Singularities of the asymptotic completion of developable M\"obius strips

TL;DR

This paper proves that developable M"obius strips in three-dimensional space must have a minimum number of singular points in their asymptotic completion, with specific bounds depending on the presence of a closed geodesic.

Contribution

It establishes lower bounds on the number of singular points in the asymptotic completion of developable M"obius strips, including cases with closed geodesics, and shows these bounds are sharp.

Findings

At least one singular point exists outside cuspidal edges.

If a closed geodesic is present, there are at least three singular points.

The bounds on singular points are proven to be sharp.

Abstract

We prove that the asymptotic completion of a developable M\"obius strip in Euclidean three-space must have at least one singular point other than cuspidal edge singularities. Moreover, if the strip contains a closed geodesic, then the number of such singular points is at least three. These lower bounds are both sharp.