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Name: Unknown
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Date: Around 1993

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)?

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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