Tightness for Polyhedral Surfaces:

For smooth surfaces, the decomposition of a tight immersion into the M+ and M- regions [More] is based on the curvature of the surface, and in particular on the number of critical points for height functions on the surface. To find an analogous result for polyhedral surfaces we use the following characterizations of polyhedral critical points:

A vertex v of M is called a local extreme vertex if v is a vertex of the convex hull of the star of v, and it is a (global) extreme vertex if it is a vertex of the convex hull of M. Note that a local extreme vertex is the local maximum for the height function on M in some direction, and that a vertex that is not locally extreme is a saddle point for every height function on M.

With these definitions, we can describe tight polyhedral surfaces as follows:

Lemma: An immersed polyhedral surface M is tight if, and only if,

  1. every local extreme vertex of M is a global extreme vertex of M,
  2. every edge of the convex hull M is an edge of M,
  3. no extreme vertex of M is in the double set of M.

Note the relationship of this description to the division into the M+ and M- regions: property (1) says that there is no local maximum inside the convex hull (i.e., all the positive curvature is on the outside), property (2) says that most of the convex envelope is present (perhaps minus the interiors of some facets) forming the analog of the M+ region, and property (3) says this region is embedded.

All three conditions are required, as can be seen from some simple examples [More].

The basic idea behind the proof is to use the fact that tightness is equivalent to the two-piece property [More] to show that an interior local extreme vertex can be cut off along with a global extreme vertex contradicting tightness, and if an edge of the convex hull is missing from the surface, then its two vertices can be cut off with one slice, again contradicting tightness.

This lemma provides a convenient method of checking a polyhedral surface for tightness; in fact, a computer can be programmed to perform this check, given the positions of the vertices and edges of the surface.

[Right] Proof of the Lemma
[Left] Tightness and homology: the modern definition
[Left] Kuiper's initial question
[Left] The polyhedral solution
[Up] Introduction

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9/29/94 dpvc@geom.umn.edu -- The Geometry Center