Critical points and curvature for embedded polyhedra

Recently a new insight into the Gauss-Bonnet Theorem and other problems in global differential geometry has come about through the connection between total curvature of embedded smooth manifolds and critical point theory for non-degenerate height functions.This paper presents an analogous program for embedded polyhedra.The methods are completely elementary, using the techniques neither of differential geometry nor of algebraic topology.As such the paper has a twofold purpose -to study global geometry of polyhedra for its own sake, and to give a deeper understanding of the theorems of global differential geometry through an elementary presentation of their finite or combinatorial content.Moreover the polyhedral theory applies to a wider class of objects, and gives a new interpretation of the relation between intrinsic and extrinsic curvature.Although the polyhedral part of the paper is relatively self-contained, the remarks which show the connection with the differentiable theory presuppose a familiarity with the classical differentiable results.For a bibliography on these and related problems, see Kuiper [4].This paper will contain no discussion of the possible convergence theorems relating the polyhedral and differentiable theories-for a presentation of this topic, expecially in the 2dimensional and convex cases we refer to A. D. Alexandrow [1].For related total curvature concepts see also .A subsequent paper of the author will deal with critical points and curvature for mappings of complexes into E ι for / > 1, and into /-dimensional manifolds.