▁line ▁and ▁area ▁elements ▁can ▁be ▁associated ▁to ▁any ▁abstract ▁Riemann ian ▁ 2 - man if old ▁in ▁a ▁local ▁chart . ▁ ▁Second ▁fundamental ▁form ▁ ▁The ▁extr insic ▁geometry ▁of ▁surfaces ▁studies ▁the ▁properties ▁of ▁surfaces ▁embedded ▁into ▁a ▁E uclidean ▁space , ▁typically ▁. ▁In ▁intr insic ▁geometry , ▁two ▁surfaces ▁are ▁" the ▁same " ▁if ▁it ▁is ▁possible ▁to ▁unfold ▁one ▁surface ▁onto ▁the ▁other ▁without ▁stretch ing ▁it , ▁i . e . ▁a ▁map ▁of ▁one ▁surface ▁onto ▁the ▁other ▁pres erving ▁distance . ▁Thus ▁a ▁cyl inder ▁is ▁locally ▁" the ▁same " ▁as ▁the ▁plane . ▁In ▁extr insic ▁geometry , ▁two ▁surfaces ▁are ▁" the ▁same " ▁if ▁they ▁are ▁con gru ent ▁in ▁the ▁ambient ▁E uclidean ▁space , ▁i . e . ▁there ▁is ▁an ▁is ometry ▁of ▁ ▁carrying ▁one ▁surface ▁onto ▁the ▁other . ▁With ▁this ▁more ▁rig id ▁definition ▁of ▁sim il itude , ▁the ▁cyl inder ▁and ▁the ▁plane ▁are ▁obviously ▁no ▁longer ▁the ▁same . ▁ ▁Although ▁the ▁primary ▁invariant ▁in ▁the ▁study ▁of ▁the ▁intr insic ▁geometry ▁of ▁surfaces ▁is ▁the ▁metric ▁( the ▁first ▁fundamental ▁form ) ▁and ▁the ▁Gaussian ▁curv ature , ▁certain ▁properties ▁of ▁surfaces ▁also ▁depend ▁on ▁an ▁embedding ▁into ▁ ▁( or ▁a ▁higher ▁dimensional ▁E uclidean ▁space ). ▁The ▁most ▁important ▁example ▁is ▁the ▁second ▁fundamental ▁form , ▁defined ▁class ically ▁as ▁follows . ▁ ▁Take ▁a ▁point ▁ ▁on ▁the ▁surface ▁in ▁a ▁local ▁chart . ▁The