Gauss-Bonnet for Spheres

Differential Geometry · Axiom Academy

EXAMPLE Gauss-Bonnet for Spheres Verifying the Gauss-Bonnet theorem for the 2-sphere Excellent work! You've verified the Gauss-Bonnet theorem for spheres. Here's what we learned: Gaussian Curvature: For a sphere of radius R , the Gaussian curvature is constant: K = 1/R² Surface Integral: The integral ∫∫ S K dA equals 4π for any sphere, regardless of its radius Euler Characteristic: The relation 4π = 2πχ holds with χ = 2 for the sphere, confirming the Gauss-Bonnet theorem Local-to-Global: The theorem connects local geometry (curvature) to global topology (Euler characteristic) Spherical Triangles: For regions on the sphere, the total curvature relates to the angle excess and Euler characteristic The Gauss-Bonnet theorem is one of the most beautiful results in differential geometry, linking intrinsic curvature to topological invariants!

This is the written version of the interactive lesson above. See the full Differential Geometry course.