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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 {α_{1}, α_{2}, … α_{m} }
be the interior angles of the curve at the points where it is not differentiable.
Then

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

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 R_{1} and inner radius R_{2}. 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