﻿ The Gauss-Bonnet Formula and its Extensions
San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
U.S.A.

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=1m(αi−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 = ∫02π(1/R)Rdθ = ∫02πdθ = 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

For this topic see HOME PAGE OF applet-magic