Torus cannot collapse to a segment

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.

New invariants of Gromov-Hausdorff limits of Riemannian surfaces with curvature bounded below

“…It follows that X is 1-dimensional and hence is either a segment or a circle. By [18], [39] tori cannot collapse to a segment. Hence X is a circle.…”

“…Assume that {X i } are homeomorphic to the 2-torus T 2 . Since by [18] and [39] torus cannot collapse to a segment, X is a circle. By Theorem 6.7(1) one has F ≡ 0.…”

“…(3) Let {X i } be homeomorphic to the torus. Katz [18] and independently Zamora [39] have shown that X must be a circle but not a segment. Theorem 6.7 says that in this case h a ≡ 0 for a = 0, 1, and h 0 = h 1 ≡ 1.…”

The Gromov–Hausdorff distance between spheres

“…Proof Note that it is enough to prove the claim for n D 1. We adapt an argument from the proof of [20,Lemma 2.3].…”

Tori can't collapse to an interval

“…Theorem 1.6. [14]. Let g n be a sequence of Riemannian metrics of sectional curvature ≥ −1 in the 2-dimensional torus M .…”