Tori Can't Collapse to an Interval

Abstract

Here we prove that under a lower sectional curvature bound, a sequence of Riemannian manifolds diffeomorphic to the standard $m$-dimensional torus cannot converge in the Gromov--Hausdorff sense to a closed interval. The proof is done by contradiction by analyzing suitable covers of a contradicting sequence, obtained from the Burago--Gromov--Perelman generalization of the Yamaguchi fibration theorem.