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