Critical points and curvature for embedded polyhedra

Abstract

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 Chern-Lashof [3]. A subsequent paper of the author will deal with critical points and curvature for mappings of complexes into Eι for / > 1, and into /-dimensional manifolds. The author wishes to thank Professor Kuiper for his interest and advice throughout the development of this research. 1. The critical point theorem Definition. A convex cell complex Mk embedded in E n is a finite collection of cells {Cr}, where each C° is a point, and each Cr is a bounded convex set with interior in some afϊine Eτ c En, such that the boundary dCτ of C r is a union of Cs with s < r, and such that if s < r and Cs Γ\Cr Φ 0, then O C dC Mk is called k-dimensional if there is a Ck in Mk but no C*+1.