Global Gauss-Bonnet Theorem

Differential Geometry · Axiom Academy

LESSON Global Gauss-Bonnet Theorem Connecting local curvature to global topology For a closed, orientable surface M without boundary, the Gauss-Bonnet theorem states: where K is the Gaussian curvature and χ(M) is the Euler characteristic of the surface. This remarkable equation tells us that integrating curvature over the entire surface yields a purely topological quantity! The Euler characteristic χ(M) is a topological invariant defined for polyhedra as: where V, E, and F are the numbers of vertices, edges, and faces. For surfaces, χ can be computed from the genus g (number of "holes"): Double torus: genus g = 2, so χ = -2 For a sphere of radius R, the Gaussian curvature is constant: K = 1/R². The surface area is 4πR². Let's verify the theorem: Since the sphere has χ = 2, we confirm: 4π = 2π(2). The theorem holds! A torus has regions of positive curvature (outer rim), negative curvature (inner rim), and zero curvature. Despite this complexity, the total curvature must equal: The positive and negative curvatures exactly cancel! This is remarkable: no matter how you deform the torus, the total curvature remains zero. 5. Proof Idea via Triangulation The proof uses a triangulation of the surface into small triangles. For each triangle with angles α, β, γ: Summing over all triangles in the triangulation and using the Euler relation V - E + F = χ, we obtain the global result. The key steps are: Triangulate the surface into F triangular faces

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