Ask A Scientist Mathematics Archive

Question: According to the Gauss-Bonnet 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 
k-manifold, 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*pi-sum of angles 
at vertex. It seems to me that this new definition of curvature allows 
vertices to concentrate curvature on sunon-regular surfaces.  This, then, 
brings me to the crux of the problem. Is it possible to generalize the Gauss
Bonnet theorem to non-regular hypersurfaces?  If X is a compact, non-regular 
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 2-sphere 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)?
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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 Gauss-Bonnet 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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