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.