Tori can't collapse to an interval

Sergio Zamora; Sergio Zamora

doi:10.3934/era.2021005

Electronic Research Archive

2021, Volume 29, Issue 4: 2637-2644. doi: 10.3934/era.2021005

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Tori can't collapse to an interval

Sergio Zamora ^,

018 McAllister Bldg, University Park, PA 16802-6402, USA

Received: 01 October 2020 Published: 11 January 2021
Primary: 53C23, 53C20; Secondary: 53C21

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.
- Alexandrov geometry,
- sectional curvature bounded below,
- Yamaguchi collapse fibration
Citation: Sergio Zamora. Tori can't collapse to an interval[J]. Electronic Research Archive, 2021, 29(4): 2637-2644. doi: 10.3934/era.2021005

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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.

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