Gauss-Bonnet for the Torus

Differential Geometry · Axiom Academy

EXAMPLE Gauss-Bonnet Theorem for the Torus Discover how positive and negative curvature perfectly cancel on a torus Excellent work! You've completed this example of the Gauss-Bonnet theorem for the torus. Here's what we learned: Topological Constraint: The Gauss-Bonnet theorem relates the total Gaussian curvature of a closed surface to its Euler characteristic, a purely topological invariant. Euler Characteristic: The torus has Euler characteristic χ = 0, which means the total curvature must integrate to zero regardless of the specific metric. Curvature Distribution: On a standard torus, regions of positive curvature (outer equator) exactly cancel with regions of negative curvature (inner equator), demonstrating the geometric realization of the topological constraint. Genus Connection: For a surface of genus g, we have χ = 2 - 2g. The torus has genus 1, giving χ = 0, which explains why there must be both positive and negative curvature regions. This beautiful theorem shows how topology constrains geometry. Any torus, regardless of how you shape it, must have total curvature equal to zero!

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