Given a geodesic triangle (a triangle formed by the arcs of three geodesics on a smooth surface),
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Given the Euler characteristic 
,
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so the integral curvature of a closed surface is not altered by a topological transformation.
Integral Curvature
See also
Gauss-Bonnet Formula, Geodesic TriangleExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Integral Curvature." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IntegralCurvature.html