▁G auss - C od az zi ▁equations ▁always ▁uniqu ely ▁determine ▁an ▁embedded ▁surface ▁locally . ▁For ▁this ▁reason ▁the ▁G auss - C od az zi ▁equations ▁are ▁often ▁called ▁the ▁fundamental ▁equations ▁for ▁embedded ▁surfaces , ▁precisely ▁ident ifying ▁where ▁the ▁intr insic ▁and ▁extr insic ▁curv atures ▁come ▁from . ▁They ▁admit ▁general izations ▁to ▁surfaces ▁embedded ▁in ▁more ▁general ▁Riemann ian ▁manif olds . ▁ ▁Sh ape ▁operator ▁ ▁The ▁differential ▁ ▁of ▁the ▁G auss ▁map ▁ ▁can ▁be ▁used ▁to ▁define ▁a ▁type ▁of ▁extr insic ▁curv ature , ▁known ▁as ▁the ▁shape ▁operator ▁or ▁We ing arten ▁map . ▁This ▁operator ▁first ▁appeared ▁implicitly ▁in ▁the ▁work ▁of ▁Wilhelm ▁Bl as ch ke ▁and ▁later ▁explicitly ▁in ▁a ▁treat ise ▁by ▁Bur ali - F ort i ▁and ▁Burg ati . ▁Since ▁at ▁each ▁point ▁ ▁of ▁the ▁surface , ▁the ▁tang ent ▁space ▁is ▁an ▁inner ▁product ▁space , ▁the ▁shape ▁operator ▁ ▁can ▁be ▁defined ▁as ▁a ▁linear ▁operator ▁on ▁this ▁space ▁by ▁the ▁formula ▁ ▁for ▁tang ent ▁vectors ▁, ▁ ▁( the ▁inner ▁product ▁makes ▁sense ▁because ▁ ▁and ▁ ▁both ▁lie ▁in ▁). ▁The ▁right ▁hand ▁side ▁is ▁symmetric ▁in ▁ ▁and ▁, ▁so ▁the ▁shape ▁operator ▁is ▁self - ad joint ▁on ▁the ▁tang ent ▁space . ▁The ▁eigenvalues ▁of ▁ ▁are ▁just ▁the ▁principal ▁curv atures ▁ ▁and ▁ ▁at ▁. ▁In ▁particular ▁the ▁determin ant ▁of ▁the ▁shape ▁operator ▁at ▁a ▁point ▁is ▁the ▁Gaussian ▁curv ature