insic ▁invariant ▁of ▁the ▁surface , ▁independent ▁of ▁any ▁embedding ▁in ▁ ▁and ▁un changed ▁under ▁coordinate ▁transformations . ▁In ▁particular ▁is omet ries ▁of ▁surfaces ▁preserve ▁Gaussian ▁curv ature . ▁ ▁G auss – Jac obi ▁equation ▁ ▁T aking ▁a ▁coordinate ▁change ▁from ▁normal ▁coordinates ▁at ▁ ▁to ▁normal ▁coordinates ▁at ▁a ▁nearby ▁point ▁, ▁yields ▁the ▁St urm – Li ou ville ▁equation ▁satisfied ▁by ▁, ▁discovered ▁by ▁G auss ▁and ▁later ▁general ised ▁by ▁Jacob i , ▁{ | ▁border =" 1 " ▁cell sp acing =" 0 " ▁cell padding =" 5 " ▁| ▁| } ▁ ▁The ▁Jacob ian ▁of ▁this ▁coordinate ▁change ▁at ▁ ▁is ▁equal ▁to ▁. ▁This ▁gives ▁another ▁way ▁of ▁establish ing ▁the ▁intr insic ▁nature ▁of ▁Gaussian ▁curv ature . ▁Because ▁ ▁can ▁be ▁interpreted ▁as ▁the ▁length ▁of ▁the ▁line ▁element ▁in ▁the ▁ ▁direction , ▁the ▁G auss – Jac obi ▁equation ▁shows ▁that ▁the ▁Gaussian ▁curv ature ▁measures ▁the ▁spread ing ▁of ▁ge odes ics ▁on ▁a ▁geometric ▁surface ▁as ▁they ▁move ▁away ▁from ▁a ▁point . ▁ ▁Lap lace – Bel tr ami ▁operator ▁On ▁a ▁surface ▁with ▁local ▁metric ▁ ▁and ▁Lap lace – Bel tr ami ▁operator ▁ ▁where ▁, ▁the ▁Gaussian ▁curv ature ▁at ▁a ▁point ▁is ▁given ▁by ▁the ▁formula ▁ ▁where ▁ ▁is ▁the ▁denotes ▁the ▁ge odes ic ▁distance ▁from ▁the ▁point . ▁Since ▁ ▁is ▁manifest ly ▁an ▁intr insic ▁invariant , ▁this ▁gives ▁yet ▁another ▁proof ▁that ▁the ▁Gaussian ▁curv