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.