

Generalizing the GaussBonnet Theorem GaussBonnet Theorem
Name: Unknown
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1993
Question:
According to the GaussBonnet theorem, if X is a compact, even
dim hypersurface in R^(k+1), then integral of K over X = Vol(S^k^)*Chi(X)/2.
However, all of the proofs I have seen of this theorem assert that X must be a
kmanifold, i.e. without vertices, edges, etc. Now, I have seen a variant
where if X is a polygonal surface, and V is the set of all vertices in X, then
sum of K over V = 2*pi*Chi(X) where K is now defined to be 2*pisum of angles
at vertex. It seems to me that this new definition of curvature allows
vertices to concentrate curvature on sunonregular surfaces. This, then,
brings me to the crux of the problem. Is it possible to generalize the Gauss
Bonnet theorem to nonregular hypersurfaces? If X is a compact, nonregular
surface, then (integral of K over X)+(sum of K over V)=2*pi = 2*pi*Chi(X). I
have tried to compute this on a 2sphere with one pole pulled to a vertex of a
cone, and the formula holds. Is it possible for you to provide a proof for
this conjecture (or disprove it)?
Replies:
Sounds like an interesting extension. However, as with most such
things, I strongly suspect somebody has already done this. Why don't you look
up GaussBonnet in several years worth of Mathematical Reviews abstract
listings to see if this is discussed anywhere. It may lead you to a recent
text on the subject, if nothing else. If you cannot find it anywhere, I would
suggest sending it in as a conjecture to one of the mathematical bulletin
journals... I am not too familiar with those things right now, but could look
up the right names if you want me to. Of course, if you can figure out a
proof, go ahead and try to publish it!
Arthur Smith
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Update: June 2012

