San José State University |
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applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley U.S.A. |
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The Gauss-Bonnet Formula and its Extensions |
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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.
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
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 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.
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
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