A piecewise-linear isometrically immersed flat Klein bottle in Euclidean 3-space
Geometric Topology
2025-04-15 v1 Differential Geometry
Abstract
We present numerical polyhedron data for the image of a piecewise-linear map from a zero-curvature Klein bottle into Euclidean 3-space such that every point in the domain has a neighborhood which is isometrically embedded. To the author's knowledge, this is the first explicit piecewise-smooth isometric immersion of a flat Klein bottle. Intuitively, the surface can be locally made from origami and but for the self-intersections has the global topology of a Klein bottle.
Cite
@article{arxiv.2504.08826,
title = {A piecewise-linear isometrically immersed flat Klein bottle in Euclidean 3-space},
author = {Stepan Paul},
journal= {arXiv preprint arXiv:2504.08826},
year = {2025}
}