, ▁but ▁it ▁also ▁contains ▁other ▁information , ▁since ▁the ▁mean ▁curv ature ▁is ▁half ▁the ▁trace ▁of ▁the ▁shape ▁operator . ▁The ▁mean ▁curv ature ▁is ▁an ▁extr insic ▁invariant . ▁In ▁intr insic ▁geometry , ▁a ▁cyl inder ▁is ▁develop able , ▁meaning ▁that ▁every ▁piece ▁of ▁it ▁is ▁intr ins ically ▁ind ist ingu ish able ▁from ▁a ▁piece ▁of ▁a ▁plane ▁since ▁its ▁G auss ▁curv ature ▁van ishes ▁ident ically . ▁Its ▁mean ▁curv ature ▁is ▁not ▁zero , ▁though ; ▁hence ▁extr ins ically ▁it ▁is ▁different ▁from ▁a ▁plane . ▁ ▁In ▁general , ▁the ▁eigen ve ctors ▁and ▁eigenvalues ▁of ▁the ▁shape ▁operator ▁at ▁each ▁point ▁determine ▁the ▁directions ▁in ▁which ▁the ▁surface ▁b ends ▁at ▁each ▁point . ▁The ▁eigenvalues ▁correspond ▁to ▁the ▁principal ▁curv atures ▁of ▁the ▁surface ▁and ▁the ▁eigen ve ctors ▁are ▁the ▁corresponding ▁principal ▁directions . ▁The ▁principal ▁directions ▁specify ▁the ▁directions ▁that ▁a ▁curve ▁embedded ▁in ▁the ▁surface ▁must ▁travel ▁to ▁have ▁maximum ▁and ▁minimum ▁curv ature , ▁these ▁being ▁given ▁by ▁the ▁principal ▁curv atures . ▁ ▁The ▁shape ▁operator ▁is ▁given ▁in ▁terms ▁of ▁the ▁components ▁of ▁the ▁first ▁and ▁second ▁fundamental ▁forms ▁by ▁the ▁We ing arten ▁equations : ▁ ▁Ge odes ic ▁curves ▁on ▁a ▁surface ▁Cur ves ▁on ▁a ▁surface ▁which ▁minim ize ▁length ▁between ▁the ▁end points ▁are ▁called ▁ge odes ics ; ▁they ▁are ▁the ▁shape ▁that ▁an ▁el astic ▁band ▁stretch ed ▁between ▁the ▁two ▁points ▁would ▁take . ▁Mathemat ically