##
Tight Polyhedral Klein Bottles, Projective Planes, and Mobius Bands

by Thomas F. Banchoff

A surface contained in E^n is *tightly embedded* if no height
function restricted to M has more than one strict local maximum.
It can be shown that a surface is tightly embedded if and only
if it has the *two-piece property* (TPP), ie. no hyperplane
in E^n divides M into more than two connected components. If M is
smoothly embedded in E^3, these properties are equivalent to the
condition of minimal total absolute curvature. A surface is said
to be embedded substantially if it lies in no affine hyperplane.

This paper presents improvements on the results of [1]. In particular,
we prove:

**Theorem K.** *There are tight substantial polyhedral embedding
of the Klein bottle K into E^5 but none int E^n for n>=6.*

**Theorem M.** *If M is tightly embedded in E^n with n=1/2 *
(5 + sqrt(49- Euler_Characteristic(M)) and Euler_Characteristic(M)<=0,
the M is embedded as a submanifold of the n simplex d^n
which contains all of the vertices and edges of d^n.*

**Theorem P.** *If P is a projective plane tightly and substantially
embedded as a polyhedral surface in E^5, then P has exactly six
extreme vertices and is quivalent under an affine transformation to
the embedding given in [1].*
**Corollary B.** *If B is a Mobius band embedded substantially in E^n
with n>=5, then n=5 and B is obtained from the tight substantial
polyhedral embedding of P by removing a convex disc from one face.*

**Corollary B'.** *If B is a Mobius band embedded substantially and
1-tightly as a polyhedral surface in E^4, then B has exactly five
extreme vertices.*