# American Institute of Mathematical Sciences

doi: 10.3934/era.2021005

## Tori can't collapse to an interval

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* Corresponding author: Sergio Zamora

Received  October 2020 Published  January 2021

Fund Project: The author would like to thank Raquel Perales and Anton Petrunin

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.

Citation: Sergio Zamora. Tori can't collapse to an interval. Electronic Research Archive, doi: 10.3934/era.2021005
##### References:

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##### References:
Flat Klein bottles can converge to an interval
The Fibration Theorem gives us a decomposition $X_n = S_1 \# S_2$
The configuration $(q_n; \tilde{p}_n ,a_n, b_n)$ violates the Alexandrov condition
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