes ▁from ▁an ▁ordinary ▁differential ▁equation , ▁first ▁considered ▁by ▁G auss ▁and ▁later ▁generalized ▁by ▁Jacob i , ▁ar ising ▁from ▁the ▁change ▁of ▁normal ▁coordinates ▁about ▁two ▁different ▁points . ▁The ▁G auss – Jac obi ▁equation ▁provides ▁another ▁way ▁of ▁computing ▁the ▁Gaussian ▁curv ature . ▁Ge omet r ically ▁it ▁explains ▁what ▁happens ▁to ▁ge odes ics ▁from ▁a ▁fixed ▁base ▁point ▁as ▁the ▁endpoint ▁var ies ▁along ▁a ▁small ▁curve ▁segment ▁through ▁data ▁recorded ▁in ▁the ▁Jacob i ▁field , ▁a ▁vector ▁field ▁along ▁the ▁ge odes ic . ▁One ▁and ▁a ▁quarter ▁centuries ▁after ▁G auss ▁and ▁Jacob i , ▁Mar ston ▁Mor se ▁gave ▁a ▁more ▁concept ual ▁interpretation ▁of ▁the ▁Jacob i ▁field ▁in ▁terms ▁of ▁second ▁derivatives ▁of ▁the ▁energy ▁function ▁on ▁the ▁infinite - dimensional ▁Hil bert ▁manifold ▁of ▁paths . ▁ ▁Ex pon ential ▁map ▁ ▁The ▁theory ▁of ▁ordinary ▁differential ▁equations ▁shows ▁that ▁if ▁ ▁is ▁smooth ▁then ▁the ▁differential ▁equation ▁ ▁with ▁initial ▁condition ▁ ▁has ▁a ▁unique ▁solution ▁for ▁ ▁sufficiently ▁small ▁and ▁the ▁solution ▁depends ▁smooth ly ▁on ▁ ▁and ▁. ▁This ▁implies ▁that ▁for ▁sufficiently ▁small ▁tang ent ▁vectors ▁ ▁at ▁a