Let S be a simply-connected section of a 2-dimensional Riemannian manifold M bounded by a piecewise differentiable curve C which consists of m pieces. Let {α₁, α₂, … α_m } be the interior angles of the curve at the points where it is not differentiable. Then

∫_SKdA + ∫_Ck_gds + Σ_i=1^m(α_i−π) = 2π

where K is the Gaussian curvature of the surface and k_g is the geodesic curvature of C.

Some Illustrations

Consider a circular disk of radius R. The Gaussian curvature is zero; the geodesic curvature is 1/R. There are no points where the boundary is not differentiable; i.e., m=0. For this case

∫_Ck_gds = ∫₀^2π(1/R)Rdθ
= ∫₀^2πdθ = 2π

Thus the formula is confirmed by this case.

Extensions

It was easy to extend the formula to surfaces which were not simply connected; i.e., that had holes. It was just a matter of keeping the directions of integration consistent.

Consider an annular ring with outer radius R₁ and inner radius R₂. A cut between the inner and outer radius produces a curve C in which there are four angles of π/2 each. The integrals of geodesic curvature on the inner and outer circles cancel out because they are traversed in opposite directions. The sum of the four interior angles is 4(π/2)=2π.

When mathematicians considered further generalization, there was the extension to the case in which S was the entire manifold. This is the Gauss-Bonnet Theorem. Then mathematicians such as Heinz Hopf considered generalization in terms of the the dimensionality of the manifold. This was a fruitful endeavor.

Extension to Surfaces with
Conical Vertices

Consider a truncated segment of half-cone with a circular cross section. Let the radius of the boundary be R. The Gaussian curvature of a cone is zero everywhere except at the vertex, where it is undefined. The geodesic curvature is not just a function of the curve; it depends also upon the surface it is embedded in. It is established elsewhere that the contribution of a conical point to the generalization of the Gauss-Bonnet is equal to its angular deficit. This just balances the deviation of the integral of geodesic curvature from 2π

Extension to Surfaces with
Sharp Ridges

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