▁on ▁, ▁one ▁of ▁the ▁simplest ▁cases ▁of ▁the ▁A ti y ah - S inger ▁index ▁theorem . ▁Another ▁related ▁result , ▁which ▁can ▁be ▁proved ▁using ▁the ▁G auss – B on net ▁theorem , ▁is ▁the ▁Po inc ar é - H opf ▁index ▁theorem ▁for ▁vector ▁fields ▁on ▁ ▁which ▁van ish ▁at ▁only ▁a ▁finite ▁number ▁of ▁points : ▁the ▁sum ▁of ▁the ▁indices ▁at ▁these ▁points ▁equals ▁the ▁E uler ▁characteristic , ▁where ▁the ▁index ▁of ▁a ▁point ▁is ▁defined ▁as ▁follows : ▁on ▁a ▁small ▁circle ▁round ▁each ▁isolated ▁zero , ▁the ▁vector ▁field ▁defines ▁a ▁map ▁into ▁the ▁unit ▁circle ; ▁the ▁index ▁is ▁just ▁the ▁w inding ▁number ▁of ▁this ▁map .) ▁ ▁Cur v ature ▁and ▁embed d ings ▁If ▁the ▁Gaussian ▁curv ature ▁of ▁a ▁surface ▁ ▁is ▁everywhere ▁positive , ▁then ▁the ▁E uler ▁characteristic ▁is ▁positive ▁so ▁ ▁is ▁home omorphic ▁( and ▁therefore ▁dif fe omorphic ) ▁to ▁. ▁If ▁in ▁addition ▁the ▁surface ▁is ▁is omet r ically ▁embedded ▁in ▁, ▁the ▁G auss ▁map ▁provides ▁an ▁explicit ▁dif fe omorphism . ▁As ▁Had am ard ▁observed , ▁in ▁this ▁case ▁the ▁surface ▁is ▁convex ; ▁this ▁criter ion ▁for ▁convex ity ▁can ▁be ▁viewed ▁as ▁a ▁ 2 - dimensional ▁general isation ▁of ▁the ▁well - known ▁second ▁derivative ▁criter ion ▁for ▁convex ity ▁of ▁plane ▁curves . ▁Hil bert ▁proved ▁that ▁every ▁is omet r ically ▁embedded ▁closed ▁surface ▁must ▁have ▁a ▁point ▁of ▁positive ▁curv