Gauss-Bonnet Theorem & Geodesic Curvature

The Gauss-Bonnet theorem is a bridge between geometry and topology: ∫∫_M K dA + ∮_∂M κ_g ds + Σ θ_i = 2πχ(M). The Gaussian curvature K integrated over a surface plus boundary geodesic curvature always equals 2π times the Euler characteristic — a topological invariant that does not change under smooth deformation.