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.

References

[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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