San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Gauss-Bonnet Formula
and its Extensions

The Formula

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, … αm } be the interior angles of the curve at the points where it is not differentiable. Then

SKdA + ∫Ckgds + Σi=1mi−π) = 2π

where K is the Gaussian curvature of the surface and kg 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

Ckgds = ∫0(1/R)Rdθ
= ∫0dθ = 2π

Thus the formula is confirmed by this case.


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

For this topic see HOME PAGE OF applet-magic