### Electronic Research Archive

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

# Tori can't collapse to an interval

• 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.

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

### Related Papers:

• 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.

 [1] On the holonomy group of locally Euclidean spaces. Ann. of Math. (1957) 65: 411-415. [2] (1964) Geometry of manifolds.Academic Press. [3] D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033 [4] Alexandrov spaces with curvature bounded below. Russian Mathematical Surveys (1992) 47: 1-58. [5] J. W. S. Cassels, An Introduction to The Geometry Of Numbers, Springer Science & Business Media (2012). [6] L. S. Charlap, Bieberbach Groups and Flat Manifolds, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4613-8687-2 [7] Filling Riemannian manifolds. J. Differential Geom. (1983) 18: 1-147. [8] Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. (1981) 53: 53-73. [9] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser Boston, Inc., Boston, MA, 2007. [10] Spin and scalar curvature in the presence of a fundamental group Ⅰ. Ann. of Math. (1980) 111: 209-230. [11] V. Kapovitch, Perelman's stability theorem, Surveys in Differential Geometry, 11 (2006), 103–136. arXiv: math/0703002. doi: 10.4310/SDG.2006.v11.n1.a5 [12] Restrictions on collapsing with a lower sectional curvature bound. Math. Z. (2005) 249: 519-539. [13] Almost nonnegative curvature, and the gradient flow on Alexandrov spaces. Ann. of Math. (2010) 171: 343-373. [14] M. G. Katz, Torus cannot collapse to a segment, J. Geom., 111 (2020), Paper No. 13, 8 pp. doi: 10.1007/s00022-020-0525-8 [15] G. Ya. Perelman, Alexandrov spaces with curvature bounded from below Ⅱ, Leningrad Branch of Steklov Institute. St. Petesburg (1991) [16] Collapsing and pinching under a lower curvature bound. Ann. of Math. (1991) 133: 317-357.
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