Torus cannot collapse to a segment
arXiv:2002.07523 · doi:10.1007/s00022-020-0525-8
Abstract
In earlier work, we analyzed the impossibility of codimension-one collapse for surfaces of negative Euler characteristic under the condition of a lower bound for the Gaussian curvature. Here we show that, under similar conditions, the torus cannot collapse to a segment. Unlike the torus, the Klein bottle can collapse to a segment; we show that in such a situation, the loops in a short basis for homology must stay a uniform distance apart.
8 pages, Journal of Geometry 111, Article number: 13 (2020)