From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Gauss-Bonnet theorem (Was Re: Yet another radians question) Date: 6 May 1999 16:07:33 GMT Newsgroups: sci.physics,sci.astro,sci.math,alt.math.moderated Keywords: piecewise-linear versions of Gauss-Bonnet curvature theorem In article <7gnojm\$dn4\$1@news.rain.org>, Nick Halloway wrote: >When trying to sum up the total curvature of a surface, you can either >look at the deviation from flatness of each point in the surface and >sum that, or you can look at how much angle deficit there is around >each point, and sum that. > >The first works for 1-surfaces in 2-space -- summing the deviation from >flatness at each point for a curve that's topologically S^1, it >works out to 2 pi. > >The second works for 2-surfaces in 3-space -- summing up the angle >deficit around each point of a 2-surface that's topologically S^2, it >works out to 4 pi. > >For n-surfaces in n+1-space, what to do? Well, you'll have deviations from "flatness" at almost all points, in general, so it isn't "summing" which you'll have to do but rather integration (unless you stay in the polyhedral -- piecewise-linear -- category). Then you find yourself asking, "gee, what can I integrate over a manifold, which measures deviation from flatness, and hope to get a topological invariant?" Answer: curvature (defined e.g. as the pullback of the area element on the sphere under the normal map n: M -> S^n, although intrinsic definitions are also possible). That's the Gauss-Bonnet theorem. I was going to object to the examples constructed, saying that in the PL category one has pushed too much degeneracy to the lower-dimensional simplexes to be able to sort through things clearly, but to my surprise as I reread "The Generalized Gauss-Bonnet Theorem and What It Means To Mankind" in Spivak's "Comprehensive Introduction to Differential Geometry", it turns out the PL approach was indeed used. Quoting page V.387: "In 1943 Allendoerfer and Weil proved a generalization of the Gauss-Bonnet formula for a polyhedral piece of a Riemannian manifold imbedded in Euclidean space; using this, they were able to obtain a proof of the general Gauss-Bonnet Theorem for [real-analytic] manifolds, by means of a triangulation." Ref.: Trans. Amer. Math. Soc. 53, (1943). 101--129 By the way, AMS last week announced the completion of its project to bring the entirety of Math Reviews online, so even the half-century-old review of this article is available to subscribers. Unfortunately those of us only marginally conversant in the ancient hieroglyphs of differential geometry in which Hassler Whitney wrote will not much aided by this particular review. (Still, the MathSciNet progress is very good news.) dave Differential geometry: http://www.math.niu.edu/~rusin/known-math/index/53-XX.html