1965 (Publication #2)

Tightly embedded 2-dimensional polyhedral manifolds

Amer. J. Math., 87 (1965), 462-472.

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ABSTRACT

This paper introduces the word "tight" to describe the geometric condition that corresponds in the smooth case to minimal total absolute curvature. For any two-dimensional surface, an embedding is tight if almost any height function has exaclty one maximum when restricted to the surface. This is equivalent to the fact that every local support hyperplane is a global support hyperplane. Although any tightly embedded smooth surface must be contained in some five-dimensional subspace, there is not such restriction for polyhedral surfaces:

Theorem A: For any n, there is a two-dimensional polyhedral surface M(n) embedded in R^n and not lying in any affine (n-1)-dimensional subspace.

In order to find examples for laarge n, it is necessary to use surfaces with high genus.

If M is a polyhedral surface tightly embedded in R^n, then n < (7 - 24 Euler characteristic of (M))/2.