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Critical Points and Curvature for Embedded Polyhedral Surfaces

by Thomas F. Banchoff

Many theorems in global geometry exploit the connection between
critical point theory and total curvature. This paper uses
the same approcah to prove the critical point theorem and uses it to
prove the Gauss-Bonnet theorem. In the case of polyhedra, a new
interpretation of Gauss's *Theorem Egregium* which relates
the extrinsic and intrinsic curvature on a surface is given.