▁this ▁angle ▁difference ▁or ▁excess . ▁His ▁formula ▁showed ▁that ▁the ▁Gaussian ▁curv ature ▁could ▁be ▁calculated ▁near ▁a ▁point ▁as ▁the ▁limit ▁of ▁area ▁over ▁angle ▁excess ▁for ▁ge odes ic ▁tri angles ▁shr inking ▁to ▁the ▁point . ▁Since ▁any ▁closed ▁surface ▁can ▁be ▁decom posed ▁up ▁into ▁ge odes ic ▁tri angles , ▁the ▁formula ▁could ▁also ▁be ▁used ▁to ▁compute ▁the ▁integral ▁of ▁the ▁curv ature ▁over ▁the ▁whole ▁surface . ▁As ▁a ▁special ▁case ▁of ▁what ▁is ▁now ▁called ▁the ▁G auss – B on net ▁theorem , ▁G auss ▁proved ▁that ▁this ▁integral ▁was ▁remark ably ▁always ▁ 2 π ▁times ▁an ▁integer , ▁a ▁topological ▁invariant ▁of ▁the ▁surface ▁called ▁the ▁E uler ▁characteristic . ▁This ▁invariant ▁is ▁easy ▁to ▁compute ▁combin atorial ly ▁in ▁terms ▁of ▁the ▁number ▁of ▁vertices , ▁edges , ▁and ▁faces ▁of ▁the ▁tri angles ▁in ▁the ▁decomposition , ▁also ▁called ▁a ▁tri ang ulation . ▁This ▁interaction ▁between ▁analysis ▁and ▁topology ▁was ▁the ▁for er un ner ▁of ▁many ▁later ▁results ▁in ▁geometry , ▁cul min ating ▁in ▁the ▁A ti y ah - S inger ▁index ▁theorem . ▁In ▁particular ▁properties ▁of ▁the ▁curv ature ▁imp ose ▁restrictions ▁on ▁the ▁topology ▁of ▁the ▁surface . ▁ ▁Ge odes ic ▁tri angles ▁G auss ▁proved ▁that , ▁if ▁ ▁is ▁a ▁ge odes ic ▁triangle ▁on ▁a ▁surface ▁with ▁angles ▁, ▁ ▁and ▁ ▁at ▁vertices ▁, ▁ ▁and ▁, ▁then ▁ ▁In ▁fact ▁taking ▁ge odes ic ▁polar ▁coordinates