Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Generalizing the Gauss-Bonnet Theorem Gauss-Bonnet Theorem
Name: Unknown
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1993


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)?



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 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



Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (help@newton.dep.anl.gov), or at Argonne's Educational Programs

NEWTON AND ASK A SCIENTIST
Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory