Theorem: 4 pi - 2 pi * chi(S^2) = 0 (Gauss-Bonnet sphere instance, chi = 2)

Theorem (Gauss-Bonnet-sphere canonical): for the unit sphere S^2 with intrinsic Gaussian curvature K = 1 and total surface area 4 pi, the Gauss-Bonnet integral gives int_{S^2} K dA = 1 * 4 pi = 4 pi. The Euler characteristic of S^2 is…