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
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
7/21/94 email@example.com --