Tightness and its consequences:

Classically, tightness is defined in terms of the total absolute curvature integral: given a mapping f of a surface M into space, the total absolute curvature is

where K is the Gaussian curvature. It can be shown that the total absolute curvature is always at least 4 - X(M) where X(M) represents the Euler characteristic of M

. When equality holds, the mapping is called tight.

This definition turns out to have several important geometric consequences and equivalent definitions:

Tightness and the convex hull.

Tightness and polar height functions.

Tightness and the two-piece property.

Tightness and homology: the modern definition.

Tightness for polyhedral surfaces.

Kuiper's original question

Introduction

7/21/94 dpvc@geom.umn.edu -- The Geometry Center