An isometric immersion of a flat Klein bottle into Euclidean 3-space

TL;DR

This paper constructs an explicit piecewise linear isometric immersion of a flat Klein bottle into three-dimensional Euclidean space, expanding the understanding of flat surface embeddings.

Contribution

It provides the first explicit isometric immersion of a flat Klein bottle into 3D space using a piecewise linear map, with verifiable inequalities ensuring local isometry and injectivity.

Findings

The map produces a self-intersecting polyhedron with zero angle defect at vertices.

Verification of local injectivity reduces to checking specific inequalities at vertices.

The construction generalizes known embeddings of flat tori and Klein bottles.

Abstract

We present an explicit piecewise linear map from a flat Klein bottle (i.e. one that is locally isometric to the Euclidean plane) into Euclidean 3-space an that is an isometric immersion -- a path isometry that is locally injective. The image is a self-intersecting polyhedron with embedded vertex figures where each vertex has zero angle defect. The construction of the map enforces the path isometry property so long as certain numerically-verifiable inequalities are satisfied, and we show that checking the local injectivity property at each vertex via another set of inequalities suffices. This work generalizes features from known piecewise linear isometric embeddings of flat tori and known piecewise smooth path isometries of flat Klein bottles, and apparently is the first explicit isometric immersion of a flat Klein bottle into $\mathbb{R}^3$.